Package 'distrEx'

distrEx – Extensions of Package distr

Description

distrEx provides some extensions of package distr:

• expectations in the form

  • E(X) for the expectation of a distribution object X

  • E(X,f) for the expectation of f(X) where X is some distribution object and f some function in X

• further functionals: var, sd, IQR, mad, median, skewness, kurtosis

• truncated moments,

• distances between distributions (Hellinger, Cramer von Mises, Kolmogorov, total variation, "convex contamination")

• lists of distributions,

• conditional distributions in factorized form

• conditional expectations in factorized form

Support for extreme value distributions has moved to package RobExtremes

Details

Package:	distrEx
Version:	2.9.6
Date:	2025-01-11
Depends:	R(>= 3.4), methods, distr(>= 2.8.0)
Imports:	startupmsg(>= 1.0.0), utils, stats
Suggests:	tcltk
LazyLoad:	yes
License:	LGPL-3
URL:	https://distr.r-forge.r-project.org/
VCS/SVNRevision:	1482

Classes

Distribution Classes
"Distribution" (from distr)
|>"UnivariateDistribution" (from distr)
|>|>"AbscontDistribution" (from distr)
|>|>|>"Gumbel"  (moved to package 'RobExtremes')
|>|>|>"Pareto"  (moved to package 'RobExtremes')
|>|>|>"GPareto" (moved to package 'RobExtremes')
|>"MultivariateDistribution"
|>|>"DiscreteMVDistribution-class"
|>"UnivariateCondDistribution"
|>|>"AbscontCondDistribution"
|>|>|>"PrognCondDistribution"
|>|>"DiscreteCondDistribution"
Condition Classes
"Condition"
|>"EuclCondition"
|>"PrognCondition"
Parameter Classes
"OptionalParameter" (from distr)
|>"Parameter" (from distr)
|>|>"LMParameter"
|>|>"GumbelParameter"
|>|>"ParetoParameter"

Functions

Integration:
GLIntegrate             Gauss-Legendre quadrature
distrExIntegrate        Integration of one-dimensional functions
Options:
distrExOptions          Function to change the global variables of the
                        package 'distrEx'
Standardization:
make01                  Centering and standardization of univariate
                        distributions

Generating Functions

Distribution Classes
ConvexContamination     Generic function for generating convex
                        contaminations
DiscreteMVDistribution
                        Generating function for
                        DiscreteMVDistribution-class
Gumbel                  Generating function for Gumbel-class
LMCondDistribution      Generating function for the conditional
                        distribution of a linear regression model.
Condition Classes
EuclCondition           Generating function for EuclCondition-class
Parameter Classes
LMParameter             Generating function for LMParameter-class

Methods

Distances:
ContaminationSize       Generic function for the computation of the
                        convex contamination (Pseudo-)distance of two
                        distributions
HellingerDist           Generic function for the computation of the
                        Hellinger distance of two distributions
KolmogorovDist          Generic function for the computation of the
                        Kolmogorov distance of two distributions
TotalVarDist            Generic function for the computation of the
                        total variation distance of two distributions
AsymTotalVarDist        Generic function for the computation of the
                        asymmetric total variation distance of two distributions
                        (for given ratio rho of negative to positive part of deviation)
OAsymTotalVarDist       Generic function for the computation of the minimal (in rho)
                        asymmetric total variation distance of two distributions
vonMisesDist            Generic function for the computation of the
                        von Mises distance of two distributions
liesInSupport           Generic function for testing the support of a
                        distribution
Functionals:
E                       Generic function for the computation of
                        (conditional) expectations
var                     Generic functions for the computation of
                        functionals
IQR                     Generic functions for the computation of
                        functionals
sd                      Generic functions for the computation of
                        functionals
mad                     Generic functions for the computation of
                        functionals
median                  Generic functions for the computation of
                        functionals
skewness                Generic functions for the computation of
                        functionals
kurtosis                Generic functions for the computation of
                        Functionals
truncated Moments:
m1df                    Generic function for the computation of clipped
                        first moments
m2df                    Generic function for the computation of clipped
                        second moments

Demos

Demos are available — see demo(package="distrEx").

Acknowledgement

G. Jay Kerns, [email protected], has provided a major contribution, in particular the functionals skewness and kurtosis are due to him.

Start-up-Banner

You may suppress the start-up banner/message completely by setting options("StartupBanner"="off") somewhere before loading this package by library or require in your R-code / R-session. If option "StartupBanner" is not defined (default) or setting options("StartupBanner"=NULL) or options("StartupBanner"="complete") the complete start-up banner is displayed. For any other value of option "StartupBanner" (i.e., not in c(NULL,"off","complete")) only the version information is displayed. The same can be achieved by wrapping the library or require call into either suppressStartupMessages() or onlytypeStartupMessages(.,atypes="version").

As for general packageStartupMessage's, you may also suppress all the start-up banner by wrapping the library or require call into suppressPackageStartupMessages() from startupmsg-version 0.5 on.

Package versions

Note: The first two numbers of package versions do not necessarily reflect package-individual development, but rather are chosen for the distrXXX family as a whole in order to ease updating "depends" information.

Note

Some functions of package stats have intentionally been masked, but completely retain their functionality — see distrExMASK(). If any of the packages e1071, moments, fBasics is to be used together with distrEx the latter must be attached after any of the first mentioned. Otherwise kurtosis() and skewness() defined as methods in distrEx may get masked.
To re-mask, you may use kurtosis <- distrEx::kurtosis; skewness <- distrEx::skewness. See also distrExMASK()

Author(s)

Matthias Kohl [email protected] and
Peter Ruckdeschel [email protected],
Maintainer: Matthias Kohl [email protected]

References

P. Ruckdeschel, M. Kohl, T. Stabla, F. Camphausen (2006): S4 Classes for Distributions, R News, 6(2), 2-6. https://CRAN.R-project.org/doc/Rnews/Rnews_2006-2.pdf a vignette for packages distr, distrSim, distrTEst,

and distrEx is included into the mere documentation package distrDoc and may be called by require("distrDoc");vignette("distr") a homepage to this package is available under
https://distr.r-forge.r-project.org/ M. Kohl (2005): Numerical Contributions to the Asymptotic Theory of Robustness. PhD Thesis. Bayreuth. Available as https://www.stamats.de/wp-content/uploads/2018/04/ThesisMKohl.pdf

See Also

distr-package

---

Absolutely continuous conditional distribution

Description

The class of absolutely continuous conditional univariate distributions.

Objects from the Class

Objects can be created by calls of the form new("AbscontCondDistribution", ...).

Slots

cond

Object of class "Condition": condition

img

Object of class "rSpace": the image space.

param

Object of class "OptionalParameter": an optional parameter.

r

Object of class "function": generates random numbers.

d

Object of class "OptionalFunction": optional conditional density function.

p

Object of class "OptionalFunction": optional conditional cumulative distribution function.

q

Object of class "OptionalFunction": optional conditional quantile function.

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "UnivariateCondDistribution", directly.
Class "Distribution", by class "UnivariateCondDistribution".

Author(s)

Matthias Kohl [email protected]

See Also

UnivariateCondDistribution-class, Distribution-class

Examples

new("AbscontCondDistribution")

---

Generic function for the computation of asymmetric total variation distance of two distributions

Description

Generic function for the computation of asymmetric total variation distance $d_v(\rho)$ of two distributions $P$ and $Q$ where the distributions may be defined for an arbitrary sample space $(\Omega,{\cal A})$. For given ratio of inlier and outlier probability $\rho$, this distance is defined as

$d_v(\rho)(P,Q)=\int (dQ-c\,dP)_+$

for $c$ defined by

$\rho \int (dQ-c\,dP)_+ = \int (dQ-c\,dP)_-$

It coincides with total variation distance for $\rho=1$.

Usage

AsymTotalVarDist(e1, e2, ...)
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
AsymTotalVarDist(e1,e2, rho = 1,
             rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,DiscreteDistribution'
AsymTotalVarDist(e1,e2, rho = 1, ...)
## S4 method for signature 'DiscreteDistribution,AbscontDistribution'
AsymTotalVarDist(e1,e2, rho = 1, ...)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
AsymTotalVarDist(e1,e2, rho = 1, ...)
## S4 method for signature 'numeric,DiscreteDistribution'
AsymTotalVarDist(e1, e2, rho = 1, ...)
## S4 method for signature 'DiscreteDistribution,numeric'
AsymTotalVarDist(e1, e2, rho  = 1, ...)
## S4 method for signature 'numeric,AbscontDistribution'
AsymTotalVarDist(e1, e2, rho = 1, asis.smooth.discretize = "discretize", 
            n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2),
            up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"),
             rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,numeric'
AsymTotalVarDist(e1, e2,  rho = 1,
            asis.smooth.discretize = "discretize", 
            n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e1),
            up.discr = getUp(e1), h.smooth = getdistrExOption("hSmooth"),
             rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
AsymTotalVarDist(e1, e2,
          rho = 1, rel.tol = .Machine$double.eps^0.3, maxiter=1000, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)

Arguments

e1	object of class "Distribution" or "numeric"
e2	object of class "Distribution" or "numeric"
asis.smooth.discretize	possible methods are "asis", "smooth" and "discretize". Default is "discretize".
n.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the number of lattice points used to discretize the abs. cont. distribution.
low.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the lower end point of the lattice used to discretize the abs. cont. distribution.
up.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the upper end point of the lattice used to discretize the abs. cont. distribution.
h.smooth	if asis.smooth.discretize is equal to "smooth" – i.e., the empirical distribution of the provided data should be smoothed – one has to specify this parameter.
rho	ratio of inlier/outlier radius
rel.tol	relative tolerance for distrExIntegrate and uniroot
maxiter	parameter for uniroot
Ngrid	How many grid points are to be evaluated to determine the range of the likelihood ratio?

,

TruncQuantile	Quantile the quantile based integration bounds (see details)
IQR.fac	Factor for the scale based integration bounds (see details)
...	further arguments to be used in particular methods – (in package distrEx: just used for distributions with a.c. parts, where it is used to pass on arguments to distrExIntegrate).
diagnostic	logical; if TRUE, the return value obtains an attribute "diagnostic" with diagnostic information on the integration, i.e., a list with entries method ("integrate" or "GLIntegrate"), call, result (the complete return value of the method), args (the args with which the method was called), and time (the time to compute the integral).

Details

For distances between absolutely continuous distributions, we use numerical integration; to determine sensible bounds we proceed as follows: by means of min(getLow(e1,eps=TruncQuantile),getLow(e2,eps=TruncQuantile)), max(getUp(e1,eps=TruncQuantile),getUp(e2,eps=TruncQuantile)) we determine quantile based bounds c(low.0,up.0), and by means of s1 <- max(IQR(e1),IQR(e2)); m1<- median(e1); m2 <- median(e2) and low.1 <- min(m1,m2)-s1*IQR.fac, up.1 <- max(m1,m2)+s1*IQR.fac we determine scale based bounds; these are combined by low <- max(low.0,low.1), up <- max(up.0,up1).

Again in the absolutely continuous case, to determine the range of the likelihood ratio, we evaluate this ratio on a grid constructed as follows: x.range <- c(seq(low, up, length=Ngrid/3), q.l(e1)(seq(0,1,length=Ngrid/3)*.999), q.l(e2)(seq(0,1,length=Ngrid/3)*.999))

Finally, for both discrete and absolutely continuous case, we clip this ratio downwards by 1e-10 and upwards by 1e10

In case we want to compute the total variation distance between (empirical) data and an abs. cont. distribution, we can specify the parameter asis.smooth.discretize to avoid trivial distances (distance = 1).

Using asis.smooth.discretize = "discretize", which is the default, leads to a discretization of the provided abs. cont. distribution and the distance is computed between the provided data and the discretized distribution.

Using asis.smooth.discretize = "smooth" causes smoothing of the empirical distribution of the provided data. This is, the empirical data is convoluted with the normal distribution Norm(mean = 0, sd = h.smooth) which leads to an abs. cont. distribution. Afterwards the distance between the smoothed empirical distribution and the provided abs. cont. distribution is computed.

Diagnostics on the involved integrations are available if argument diagnostic is TRUE. Then there is attribute diagnostic attached to the return value, which may be inspected and accessed through showDiagnostic and getDiagnostic.

Value

Asymmetric Total variation distance of e1 and e2

Methods

e1 = "AbscontDistribution", e2 = "AbscontDistribution":

total variation distance of two absolutely continuous univariate distributions which is computed using distrExIntegrate.

e1 = "AbscontDistribution", e2 = "DiscreteDistribution":

total variation distance of absolutely continuous and discrete univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":

total variation distance of two discrete univariate distributions which is computed using support and sum.

e1 = "DiscreteDistribution", e2 = "AbscontDistribution":

total variation distance of discrete and absolutely continuous univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "numeric", e2 = "DiscreteDistribution":

Total variation distance between (empirical) data and a discrete distribution.

e1 = "DiscreteDistribution", e2 = "numeric":

Total variation distance between (empirical) data and a discrete distribution.

e1 = "numeric", e2 = "AbscontDistribution":

Total variation distance between (empirical) data and an abs. cont. distribution.

e1 = "AbscontDistribution", e1 = "numeric":

Total variation distance between (empirical) data and an abs. cont. distribution.

e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":

Total variation distance of mixed discrete and absolutely continuous univariate distributions.

Author(s)

Peter Ruckdeschel [email protected]

References

to be filled; Agostinelli, C and Ruckdeschel, P. (2009): A simultaneous inlier and outlier model by asymmetric total variation distance.

See Also

TotalVarDist-methods, ContaminationSize, KolmogorovDist, HellingerDist, Distribution-class

Examples

AsymTotalVarDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
                 mixCoeff=c(0.2,0.8)), rho=0.3)
AsymTotalVarDist(Norm(), Td(10), rho=0.3)
AsymTotalVarDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100), rho=0.3) # mutually singular
AsymTotalVarDist(Pois(10), Binom(size = 20), rho=0.3) 

x <- rnorm(100)
AsymTotalVarDist(Norm(), x, rho=0.3)
AsymTotalVarDist(x, Norm(), asis.smooth.discretize = "smooth", rho=0.3)

y <- (rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5)
AsymTotalVarDist(y, Norm(), rho=0.3)
AsymTotalVarDist(y, Norm(), asis.smooth.discretize = "smooth", rho=0.3)

AsymTotalVarDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5), rho=0.3)

---

Conditions

Description

The class of conditions.

Objects from the Class

Objects can be created by calls of the form new("Condition", ...).

Slots

name

Object of class "character": name of the condition

Methods

name

signature(object = "Condition"): accessor function for slot name.

name<-

signature(object = "Condition"): replacement function for slot name.

Author(s)

Matthias Kohl [email protected]

See Also

UnivariateCondDistribution-class

Examples

new("Condition")

---

Generic Function for the Computation of the Convex Contamination (Pseudo-)Distance of Two Distributions

Description

Generic function for the computation of convex contamination (pseudo-)distance of two probability distributions $P$ and $Q$. That is, the minimal size $\varepsilon\in [0,1]$ is computed such that there exists some probability distribution $R$ with

$Q = (1-\varepsilon)P + \varepsilon R$

Usage

ContaminationSize(e1, e2, ...)
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
ContaminationSize(e1,e2)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
ContaminationSize(e1,e2)
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
ContaminationSize(e1,e2)

Arguments

e1	object of class "Distribution"
e2	object of class "Distribution"
...	further arguments to be used in particular methods (not in package distrEx)

Details

Computes the distance from e1 to e2 respectively $P$ to $Q$. This is not really a distance as it is not symmetric!

Value

A list containing the following components:

e1	object of class "Distribution"; ideal distribution
e2	object of class "Distribution"; 'contaminated' distribution
size.of.contamination	size of contamination

Methods

e1 = "AbscontDistribution", e2 = "AbscontDistribution":

convex contamination (pseudo-)distance of two absolutely continuous univariate distributions.

e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":

convex contamination (pseudo-)distance of two discrete univariate distributions.

e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":

convex contamination (pseudo-)distance of two discrete univariate distributions.

Author(s)

Matthias Kohl [email protected],
Peter Ruckdeschel [email protected]

References

Huber, P.J. (1981) Robust Statistics. New York: Wiley.

See Also

KolmogorovDist, TotalVarDist, HellingerDist, Distribution-class

Examples

ContaminationSize(Norm(), Norm(mean=0.1))
ContaminationSize(Pois(), Pois(1.5))

---

Generic Function for Generating Convex Contaminations

Description

Generic function for generating convex contaminations. This is also known as gross error model. Given two distributions $P$ (ideal distribution), $R$ (contaminating distribution) and the size $\varepsilon\in [0,1]$ the convex contaminated distribution

$Q = (1-\varepsilon)P + \varepsilon R$

is generated.

Usage

ConvexContamination(e1, e2, size)

Arguments

e1	object of class "Distribution": ideal distribution
e2	object of class "Distribution": contaminating distribution
size	size of contamination (amount of gross errors)

Value

Object of class "Distribution".

Methods

e1 = "UnivariateDistribution", e2 = "UnivariateDistribution", size = "numeric":

convex combination of two univariate distributions

e1 = "AbscontDistribution", e2 = "AbscontDistribution", size = "numeric":

convex combination of two absolutely continuous univariate distributions

e1 = "DiscreteDistribution", e2 = "DiscreteDistribution", size = "numeric":

convex combination of two discrete univariate distributions

e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution", size = "numeric":

convex combination of two univariate distributions which may be coerced to "UnivarLebDecDistribution".

Author(s)

Matthias Kohl [email protected]

References

Huber, P.J. (1981) Robust Statistics. New York: Wiley.

See Also

ContaminationSize, Distribution-class

Examples

# Convex combination of two normal distributions
C1 <- ConvexContamination(e1 = Norm(), e2 = Norm(mean = 5), size = 0.1)
plot(C1)

---

Generic function for the computation of the Cramer - von Mises distance of two distributions

Description

Generic function for the computation of the Cramer - von Mises distance $d_\mu$ of two distributions $P$ and $Q$ where the distributions are defined on a finite-dimensional Euclidean space $(\R^m,{\cal B}^m)$ with ${\cal B}^m$ the Borel-$\sigma$-algebra on $R^m$. The Cramer - von Mises distance is defined as

$d_\mu(P,Q)^2=\int\,(P(\{y\in\R^m\,|\,y\le x\})-Q(\{y\in\R^m\,|\,y\le x\}))^2\,\mu(dx)$

where $\le$ is coordinatewise on $\R^m$.

Usage

CvMDist(e1, e2, ...)
## S4 method for signature 'UnivariateDistribution,UnivariateDistribution'
CvMDist(e1, e2, mu = e1, useApply = FALSE, ..., diagnostic = FALSE)
## S4 method for signature 'numeric,UnivariateDistribution'
CvMDist(e1, e2, mu = e1, ..., diagnostic = FALSE)

Arguments

e1	object of class "Distribution" or class "numeric"
e2	object of class "Distribution"
...	further arguments to be used e.g. by E()
useApply	logical; to be passed to E()
mu	object of class "Distribution"; integration measure; defaulting to e2
diagnostic	logical; if TRUE, the return value obtains an attribute "diagnostic" with diagnostic information on the integration, i.e., a list with entries method ("integrate" or "GLIntegrate"), call, result (the complete return value of the method), args (the args with which the method was called), and time (the time to compute the integral).

Details

Diagnostics on the involved integrations are available if argument diagnostic is TRUE. Then there is attribute diagnostic attached to the return value, which may be inspected and accessed through showDiagnostic and getDiagnostic.

Value

Cramer - von Mises distance of e1 and e2

Methods

e1 = "UnivariateDistribution", e2 = "UnivariateDistribution":

Cramer - von Mises distance of two univariate distributions.

e1 = "numeric", e2 = "UnivariateDistribution":

Cramer - von Mises distance between the empirical formed from a data set (e1) and a univariate distribution.

Author(s)

Matthias Kohl [email protected],
Peter Ruckdeschel [email protected]

References

Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.

See Also

ContaminationSize, TotalVarDist, HellingerDist, KolmogorovDist, Distribution-class

Examples

CvMDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
                 mixCoeff=c(0.2,0.8)))
CvMDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
                 mixCoeff=c(0.2,0.8)),mu=Norm())
CvMDist(Norm(), Td(10))
CvMDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100))
CvMDist(Pois(10), Binom(size = 20)) 
CvMDist(rnorm(100),Norm())
CvMDist((rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5), Norm())
CvMDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5))
CvMDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5), mu = Pois())

---

Methods for Function dim in Package ‘distrEx’

Description

dim-methods

Methods

dim

signature(object = "DiscreteMVDistribution"): returns the dimension of the distribution

See Also

dim-methods,
dim

---

Discrete conditional distribution

Description

The class of discrete conditional univariate distributions.

Objects from the Class

Objects can be created by calls of the form new("DiscreteCondDistribution", ...).

Slots

support

Object of class "function": conditional support.

cond

Object of class "Condition": condition

img

Object of class "rSpace": the image space.

param

Object of class "OptionalParameter": an optional parameter.

r

Object of class "function": generates random numbers.

d

Object of class "OptionalFunction": optional conditional density function.

p

Object of class "OptionalFunction": optional conditional cumulative distribution function.

q

Object of class "OptionalFunction": optional conditional quantile function.

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "UnivariateCondDistribution", directly.
Class "Distribution", by class "UnivariateCondDistribution".

Author(s)

Matthias Kohl [email protected]

See Also

UnivariateCondDistribution-class

Examples

new("DiscreteCondDistribution")

---

Generating function for multivariate discrete distribution

Description

Generates an object of class "DiscreteMVDistribution".

Usage

DiscreteMVDistribution(supp, prob, Symmetry = NoSymmetry())

Arguments

supp	numeric matrix whose rows form the support of the discrete multivariate distribution.
prob	vector of probability weights for the elements of supp.
Symmetry	you may help R in calculations if you tell it whether the distribution is non-symmetric (default) or symmetric with respect to a center.

Details

Typical usages are

    DiscreteMVDistribution(supp, prob)
    DiscreteMVDistribution(supp)
  

Identical rows are collapsed to unique support values. If prob is missing, all elements in supp are equally weighted.

Value

Object of class "DiscreteMVDistribution"

Author(s)

Matthias Kohl [email protected]

See Also

DiscreteMVDistribution-class

Examples

# Dirac-measure at (0,0,0)
D1 <- DiscreteMVDistribution(supp = c(0,0,0))
support(D1)

# simple discrete distribution
D2 <- DiscreteMVDistribution(supp = matrix(c(0,1,0,2,2,1,1,0), ncol=2), 
                prob = c(0.3, 0.2, 0.2, 0.3))
support(D2)
r(D2)(10)

---

Discrete Multivariate Distributions

Description

The class of discrete multivariate distributions.

Objects from the Class

Objects can be created by calls of the form new("DiscreteMVDistribution", ...). More frequently they are created via the generating function DiscreteMVDistribution.

Slots

img

Object of class "rSpace". Image space of the distribution. Usually an object of class "EuclideanSpace".

param

Object of class "OptionalParameter". Optional parameter of the multivariate distribution.

r

Object of class "function": generates (pseudo-)random numbers

d

Object of class "OptionalFunction": optional density function

p

Object of class "OptionalFunction": optional cumulative distribution function

q

Object of class "OptionalFunction": optional quantile function

support

numeric matrix whose rows form the support of the distribution

.finSupport

logical: (later on to be) used internally to check whether the true support is finite; the element in the 1st row and ith column indicates whether the ith marginal distribution has a finite left endpoint, and the element in the 2nd row and ith column if it is has a finite right endpoint); not yet further used.

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Extends

Class "MultivariateDistribution", directly.
Class "Distribution", by class "MultivariateDistribution".

Methods

support

signature(object = "DiscreteMVDistribution"): accessor function for slot support.

Author(s)

Matthias Kohl [email protected]

See Also

Distribution-class, MultivariateDistribution-class, DiscreteMVDistribution, E-methods

Examples

(D1 <- new("MultivariateDistribution")) # Dirac measure in (0,0)
r(D1)(5)

(D2 <- DiscreteMVDistribution(supp = matrix(c(1:5, rep(3, 5)), ncol=2, byrow=TRUE)))
support(D2)
r(D2)(10)
d(D2)(support(D2))
p(D2)(lower = c(1,1), upper = c(3,3))
q(D2)
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
param(D2)
img(D2)

e1 <- E(D2) # expectation

---

Integration of One-Dimensional Functions

Description

Numerical integration via integrate. In case integrate fails a Gauss-Legendre quadrature is performed.

Usage

distrExIntegrate(f, lower, upper, subdivisions = 100, 
                 rel.tol = .Machine$double.eps^0.25, 
                 abs.tol = rel.tol, stop.on.error = TRUE, 
                 distr, order, ..., diagnostic = FALSE)
showDiagnostic(x, what, withNonShows = FALSE, ...)
getDiagnostic(x, what, reorganized=TRUE)
## S3 method for class 'DiagnosticClass'
print(x, what, withNonShows = FALSE, xname, ...)

Arguments

f	an R function taking a numeric first argument and returning a numeric vector of the same length. Returning a non-finite element will generate an error.
lower	lower limit of integration. Can be -Inf.
upper	upper limit of integration. Can be Inf.
subdivisions	the maximum number of subintervals.
rel.tol	relative accuracy requested.
abs.tol	absolute accuracy requested.
stop.on.error	logical. If TRUE (the default) an error stops the function. If false some errors will give a result with a warning in the message component.
distr	object of class UnivariateDistribution.
order	order of Gauss-Legendre quadrature.
diagnostic	logical; if TRUE, the return value obtains an attribute "diagnostic" with diagnostic information on the integration, i.e., a list with entries method ("integrate" or "GLIntegrate"), call, result (the complete return value of the method), args (the args with which the method was called), and time (the time to compute the integral).
...	In case of integrators: additional arguments to be passed to f. Remember to use argument names not matching those of integrate and GLIntegrate! In case of showDiagnostic, print.DiagnosticClass: additional arguments to be passed on to print methods called for particular items in the diagnostic list.
x	the item for which the diagnostic is to be shown.
what	a character vector with all the diagnostic items to be selected/shown. If empty or missing all items are selected/shown.
withNonShows	internally we distinguish items which are easily printed (first kind) (numeric, logical, character) and more difficult ones (second kind), e.g., calls, functions, lists. The distinction is made according to the list item name. If withNonShows==TRUE one also attempts to show the selected items of the second kind, otherwise they are not shown (but returned).
xname	an optional name for the diagnostic object to be shown.
reorganized	should the diagnostic information be reorganized (using internal function .reorganizeDiagnosticList?

Details

distrExIntegrate calls integrate. In case integrate returns an error a Gauss-Legendre integration is performed using GLIntegrate. If lower or (and) upper are infinite the GLIntegrateTruncQuantile, respectively the 1-GLIntegrateTruncQuantile quantile of distr is used instead.

distrExIntegrate is called from many places in the distr and robast families of packages. At every such instance, diagnostic information can be collected (setting a corresponding argument diagnostic to TRUE in the calling function. This diagnostic information is originally stored in a tree like list structure of S3 class DiagnosticClass which is then attached as attribute diagnostic to the respective object. It can be inspected and accessed through showDiagnostic and getDiagnostic. More specifically, for any object with attribute diagnostic, showDiagnostic shows the diagnostic collected during integration, and getDiagnostic returns the diagnostic collected during integration. To this end, print.DiagnosticClass is an S3 method for print for objects of S3 class DiagnosticClass.

Value

The value of distrExIntegrate is a numeric approximation of the integral. If argument diagnostic==TRUE in distrExIntegrate, the return value has an attribute diagnostic of S3 class DiagnosticClass containing diagnostic information on the integration.

showDiagnostic, getDiagnostic, print.DiagnosticClass all return (invisibly) a list with the selected items, reorganized by internal function .reorganizeDiagnosticList, respectively, in case of argument reorganized==FALSE, getDiagnostic returns (invisibly) the diagnostic information as is.

Author(s)

Matthias Kohl [email protected]

References

Based on QUADPACK routines dqags and dqagi by R. Piessens and E. deDoncker-Kapenga, available from Netlib.

R. Piessens, E. deDoncker-Kapenga, C. Uberhuber, D. Kahaner (1983) Quadpack: a Subroutine Package for Automatic Integration. Springer Verlag.

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery (1992) Numerical Recipies in C. The Art of Scientific Computing. Second Edition. Cambridge University Press.

See Also

integrate, GLIntegrate, distrExOptions

Examples

fkt <- function(x){x*dchisq(x+1, df = 1)}
integrate(fkt, lower = -1, upper = 3)
GLIntegrate(fkt, lower = -1, upper = 3)
try(integrate(fkt, lower = -1, upper = 5))
distrExIntegrate(fkt, lower = -1, upper = 5)

---

Masking of/by other functions in package "distrEx"

Description

Provides information on the (intended) masking of and (non-intended) masking by other other functions in package distrEx

Usage

distrExMASK(library = NULL)

Arguments

library

a character vector with path names of R libraries, or NULL. The default value of NULL corresponds to all libraries currently known. If the default is used, the loaded packages are searched before the libraries

Value

no value is returned

Author(s)

Peter Ruckdeschel [email protected]

Examples

## IGNORE_RDIFF_BEGIN
distrExMASK()
## IGNORE_RDIFF_END

---

Moved functionality from package "distrEx"

Description

Provides information on moved of functionality from package distrEx.

Usage

distrExMOVED(library = NULL)

Arguments

library

a character vector with path names of R libraries, or NULL. The default value of NULL corresponds to all libraries currently known. If the default is used, the loaded packages are searched before the libraries

Value

no value is returned

Author(s)

Peter Ruckdeschel [email protected]

Examples

## IGNORE_RDIFF_BEGIN
distrExMOVED()
## IGNORE_RDIFF_END

---

Function to change the global variables of the package ‘distrEx’

Description

With distrExOptions you can inspect and change the global variables of the package distrEx.

Usage

distrExOptions(...)
distrExoptions(...)
getdistrExOption(x)

Arguments

...	any options can be defined, using name = value or by passing a list of such tagged values.
x	a character string holding an option name.

Value

distrExOptions() returns a list of the global variables.
distrExOptions(x) returns the global variable x.
getdistrExOption(x) returns the global variable x.
distrExOptions(x=y) sets the value of the global variable x to y.

distrExoptions

For compatibility with spelling in package distr, distrExoptions is just a synonym to distrExOptions.

Global Options

MCIterations:

number of Monte-Carlo iterations used for crude Monte-Carlo integration; defaults to 1e5.

GLIntegrateTruncQuantile:

If integrate fails and there are infinite integration limits, the function GLIntegrate is called inside of distrExIntegrate with the corresponding quantiles GLIntegrateTruncQuantile respectively, 1 - GLIntegrateTruncQuantile as finite integration limits; defaults to 10*.Machine$double.eps.

GLIntegrateOrder:

The order used for the Gauss-Legendre integration inside of distrExIntegrate; defaults to 500.

ElowerTruncQuantile:

The lower limit of integration used inside of E which corresponds to the ElowerTruncQuantile-quantile; defaults to 1e-7.

EupperTruncQuantile:

The upper limit of integration used inside of E which corresponds to the (1-ElowerTruncQuantile)-quantile; defaults to 1e-7.

ErelativeTolerance:

The relative tolerance used inside of E when calling distrExIntegrate; defaults to .Machine$double.eps^0.25.

m1dfLowerTruncQuantile:

The lower limit of integration used inside of m1df which corresponds to the m1dfLowerTruncQuantile-quantile; defaults to 0.

m1dfRelativeTolerance:

The relative tolerance used inside of m1df when calling distrExIntegrate; defaults to .Machine$double.eps^0.25.

m2dfLowerTruncQuantile:

The lower limit of integration used inside of m2df which corresponds to the m2dfLowerTruncQuantile-quantile; defaults to 0.

m2dfRelativeTolerance:

The relative tolerance used inside of m2df when calling distrExIntegrate; defaults to .Machine$double.eps^0.25.

nDiscretize:

number of support values used for the discretization of objects of class "AbscontDistribution"; defaults to 100.

hSmooth:

smoothing parameter to smooth objects of class "DiscreteDistribution". This is done via convolution with the normal distribution Norm(mean = 0, sd = hSmooth); defaults to 0.05.

IQR.fac:

for determining sensible integration ranges, we use both quantile and scale based methods; for the scale based method we use the median of the distribution $\pm$ IQR.fac$\times$ the IQR; defaults to 15.

propagate.names.functionals

should names obtained from parameter coordinates be propagated to return values of specific S4 methods for functionals; defaults to TRUE.

Author(s)

Matthias Kohl [email protected]

See Also

options, getOption

Examples

distrExOptions()
distrExOptions("ElowerTruncQuantile")
distrExOptions("ElowerTruncQuantile" = 1e-6)
# or
distrExOptions(ElowerTruncQuantile = 1e-6)
getdistrExOption("ElowerTruncQuantile")

---

Generic Function for the Computation of (Conditional) Expectations

Description

Generic function for the computation of (conditional) expectations.

Usage

E(object, fun, cond, ...)

## S4 method for signature 'UnivariateDistribution,missing,missing'
E(object, 
             low = NULL, upp = NULL, Nsim = getdistrExOption("MCIterations"), ...)

## S4 method for signature 'UnivariateDistribution,function,missing'
E(object, fun, 
        useApply = TRUE, low = NULL, upp = NULL,
        Nsim = getdistrExOption("MCIterations"), ...)

## S4 method for signature 'AbscontDistribution,missing,missing'
E(object, low = NULL, upp = NULL,
             rel.tol= getdistrExOption("ErelativeTolerance"),
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"),
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"),
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE)

## S4 method for signature 'AbscontDistribution,function,missing'
E(object, fun, useApply = TRUE,
             low = NULL, upp = NULL, 
             rel.tol= getdistrExOption("ErelativeTolerance"), 
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE)

## S4 method for signature 'UnivarMixingDistribution,missing,missing'
E(object, low = NULL, 
             upp = NULL, rel.tol= getdistrExOption("ErelativeTolerance"), 
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE)

## S4 method for signature 'UnivarMixingDistribution,function,missing'
E(object, fun,
             useApply = TRUE, low = NULL, upp = NULL,
             rel.tol= getdistrExOption("ErelativeTolerance"), 
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE)

## S4 method for signature 'UnivarMixingDistribution,missing,ANY'
E(object, cond, low = NULL, 
             upp = NULL, rel.tol= getdistrExOption("ErelativeTolerance"), 
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE)

## S4 method for signature 'UnivarMixingDistribution,function,ANY'
E(object, fun, cond,
             useApply = TRUE, low = NULL, upp = NULL,
             rel.tol= getdistrExOption("ErelativeTolerance"),
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE)

## S4 method for signature 'DiscreteDistribution,function,missing'
E(object, fun, useApply = TRUE, 
             low = NULL, upp = NULL, ...)

## S4 method for signature 'AffLinDistribution,missing,missing'
E(object, low = NULL, upp = NULL,
             ..., diagnostic = FALSE)

## S4 method for signature 'AffLinUnivarLebDecDistribution,missing,missing'
E(object, low = NULL,
             upp = NULL, ..., diagnostic = FALSE)

## S4 method for signature 'MultivariateDistribution,missing,missing'
E(object, 
             Nsim = getdistrExOption("MCIterations"), ...)
## S4 method for signature 'MultivariateDistribution,function,missing'
E(object, fun,
             useApply = TRUE, Nsim = getdistrExOption("MCIterations"), ...)

## S4 method for signature 'DiscreteMVDistribution,missing,missing'
E(object, low = NULL,
             upp = NULL, ...)

## S4 method for signature 'DiscreteMVDistribution,function,missing'
E(object, fun, 
             useApply = TRUE, ...)

## S4 method for signature 'AbscontCondDistribution,missing,numeric'
E(object, cond,
             useApply = TRUE, low = NULL, upp = NULL,
             rel.tol= getdistrExOption("ErelativeTolerance"), 
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE)

## S4 method for signature 'DiscreteCondDistribution,missing,numeric'
E(object, cond,
             useApply = TRUE, low = NULL, upp = NULL, ...)

## S4 method for signature 'UnivariateCondDistribution,function,numeric'
E(object, fun, cond, 
              withCond = FALSE, useApply = TRUE, low = NULL, upp = NULL,
              Nsim = getdistrExOption("MCIterations"), ...)

## S4 method for signature 'AbscontCondDistribution,function,numeric'
E(object, fun, cond, 
               withCond = FALSE, useApply = TRUE, low = NULL, upp = NULL,
             rel.tol= getdistrExOption("ErelativeTolerance"), 
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac")
             , ..., diagnostic = FALSE)

## S4 method for signature 'DiscreteCondDistribution,function,numeric'
E(object, fun, cond, 
             withCond = FALSE, useApply = TRUE, low = NULL, upp = NULL,...)

## S4 method for signature 'UnivarLebDecDistribution,missing,missing'
E(object, low = NULL,
             upp = NULL, rel.tol= getdistrExOption("ErelativeTolerance"),
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE )
## S4 method for signature 'UnivarLebDecDistribution,function,missing'
E(object, fun, 
             useApply = TRUE, low = NULL, upp = NULL,
             rel.tol= getdistrExOption("ErelativeTolerance"),
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE )
## S4 method for signature 'UnivarLebDecDistribution,missing,ANY'
E(object, cond, low = NULL,
             upp = NULL, rel.tol= getdistrExOption("ErelativeTolerance"),
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE )
## S4 method for signature 'UnivarLebDecDistribution,function,ANY'
E(object, fun, cond, 
             useApply = TRUE, low = NULL, upp = NULL,
             rel.tol= getdistrExOption("ErelativeTolerance"), 
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE )

## S4 method for signature 'AcDcLcDistribution,ANY,ANY'
E(object, fun, cond, low = NULL,
             upp = NULL, rel.tol= getdistrExOption("ErelativeTolerance"),
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"), ..., diagnostic = FALSE)
## S4 method for signature 'CompoundDistribution,missing,missing'
E(object, low = NULL,
             upp = NULL, ..., diagnostic = FALSE)

## S4 method for signature 'Arcsine,missing,missing'
E(object, low = NULL, upp = NULL, ..., diagnostic = FALSE)
## S4 method for signature 'Beta,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...,
             diagnostic = FALSE)
## S4 method for signature 'Binom,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...)
## S4 method for signature 'Cauchy,missing,missing'
E(object, low = NULL, upp = NULL, ..., diagnostic = FALSE)
## S4 method for signature 'Cauchy,function,missing'
E(object, fun, low = NULL, upp = NULL,
             rel.tol = getdistrExOption("ErelativeTolerance"),
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"),
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"),
             IQR.fac = max(1e4,getdistrExOption("IQR.fac")),
             ..., diagnostic = FALSE)
## S4 method for signature 'Chisq,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...)
## S4 method for signature 'Dirac,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...)
## S4 method for signature 'DExp,missing,missing'
E(object, low = NULL, upp = NULL, ..., diagnostic = FALSE)
## S4 method for signature 'Exp,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...)
## S4 method for signature 'Fd,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...,
             diagnostic = FALSE)
## S4 method for signature 'Gammad,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...,
             diagnostic = FALSE)
## S4 method for signature 'Gammad,function,missing'
E(object, fun, low = NULL, upp = NULL,
             rel.tol = getdistrExOption("ErelativeTolerance"),
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"),
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"),
             IQR.fac = max(1e4,getdistrExOption("IQR.fac")), ..., diagnostic = FALSE)
## S4 method for signature 'Geom,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...)
## S4 method for signature 'Hyper,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...)
## S4 method for signature 'Logis,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...,
             diagnostic = FALSE)
## S4 method for signature 'Lnorm,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...,
             diagnostic = FALSE)
## S4 method for signature 'Nbinom,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...)
## S4 method for signature 'Norm,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...)
## S4 method for signature 'Pois,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...)
## S4 method for signature 'Unif,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...,
             diagnostic = FALSE)
## S4 method for signature 'Td,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...,
             diagnostic = FALSE)
## S4 method for signature 'Weibull,missing,missing'
E(object, low = NULL, upp = NULL,
             propagate.names=getdistrExOption("propagate.names.functionals"), ...,
             diagnostic = FALSE)
## S4 method for signature 'Weibull,function,missing'
E(object, fun, low = NULL, upp = NULL,
             rel.tol = getdistrExOption("ErelativeTolerance"),
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"),
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"),
             IQR.fac = max(1e4,getdistrExOption("IQR.fac")), ..., diagnostic = FALSE)
.qtlIntegrate(object, fun, low = NULL, upp = NULL,
             rel.tol= getdistrExOption("ErelativeTolerance"),
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"),
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"),
             IQR.fac = max(1e4,getdistrExOption("IQR.fac")), ...,
             .withLeftTail = FALSE, .withRightTail = FALSE, diagnostic = FALSE)

Arguments

object	object of class "Distribution"
fun	if missing the (conditional) expectation is computed else the (conditional) expection of fun is computed.
cond	if not missing the conditional expectation given cond is computed.
Nsim	number of MC simulations used to determine the expectation.
rel.tol	relative tolerance for distrExIntegrate.
low	lower bound of integration range.
upp	upper bound of integration range.
lowerTruncQuantile	lower quantile for quantile based integration range.
upperTruncQuantile	upper quantile for quantile based integration range.
IQR.fac	factor for scale based integration range (i.e.; median of the distribution $\pm$IQR.fac$\times$IQR).
...	additional arguments to fun
useApply	logical: should sapply, respectively apply be used to evaluate fun.
withCond	logical: is cond in the argument list of fun.
.withLeftTail	logical: should left tail (falling into quantile range [0,0.02]) be computed separately to enhance accuracy?
.withRightTail	logical: should right tail (falling into quantile range [0.98,1]) be computed separately to enhance accuracy?
diagnostic	logical; if TRUE, the return value obtains an attribute "diagnostic" with diagnostic information on the integration, i.e., a list with entries method ("integrate" or "GLIntegrate"), call, result (the complete return value of the method), args (the args with which the method was called), and time (the time to compute the integral).
propagate.names	logical: should names obtained from parameter coordinates be propagated to return values of specific S4 methods for functionals; defaults to the value of the respective distrExoption propagate.names.functionals.

Details

The precision of the computations can be controlled via certain global options; cf. distrExOptions. Also note that arguments low and upp should be given as named arguments in order to prevent them to be matched by arguments fun or cond. Also the result, when arguments low or upp is given, is the unconditional value of the expectation; no conditioning with respect to low <= object <= upp is done.

For the Cauchy, the Gamma and Weibull distribution for integration with missing argument cond but given argument fun, we use integration on [0,1] (i.e, via the respective probability transformation). This done via helper function .qtlIntegrate, where both arguments .withLeftTail and .withRightTail are TRUE for the Cauchy and Gamma distributions, and only .withRightTail ist TRUE for the Weibull distribution.

Diagnostics on the involved integrations are available if argument diagnostic is TRUE. Then there is attribute diagnostic attached to the return value, which may be inspected and accessed through showDiagnostic and getDiagnostic.

Value

The (conditional) expectation is computed.

Methods

object = "UnivariateDistribution", fun = "missing", cond = "missing":

expectation of univariate distributions using crude Monte-Carlo integration.

object = "AbscontDistribution", fun = "missing", cond = "missing":

expectation of absolutely continuous univariate distributions using distrExIntegrate.

object = "DiscreteDistribution", fun = "missing", cond = "missing":

expectation of discrete univariate distributions using support and sum.

object = "MultivariateDistribution", fun = "missing", cond = "missing":

expectation of multivariate distributions using crude Monte-Carlo integration.

object = "DiscreteMVDistribution", fun = "missing", cond = "missing":

expectation of discrete multivariate distributions. The computation is based on support and sum.

object = "UnivariateDistribution", fun = "missing", cond = "missing":

expectation of univariate Lebesgue decomposed distributions by separate calculations for discrete and absolutely continuous part.

object = "AffLinDistribution", fun = "missing", cond = "missing":

expectation of an affine linear transformation $aX+b$ as $a E[X]+b$ for X either "DiscreteDistribution" or "AbscontDistribution".

object = "AffLinUnivarLebDecDistribution", fun = "missing", cond = "missing":

expectation of an affine linear transformation $aX+b$ as $a E[X]+b$ for X either "UnivarLebDecDistribution".

object = "UnivariateDistribution", fun = "function", cond = "missing":

expectation of fun under univariate distributions using crude Monte-Carlo integration.

object = "UnivariateDistribution", fun = "function", cond = "missing":

expectation of fun under univariate Lebesgue decomposed distributions by separate calculations for discrete and absolutely continuous part.

object = "AbscontDistribution", fun = "function", cond = "missing":

expectation of fun under absolutely continuous univariate distributions using distrExIntegrate.

object = "DiscreteDistribution", fun = "function", cond = "missing":

expectation of fun under discrete univariate distributions using support and sum.

object = "MultivariateDistribution", fun = "function", cond = "missing":

expectation of multivariate distributions using crude Monte-Carlo integration.

object = "DiscreteMVDistribution", fun = "function", cond = "missing":

expectation of fun under discrete multivariate distributions. The computation is based on support and sum.

object = "UnivariateCondDistribution", fun = "missing", cond = "numeric":

conditional expectation for univariate conditional distributions given cond. The integral is computed using crude Monte-Carlo integration.

object = "AbscontCondDistribution", fun = "missing", cond = "numeric":

conditional expectation for absolutely continuous, univariate conditional distributions given cond. The computation is based on distrExIntegrate.

object = "DiscreteCondDistribution", fun = "missing", cond = "numeric":

conditional expectation for discrete, univariate conditional distributions given cond. The computation is based on support and sum.

object = "UnivariateCondDistribution", fun = "function", cond = "numeric":

conditional expectation of fun under univariate conditional distributions given cond. The integral is computed using crude Monte-Carlo integration.

object = "AbscontCondDistribution", fun = "function", cond = "numeric":

conditional expectation of fun under absolutely continuous, univariate conditional distributions given cond. The computation is based on distrExIntegrate.

object = "DiscreteCondDistribution", fun = "function", cond = "numeric":

conditional expectation of fun under discrete, univariate conditional distributions given cond. The computation is based on support and sum.

object = "UnivarLebDecDistribution", fun = "missing", cond = "missing":

expectation by separate evaluation of expectation of discrete and abs. continuous part and subsequent weighting.

object = "UnivarLebDecDistribution", fun = "function", cond = "missing":

expectation by separate evaluation of expectation of discrete and abs. continuous part and subsequent weighting.

object = "UnivarLebDecDistribution", fun = "missing", cond = "ANY":

expectation by separate evaluation of expectation of discrete and abs. continuous part and subsequent weighting.

object = "UnivarLebDecDistribution", fun = "function", cond = "ANY":

expectation by separate evaluation of expectation of discrete and abs. continuous part and subsequent weighting.

object = "UnivarMixingDistribution", fun = "missing", cond = "missing":

expectation is computed component-wise with subsequent weighting acc. to mixCoeff.

object = "UnivarMixingDistribution", fun = "function", cond = "missing":

expectation is computed component-wise with subsequent weighting acc. to mixCoeff.

object = "UnivarMixingDistribution", fun = "missing", cond = "ANY":

expectation is computed component-wise with subsequent weighting acc. to mixCoeff.

object = "UnivarMixingDistribution", fun = "function", cond = "ANY":

expectation is computed component-wise with subsequent weighting acc. to mixCoeff.

object = "AcDcLcDistribution", fun = "ANY", cond = "ANY":

expectation by first coercing to class "UnivarLebDecDistribution" and using the corresponding method.

object = "CompoundDistribution", fun = "missing", cond = "missing":

if we are in i.i.d. situation (i.e., slot SummandsDistr is of class UnivariateDistribution) the formula $E[N]E[S]$ for $N$ the frequency distribution and $S$ the summand distribution; else we coerce to "UnivarLebDecDistribution".

object = "Arcsine", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Beta", fun = "missing", cond = "missing":

for noncentrality 0 exact evaluation using explicit expressions.

object = "Binom", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Cauchy", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Chisq", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Dirac", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "DExp", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Exp", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Fd", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Gammad", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Gammad", fun = "function", cond = "missing":

use integration over the quantile range for numerical integration via helper function .qtlIntegrate.

object = "Geom", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Hyper", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Logis", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Lnorm", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Nbinom", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Norm", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Pois", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Unif", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Td", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Weibull", fun = "missing", cond = "missing":

exact evaluation using explicit expressions.

object = "Weibull", fun = "function", cond = "missing":

use integration over the quantile range for numerical integration via helper function .qtlIntegrate.

Author(s)

Matthias Kohl [email protected] and Peter Ruckdeschel [email protected]

See Also

distrExIntegrate, m1df, m2df, Distribution-class

Examples

# mean of Exp(1) distribution
E <- Exp() 

E(E) ## uses explicit terms
E(as(E,"AbscontDistribution")) ## uses numerical integration
E(as(E,"UnivariateDistribution")) ## uses simulations
E(E, fun = function(x){2*x^2}) ## uses simulations

# the same operator for discrete distributions:
P <- Pois(lambda=2)

E(P) ## uses explicit terms
E(as(P,"DiscreteDistribution")) ## uses sums
E(as(P,"UnivariateDistribution")) ## uses simulations
E(P, fun = function(x){2*x^2}) ## uses simulations


# second moment of N(1,4)
E(Norm(mean=1, sd=2), fun = function(x){x^2})
E(Norm(mean=1, sd=2), fun = function(x){x^2}, useApply = FALSE)

# conditional distribution of a linear model
D1 <- LMCondDistribution(theta = 1) 
E(D1, cond = 1)
E(Norm(mean=1))
E(D1, function(x){x^2}, cond = 1)
E(Norm(mean=1), fun = function(x){x^2})
E(D1, function(x, cond){cond*x^2}, cond = 2, withCond = TRUE, useApply = FALSE)
E(Norm(mean=2), function(x){2*x^2})

E(as(Norm(mean=2),"AbscontDistribution"))
### somewhat less accurate:
E(as(Norm(mean=2),"AbscontDistribution"), 
     lowerTruncQuantil=1e-4,upperTruncQuantil=1e-4, IQR.fac= 4)
### even less accurate:
E(as(Norm(mean=2),"AbscontDistribution"), 
     lowerTruncQuantil=1e-2,upperTruncQuantil=1e-2, IQR.fac= 4)
### no good idea, but just as an example:
E(as(Norm(mean=2),"AbscontDistribution"), 
     lowerTruncQuantil=1e-2,upperTruncQuantil=1e-2, IQR.fac= .1)

### truncation of integration range; see also m1df...
E(Norm(mean=2), low=2,upp=4)

E(Cauchy())
E(Cauchy(),upp=3,low=-2)
# some Lebesgue decomposed distribution 
mymix <- UnivarLebDecDistribution(acPart = Norm(), discretePart = Binom(4,.4),
         acWeight = 0.4)
E(mymix)

---

Generating function for mulitvariate discrete distribution

Description

Generates an object of class "DiscreteMVDistribution".

Usage

EmpiricalMVDistribution(data, Symmetry = NoSymmetry())

Arguments

data	numeric matrix with data where the rows are interpreted as observations.
Symmetry	you may help R in calculations if you tell it whether the distribution is non-symmetric (default) or symmetric with respect to a center.

Details

The function is a simple utility function providing a wrapper to the generating function DiscreteMVDistribution.

Typical usages are

    EmpiricalMVDistribution(data)
  

Identical rows are collapsed to unique support values. If prob is missing, all elements in supp are equally weighted.

Value

Object of class "DiscreteMVDistribution"

Author(s)

Matthias Kohl [email protected]

See Also

DiscreteMVDistribution

Examples

## generate some data
X <- matrix(rnorm(50), ncol = 5)

## empirical distribution of X
D1 <- EmpiricalMVDistribution(data = X)
support(D1)
r(D1)(10)

---

Generating function for EuclCondition-class

Description

Generates an object of class "EuclCondition".

Usage

EuclCondition(dimension)

Arguments

dimension

positive integer: dimension of the Euclidean space

Value

Object of class "EuclCondition"

Author(s)

Matthias Kohl [email protected]

See Also

EuclCondition-class

Examples

EuclCondition(dimension = 3)

## The function is currently defined as
function(dimension){
    new("EuclCondition", Range = EuclideanSpace(dimension = dimension))
}

---

Conditioning by an Euclidean space.

Description

Conditioning by an Euclidean space.

Objects from the Class

Objects can be created by calls of the form new("EuclCondition", ...). More frequently they are created via the generating function EuclCondition.

Slots

Range

Object of class "EuclideanSpace".

name

Object of class "character": name of condition.

Extends

Class "Condition", directly.

Methods

Range

signature(object = "EuclCondition") accessor function for slot Range.

show

signature(object = "EuclCondition")

Author(s)

Matthias Kohl [email protected]

See Also

Condition-class, EuclCondition

Examples

new("EuclCondition")

---

Gauss-Legendre Quadrature

Description

Gauss-Legendre quadrature over a finite interval.

Usage

GLIntegrate(f, lower, upper, order = 500, ...)

Arguments

f	an R function taking a numeric first argument and returning a numeric vector of the same length. Returning a non-finite element will generate an error.
lower	finite lower limit of integration.
upper	finite upper limit of integration.
order	order of Gauss-Legendre quadrature.
...	additional arguments to be passed to f. Remember to use argument names not matching those of GLIntegrate!

Details

In case order = 100, 500, 1000 saved abscissas and weights are used. Otherwise the corresponding abscissas and weights are computed using the algorithm given in Section 4.5 of Press et al. (1992).

Value

Estimate of the integral.

Author(s)

Matthias Kohl [email protected]

References

W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery (1992) Numerical Recipies in C. The Art of Scientific Computing. Second Edition. Cambridge University Press.

See Also

integrate, distrExIntegrate

Examples

integrate(dnorm, -1.96, 1.96)
GLIntegrate(dnorm, -1.96, 1.96)

---

Generic function for the computation of the Hellinger distance of two distributions

Description

Generic function for the computation of the Hellinger distance $d_h$ of two distributions $P$ and $Q$ which may be defined for an arbitrary sample space $(\Omega,{\cal A})$. The Hellinger distance is defined as

$d_h(P,Q)=\frac{1}{2}\int|\sqrt{dP}\,-\sqrt{dQ}\,|^2$

where $\sqrt{dP}$, respectively $\sqrt{dQ}$ denotes the square root of the densities.

Usage

HellingerDist(e1, e2, ...)
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
HellingerDist(e1,e2, 
                        rel.tol=.Machine$double.eps^0.3, 
                        TruncQuantile = getdistrOption("TruncQuantile"), 
                        IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,DiscreteDistribution'
HellingerDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,AbscontDistribution'
HellingerDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
HellingerDist(e1,e2, ...)
## S4 method for signature 'numeric,DiscreteDistribution'
HellingerDist(e1, e2, ...)
## S4 method for signature 'DiscreteDistribution,numeric'
HellingerDist(e1, e2, ...)
## S4 method for signature 'numeric,AbscontDistribution'
HellingerDist(e1, e2, asis.smooth.discretize = "discretize", 
            n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2),
            up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"),
                        rel.tol=.Machine$double.eps^0.3, 
                        TruncQuantile = getdistrOption("TruncQuantile"), 
                        IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,numeric'
HellingerDist(e1, e2, asis.smooth.discretize = "discretize", 
            n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e1),
            up.discr = getUp(e1), h.smooth = getdistrExOption("hSmooth"), 
                        rel.tol=.Machine$double.eps^0.3, 
                        TruncQuantile = getdistrOption("TruncQuantile"), 
                        IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
HellingerDist(e1,e2, 
                        rel.tol=.Machine$double.eps^0.3, 
                        TruncQuantile = getdistrOption("TruncQuantile"), 
                        IQR.fac = 15, ..., diagnostic = FALSE)

Arguments

e1	object of class "Distribution" or class "numeric"
e2	object of class "Distribution" or class "numeric"
asis.smooth.discretize	possible methods are "asis", "smooth" and "discretize". Default is "discretize".
n.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the number of lattice points used to discretize the abs. cont. distribution.
low.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the lower end point of the lattice used to discretize the abs. cont. distribution.
up.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the upper end point of the lattice used to discretize the abs. cont. distribution.
h.smooth	if asis.smooth.discretize is equal to "smooth" – i.e., the empirical distribution of the provided data should be smoothed – one has to specify this parameter.
rel.tol	relative accuracy requested in integration
TruncQuantile	Quantile the quantile based integration bounds (see details)
IQR.fac	Factor for the scale based integration bounds (see details)
...	further arguments to be used in particular methods – (in package distrEx: just used for distributions with a.c. parts, where it is used to pass on arguments to distrExIntegrate).
diagnostic	logical; if TRUE, the return value obtains an attribute "diagnostic" with diagnostic information on the integration, i.e., a list with entries method ("integrate" or "GLIntegrate"), call, result (the complete return value of the method), args (the args with which the method was called), and time (the time to compute the integral).

Details

For distances between absolutely continuous distributions, we use numerical integration; to determine sensible bounds we proceed as follows: by means of min(getLow(e1,eps=TruncQuantile),getLow(e2,eps=TruncQuantile)), max(getUp(e1,eps=TruncQuantile),getUp(e2,eps=TruncQuantile)) we determine quantile based bounds c(low.0,up.0), and by means of s1 <- max(IQR(e1),IQR(e2)); m1<- median(e1); m2 <- median(e2) and low.1 <- min(m1,m2)-s1*IQR.fac, up.1 <- max(m1,m2)+s1*IQR.fac we determine scale based bounds; these are combined by low <- max(low.0,low.1), up <- max(up.0,up1).

In case we want to compute the Hellinger distance between (empirical) data and an abs. cont. distribution, we can specify the parameter asis.smooth.discretize to avoid trivial distances (distance = 1).

Using asis.smooth.discretize = "discretize", which is the default, leads to a discretization of the provided abs. cont. distribution and the distance is computed between the provided data and the discretized distribution.

Using asis.smooth.discretize = "smooth" causes smoothing of the empirical distribution of the provided data. This is, the empirical data is convoluted with the normal distribution Norm(mean = 0, sd = h.smooth) which leads to an abs. cont. distribution. Afterwards the distance between the smoothed empirical distribution and the provided abs. cont. distribution is computed.

Diagnostics on the involved integrations are available if argument diagnostic is TRUE. Then there is attribute diagnostic attached to the return value, which may be inspected and accessed through showDiagnostic and getDiagnostic.

Value

Hellinger distance of e1 and e2

Methods

e1 = "AbscontDistribution", e2 = "AbscontDistribution":

Hellinger distance of two absolutely continuous univariate distributions which is computed using distrExintegrate.

e1 = "AbscontDistribution", e2 = "DiscreteDistribution":

Hellinger distance of absolutely continuous and discrete univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":

Hellinger distance of two discrete univariate distributions which is computed using support and sum.

e1 = "DiscreteDistribution", e2 = "AbscontDistribution":

Hellinger distance of discrete and absolutely continuous univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "numeric", e2 = "DiscreteDistribution":

Hellinger distance between (empirical) data and a discrete distribution.

e1 = "DiscreteDistribution", e2 = "numeric":

Hellinger distance between (empirical) data and a discrete distribution.

e1 = "numeric", e2 = "AbscontDistribution":

Hellinger distance between (empirical) data and an abs. cont. distribution.

e1 = "AbscontDistribution", e1 = "numeric":

Hellinger distance between (empirical) data and an abs. cont. distribution.

e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":

Hellinger distance of mixed discrete and absolutely continuous univariate distributions.

Author(s)

Matthias Kohl [email protected],
Peter Ruckdeschel [email protected]

References

Huber, P.J. (1981) Robust Statistics. New York: Wiley.

Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.

See Also

distrExIntegrate, ContaminationSize, TotalVarDist, KolmogorovDist, Distribution-class

Examples

HellingerDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
                 mixCoeff=c(0.2,0.8)))
HellingerDist(Norm(), Td(10))
HellingerDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100)) # mutually singular
HellingerDist(Pois(10), Binom(size = 20)) 

x <- rnorm(100)
HellingerDist(Norm(), x)
HellingerDist(x, Norm(), asis.smooth.discretize = "smooth")

y <- (rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5)
HellingerDist(y, Norm())
HellingerDist(y, Norm(), asis.smooth.discretize = "smooth")

HellingerDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5))

---

Generic function for the computation of the Kolmogorov distance of two distributions

Description

Generic function for the computation of the Kolmogorov distance $d_\kappa$ of two distributions $P$ and $Q$ where the distributions are defined on a finite-dimensional Euclidean space $(\R^m,{\cal B}^m)$ with ${\cal B}^m$ the Borel-$\sigma$-algebra on $R^m$. The Kolmogorov distance is defined as

$d_\kappa(P,Q)=\sup\{|P(\{y\in\R^m\,|\,y\le x\})-Q(\{y\in\R^m\,|\,y\le x\})| | x\in\R^m\}$

where $\le$ is coordinatewise on $\R^m$.

Usage

KolmogorovDist(e1, e2, ...)
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
KolmogorovDist(e1,e2, ...)
## S4 method for signature 'AbscontDistribution,DiscreteDistribution'
KolmogorovDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,AbscontDistribution'
KolmogorovDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
KolmogorovDist(e1,e2, ...)
## S4 method for signature 'numeric,UnivariateDistribution'
KolmogorovDist(e1, e2, ...)
## S4 method for signature 'UnivariateDistribution,numeric'
KolmogorovDist(e1, e2, ...)
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
KolmogorovDist(e1, e2, ...)

Arguments

e1	object of class "Distribution" or class "numeric"
e2	object of class "Distribution" or class "numeric"
...	further arguments to be used in particular methods (not in package distrEx)

Value

Kolmogorov distance of e1 and e2

Methods

e1 = "AbscontDistribution", e2 = "AbscontDistribution":

Kolmogorov distance of two absolutely continuous univariate distributions which is computed using a union of a (pseudo-)random and a deterministic grid.

e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":

Kolmogorov distance of two discrete univariate distributions. The distance is attained at some point of the union of the supports of e1 and e2.

e1 = "AbscontDistribution", e2 = "DiscreteDistribution":

Kolmogorov distance of absolutely continuous and discrete univariate distributions. It is computed using a union of a (pseudo-)random and a deterministic grid in combination with the support of e2.

e1 = "DiscreteDistribution", e2 = "AbscontDistribution":

Kolmogorov distance of discrete and absolutely continuous univariate distributions. It is computed using a union of a (pseudo-)random and a deterministic grid in combination with the support of e1.

e1 = "numeric", e2 = "UnivariateDistribution":

Kolmogorov distance between (empirical) data and a univariate distribution. The computation is based on ks.test.

e1 = "UnivariateDistribution", e2 = "numeric":

Kolmogorov distance between (empirical) data and a univariate distribution. The computation is based on ks.test.

e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":

Kolmogorov distance of mixed discrete and absolutely continuous univariate distributions. It is computed using a union of the discrete part, a (pseudo-)random and a deterministic grid in combination with the support of e1.

Author(s)

Matthias Kohl [email protected],
Peter Ruckdeschel [email protected]

References

Huber, P.J. (1981) Robust Statistics. New York: Wiley.

Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.

See Also

ContaminationSize, TotalVarDist, HellingerDist, Distribution-class

Examples

KolmogorovDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
                 mixCoeff=c(0.2,0.8)))
KolmogorovDist(Norm(), Td(10))
KolmogorovDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100))
KolmogorovDist(Pois(10), Binom(size = 20)) 
KolmogorovDist(Norm(), rnorm(100))
KolmogorovDist((rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5), Norm())
KolmogorovDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5))

---

Generic Function for Testing the Support of a Distribution

Description

The function tests if x lies in the support of the distribution object.

Usage

## S4 method for signature 'DiscreteMVDistribution,numeric'
liesInSupport(object, x, checkFin = FALSE)
## S4 method for signature 'DiscreteMVDistribution,matrix'
liesInSupport(object, x, checkFin = FALSE)

Arguments

object	object of class "Distribution"
x	numeric vector or matrix
checkFin	logical: in case FALSE, we simply check whether x lies exactly in the numerical support (of finitely many support points); later on we might try to mimick the univariate case more closely in case TRUE, but so far this is not yet used.

Value

logical vector

Methods

object = "DiscreteMVDistribution", x = "numeric":

does x lie in the support of object.

object = "DiscreteMVDistribution", x = "matrix":

does x lie in the support of object.

Author(s)

Matthias Kohl [email protected]

See Also

Distribution-class

Examples

M <- matrix(rpois(30, lambda = 10), ncol = 3)
D1 <- DiscreteMVDistribution(M)
M1 <- rbind(r(D1)(10), matrix(rpois(30, lam = 10), ncol = 3))
liesInSupport(D1, M1)

---

Generating function for the conditional distribution of a linear regression model.

Description

Generates an object of class "AbscontCondDistribution" which is the conditional distribution of a linear regression model (given the regressor).

Usage

LMCondDistribution(Error = Norm(), theta = 0, intercept = 0, scale = 1)

Arguments

Error	Object of class "AbscontDistribution": error distribution.
theta	numeric vector: regression parameter.
intercept	real number: intercept parameter.
scale	positive real number: scale parameter.

Value

Object of class "AbscontCondDistribution"

Author(s)

Matthias Kohl [email protected]

See Also

AbscontCondDistribution-class, E-methods

Examples

# normal error distribution
(D1 <- LMCondDistribution(theta = 1)) # corresponds to Norm(cond, 1)
plot(D1)
r(D1)
d(D1)
p(D1)
q(D1)
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
param(D1)
cond(D1)

d(D1)(0, cond = 1)
d(Norm(mean=1))(0)

E(D1, cond = 1)
E(D1, function(x){x^2}, cond = 2)
E(Norm(mean=2), function(x){x^2})

---

Generating function for LMParameter-class

Description

Generates an object of class "LMParameter".

Usage

LMParameter(theta = 0, intercept = 0, scale = 1)

Arguments

theta	numeric vector: regression parameter (default =0).
intercept	real number: intercept parameter (default =0).
scale	positive real number: scale parameter (default =1).

Value

Object of class "LMParameter"

Author(s)

Matthias Kohl [email protected]

See Also

LMParameter-class

Examples

LMParameter(theta = c(1,1), intercept = 2, scale = 0.5)

## The function is currently defined as
function(theta = 0, intercept = 0, scale = 1){
    new("LMParameter", theta = theta, intercept = intercept, scale = 1)
}

---

Parameter of a linear regression model

Description

Parameter of a linear regression model

$y = \mu + x^\tau\theta + \sigma u$

with intercept $\mu$, regression parameter $\theta$ and error scale $\sigma$.

Objects from the Class

Objects can be created by calls of the form new("LMParameter", ...). More frequently they are created via the generating function LMParameter.

Slots

theta

numeric vector: regression parameter.

intercept

real number: intercept parameter.

scale

positive real number: scale paramter.

name

character vector: the default name is “parameter of a linear regression model”.

Extends

Class "Parameter", directly.
Class "OptionalParameter", by class "Parameter".

Methods

show

signature(object = "LMParameter")

Author(s)

Matthias Kohl [email protected]

See Also

Parameter-class, LMParameter

Examples

new("LMParameter")

---

Generic Function for the Computation of Clipped First Moments

Description

Generic function for the computation of clipped first moments. The moments are clipped at upper.

Usage

m1df(object, upper, ...)
## S4 method for signature 'AbscontDistribution'
m1df(object, upper, 
             lowerTruncQuantile = getdistrExOption("m1dfLowerTruncQuantile"),
             rel.tol = getdistrExOption("m1dfRelativeTolerance"), ...)

Arguments

object	object of class "Distribution"
upper	clipping bound
rel.tol	relative tolerance for distrExIntegrate.
lowerTruncQuantile	lower quantile for quantile based integration range.
...	additional arguments to E

Details

The precision of the computations can be controlled via certain global options; cf. distrExOptions.

Value

The first moment of object clipped at upper is computed.

Methods

object = "UnivariateDistribution":

uses call E(object, upp=upper, ...).

object = "AbscontDistribution":

clipped first moment for absolutely continuous univariate distributions which is computed using integrate.

object = "LatticeDistribution":

clipped first moment for discrete univariate distributions which is computed using support and sum.

object = "AffLinDistribution":

clipped first moment for affine linear distributions which is computed on basis of slot X0.

object = "Binom":

clipped first moment for Binomial distributions which is computed using pbinom.

object = "Pois":

clipped first moment for Poisson distributions which is computed using ppois.

object = "Norm":

clipped first moment for normal distributions which is computed using dnorm and pnorm.

object = "Exp":

clipped first moment for exponential distributions which is computed using pexp.

object = "Chisq":

clipped first moment for $\chi^2$ distributions which is computed using pchisq.

Author(s)

Matthias Kohl [email protected]

See Also

distrExIntegrate, m2df, E

Examples

# standard normal distribution
N1 <- Norm()
m1df(N1, 0)

# Poisson distribution
P1 <- Pois(lambda=2)
m1df(P1, 3)
m1df(P1, 3, fun = function(x)sin(x))

# absolutely continuous distribution
D1 <- Norm() + Exp() # convolution
m1df(D1, 2)
m1df(D1, Inf)
E(D1)

---

Generic function for the computation of clipped second moments

Description

Generic function for the computation of clipped second moments. The moments are clipped at upper.

Usage

m2df(object, upper, ...)
## S4 method for signature 'AbscontDistribution'
m2df(object, upper, 
             lowerTruncQuantile = getdistrExOption("m2dfLowerTruncQuantile"),
             rel.tol = getdistrExOption("m2dfRelativeTolerance"), ...)

Arguments

object	object of class "Distribution"
upper	clipping bound
rel.tol	relative tolerance for distrExIntegrate.
lowerTruncQuantile	lower quantile for quantile based integration range.
...	additional arguments to E

Details

The precision of the computations can be controlled via certain global options; cf. distrExOptions.

Value

The second moment of object clipped at upper is computed.

Methods

object = "UnivariateDistribution":

uses call E(object, upp=upper, fun = function, ...).

object = "AbscontDistribution":

clipped second moment for absolutely continuous univariate distributions which is computed using integrate.

object = "LatticeDistribution":

clipped second moment for discrete univariate distributions which is computed using support and sum.

object = "AffLinDistribution":

clipped second moment for affine linear distributions which is computed on basis of slot X0.

object = "Binom":

clipped second moment for Binomial distributions which is computed using pbinom.

object = "Pois":

clipped second moment for Poisson distributions which is computed using ppois.

object = "Norm":

clipped second moment for normal distributions which is computed using dnorm and pnorm.

object = "Exp":

clipped second moment for exponential distributions which is computed using pexp.

object = "Chisq":

clipped second moment for $\chi^2$ distributions which is computed using pchisq.

Author(s)

Matthias Kohl [email protected]

See Also

m2df-methods, E-methods

Examples

# standard normal distribution
N1 <- Norm()
m2df(N1, 0)

# Poisson distribution
P1 <- Pois(lambda=2)
m2df(P1, 3)
m2df(P1, 3, fun = function(x)sin(x))

# absolutely continuous distribution
D1 <- Norm() + Exp() # convolution
m2df(D1, 2)
m2df(D1, Inf)
E(D1, function(x){x^2})

---

Centering and Standardization of Univariate Distributions

Description

The function make01 produces a new centered and standardized univariate distribution.

Usage

make01(x)

Arguments

x	an object of class "UnivariateDistribution"

Details

Thanks to the functionals provided in this package, the code is a one-liner: (x-E(x))/sd(x).

Value

Object of class "UnivariateDistribution" with expectation 0 and variance 1.

Author(s)

Peter Ruckdeschel [email protected]

See Also

E, Var

Examples

X <- sin(exp(2*log(abs( Norm())))) ## something weird
X01 <- make01(X)
print(X01)
plot(X01)
sd(X01); E(X01)

---

Multivariate Distributions

Description

The class of multivariate distributions. One has at least to specify the image space of the distribution and a function generating (pseudo-)random numbers. The slot q is usually filled with NULL for dimensions > 1.

Objects from the Class

Objects can be created by calls of the form new("MultivariateDistribution", ...).

Slots

img

Object of class "rSpace". Image space of the distribution. Usually an object of class "EuclideanSpace".

param

Object of class "OptionalParameter". Optional parameter of the multivariate distribution.

r

Object of class "function": generates (pseudo-)random numbers

d

Object of class "OptionalFunction": optional density function

p

Object of class "OptionalFunction": optional cumulative distribution function

q

Object of class "OptionalFunction": optional quantile function

.withArith

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim

logical: used internally to issue warnings as to accuracy

.logExact

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Symmetry

object of class "DistributionSymmetry"; used internally to avoid unnecessary calculations.

Extends

Class "Distribution", directly.

Methods

show

signature(object = "MultivariateDistribution")

plot

signature(object = "MultivariateDistribution"): not yet implemented.

Author(s)

Matthias Kohl [email protected]

See Also

Distribution-class

Examples

# Dirac-measure in (0,0)
new("MultivariateDistribution")

---

Generic function for the computation of (minimal) asymmetric total variation distance of two distributions

Description

Generic function for the computation of (minimal) asymmetric total variation distance $d_v^\ast$ of two distributions $P$ and $Q$ where the distributions may be defined for an arbitrary sample space $(\Omega,{\cal A})$. This distance is defined as

$d_v^\ast(P,Q)=\min_c \int |dQ-c\,dP|$

Usage

OAsymTotalVarDist(e1, e2, ...)
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
OAsymTotalVarDist(e1,e2, 
             rel.tol = .Machine$double.eps^0.3,  Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,DiscreteDistribution'
OAsymTotalVarDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,AbscontDistribution'
OAsymTotalVarDist(e1,e2,  ...)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
OAsymTotalVarDist(e1,e2, ...)
## S4 method for signature 'numeric,DiscreteDistribution'
OAsymTotalVarDist(e1, e2,  ...)
## S4 method for signature 'DiscreteDistribution,numeric'
OAsymTotalVarDist(e1, e2,  ...)
## S4 method for signature 'numeric,AbscontDistribution'
OAsymTotalVarDist(e1, e2,  asis.smooth.discretize = "discretize", 
            n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2),
            up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"),
             rel.tol = .Machine$double.eps^0.3, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,numeric'
OAsymTotalVarDist(e1, e2, 
            asis.smooth.discretize = "discretize", 
            n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e1),
            up.discr = getUp(e1), h.smooth = getdistrExOption("hSmooth"),
             rel.tol = .Machine$double.eps^0.3, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
OAsymTotalVarDist(e1, e2,
             rel.tol = .Machine$double.eps^0.3, Ngrid = 10000,
             TruncQuantile = getdistrOption("TruncQuantile"),
             IQR.fac = 15, ..., diagnostic = FALSE)

Arguments

e1	object of class "Distribution" or "numeric"
e2	object of class "Distribution" or "numeric"
asis.smooth.discretize	possible methods are "asis", "smooth" and "discretize". Default is "discretize".
n.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the number of lattice points used to discretize the abs. cont. distribution.
low.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the lower end point of the lattice used to discretize the abs. cont. distribution.
up.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the upper end point of the lattice used to discretize the abs. cont. distribution.
h.smooth	if asis.smooth.discretize is equal to "smooth" – i.e., the empirical distribution of the provided data should be smoothed – one has to specify this parameter.
rel.tol	relative tolerance for distrExIntegrate and uniroot
Ngrid	How many grid points are to be evaluated to determine the range of the likelihood ratio?

,

TruncQuantile	Quantile the quantile based integration bounds (see details)
IQR.fac	Factor for the scale based integration bounds (see details)
...	further arguments to be used in particular methods – (in package distrEx: just used for distributions with a.c. parts, where it is used to pass on arguments to distrExIntegrate).
diagnostic	logical; if TRUE, the return value obtains an attribute "diagnostic" with diagnostic information on the integration, i.e., a list with entries method ("integrate" or "GLIntegrate"), call, result (the complete return value of the method), args (the args with which the method was called), and time (the time to compute the integral).

Details

For distances between absolutely continuous distributions, we use numerical integration; to determine sensible bounds we proceed as follows: by means of min(getLow(e1,eps=TruncQuantile),getLow(e2,eps=TruncQuantile)), max(getUp(e1,eps=TruncQuantile),getUp(e2,eps=TruncQuantile)) we determine quantile based bounds c(low.0,up.0), and by means of s1 <- max(IQR(e1),IQR(e2)); m1<- median(e1); m2 <- median(e2) and low.1 <- min(m1,m2)-s1*IQR.fac, up.1 <- max(m1,m2)+s1*IQR.fac we determine scale based bounds; these are combined by low <- max(low.0,low.1), up <- max(up.0,up1).

Again in the absolutely continuous case, to determine the range of the likelihood ratio, we evaluate this ratio on a grid constructed as follows: x.range <- c(seq(low, up, length=Ngrid/3), q.l(e1)(seq(0,1,length=Ngrid/3)*.999), q.l(e2)(seq(0,1,length=Ngrid/3)*.999))

Finally, for both discrete and absolutely continuous case, we clip this ratio downwards by 1e-10 and upwards by 1e10

In case we want to compute the total variation distance between (empirical) data and an abs. cont. distribution, we can specify the parameter asis.smooth.discretize to avoid trivial distances (distance = 1).

Using asis.smooth.discretize = "discretize", which is the default, leads to a discretization of the provided abs. cont. distribution and the distance is computed between the provided data and the discretized distribution.

Using asis.smooth.discretize = "smooth" causes smoothing of the empirical distribution of the provided data. This is, the empirical data is convoluted with the normal distribution Norm(mean = 0, sd = h.smooth) which leads to an abs. cont. distribution. Afterwards the distance between the smoothed empirical distribution and the provided abs. cont. distribution is computed.

Diagnostics on the involved integrations are available if argument diagnostic is TRUE. Then there is attribute diagnostic attached to the return value, which may be inspected and accessed through showDiagnostic and getDiagnostic.

Value

OAsymmetric Total variation distance of e1 and e2

Methods

e1 = "AbscontDistribution", e2 = "AbscontDistribution":

total variation distance of two absolutely continuous univariate distributions which is computed using distrExIntegrate.

e1 = "AbscontDistribution", e2 = "DiscreteDistribution":

total variation distance of absolutely continuous and discrete univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":

total variation distance of two discrete univariate distributions which is computed using support and sum.

e1 = "DiscreteDistribution", e2 = "AbscontDistribution":

total variation distance of discrete and absolutely continuous univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "numeric", e2 = "DiscreteDistribution":

Total variation distance between (empirical) data and a discrete distribution.

e1 = "DiscreteDistribution", e2 = "numeric":

Total variation distance between (empirical) data and a discrete distribution.

e1 = "numeric", e2 = "AbscontDistribution":

Total variation distance between (empirical) data and an abs. cont. distribution.

e1 = "AbscontDistribution", e1 = "numeric":

Total variation distance between (empirical) data and an abs. cont. distribution.

e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":

Total variation distance of mixed discrete and absolutely continuous univariate distributions.

Author(s)

Peter Ruckdeschel [email protected]

References

to be filled; Agostinelli, C and Ruckdeschel, P. (2009): A simultaneous inlier and outlier model by asymmetric total variation distance.

See Also

TotalVarDist-methods, ContaminationSize, KolmogorovDist, HellingerDist, Distribution-class

Examples

OAsymTotalVarDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
                 mixCoeff=c(0.2,0.8)))
OAsymTotalVarDist(Norm(), Td(10))
OAsymTotalVarDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100)) # mutually singular
OAsymTotalVarDist(Pois(10), Binom(size = 20)) 

x <- rnorm(100)
OAsymTotalVarDist(Norm(), x)
OAsymTotalVarDist(x, Norm(), asis.smooth.discretize = "smooth")

y <- (rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5)
OAsymTotalVarDist(y, Norm())
OAsymTotalVarDist(y, Norm(), asis.smooth.discretize = "smooth")

OAsymTotalVarDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5))

---

Methods for Function plot in Package ‘distrEx’

Description

plot-methods

Usage

plot(x, y, ...)
## S4 method for signature 'UnivariateCondDistribution,missing'
plot(x, y, ...)
## S4 method for signature 'MultivariateDistribution,missing'
plot(x, y, ...)

Arguments

x	object of class "UnivariateCondDistribution" or class "MultivariateDistribution": distribution(s) which should be plotted
y	missing
...	addtional arguments

Details

upto now only warnings are issued that the corresponding method is not yet implemented;

---

Generating function for PrognCondDistribution-class

Description

Generates an object of class "PrognCondDistribution".

Usage

PrognCondDistribution(Regr, Error,
             rel.tol= getdistrExOption("ErelativeTolerance"), 
             lowerTruncQuantile = getdistrExOption("ElowerTruncQuantile"), 
             upperTruncQuantile = getdistrExOption("EupperTruncQuantile"), 
             IQR.fac = getdistrExOption("IQR.fac"))

Arguments

Regr	object of class AbscontDistribution; the distribution of X.
Error	object of class AbscontDistribution; the distribution of eps.
rel.tol	relative tolerance for distrExIntegrate.
lowerTruncQuantile	lower quantile for quantile based integration range.
upperTruncQuantile	upper quantile for quantile based integration range.
IQR.fac	factor for scale based integration range (i.e.; median of the distribution $\pm$IQR.fac$\times$IQR).

Details

For independent r.v.'s X,E with univariate, absolutely continuous (a.c.) distributions Regr and Error, respectively, PrognCondDistribution() returns the (factorized, conditional) posterior distribution of X given X+E=y. as an object of class PrognCondDistribution.

Value

Object of class "PrognCondDistribution"

Author(s)

Peter Ruckdeschel [email protected],

See Also

PrognCondDistribution-class; demo(‘Prognose.R’).

Examples

PrognCondDistribution(Error = ConvexContamination(Norm(), Norm(4,1), size=0.1))

---

Posterior distribution in convolution

Description

The posterior distribution of X given (X+E)=y

Objects from the Class

Objects can be created by calls of the form PrognCondDistribution where Regr and error are the respective (a.c.) distributions of X and E and the other arguments control accuracy in integration.

Slots

cond:

Object of class "PrognCondition": condition

img:

Object of class "rSpace": the image space.

param:

Object of class "OptionalParameter": an optional parameter.

r:

Object of class "function": generates random numbers.

d:

Object of class "OptionalFunction": optional conditional density function.

p:

Object of class "OptionalFunction": optional conditional cumulative distribution function.

q:

Object of class "OptionalFunction": optional conditional quantile function.

gaps:

(numeric) matrix or NULL

.withArith:

logical: used internally to issue warnings as to interpretation of arithmetics

.withSim:

logical: used internally to issue warnings as to accuracy

.logExact:

logical: used internally to flag the case where there are explicit formulae for the log version of density, cdf, and quantile function

.lowerExact:

logical: used internally to flag the case where there are explicit formulae for the lower tail version of cdf and quantile function

Extends

Class "AbscontCondDistribution", directly.
Class "Distribution", by classes "UnivariateCondDistribution" and "AbscontCondDistribution".

Author(s)

Matthias Kohl [email protected]

See Also

PrognCondition-class, UnivariateCondDistribution-class AbscontCondDistribution-class, Distribution-class

Examples

PrognCondDistribution()

---

Conditions of class 'PrognCondition'

Description

The class PrognCondition realizes the condition that X+E=y in a convolution setup

Usage

PrognCondition(range = EuclideanSpace())

Arguments

range

an object of class "EuclideanSpace"

Value

Object of class "PrognCondition"

Objects from the Class

Objects can be created by calls of the form PrognCondition(range).

Slots

name

Object of class "character": name of the PrognCondition

range

Object of class "EuclideanSpace": range of the PrognCondition

Extends

Class "Condition", directly.

Methods

show

signature(object = "PrognCondition")

Author(s)

Matthias Kohl [email protected]

See Also

PrognCondDistribution-class,Condition-class

Examples

PrognCondition()

---

Generic function for the computation of the total variation distance of two distributions

Description

Generic function for the computation of the total variation distance $d_v$ of two distributions $P$ and $Q$ where the distributions may be defined for an arbitrary sample space $(\Omega,{\cal A})$. The total variation distance is defined as

$d_v(P,Q)=\sup_{B\in{\cal A}}|P(B)-Q(B)|$

Usage

TotalVarDist(e1, e2, ...)
## S4 method for signature 'AbscontDistribution,AbscontDistribution'
TotalVarDist(e1,e2, 
                        rel.tol=.Machine$double.eps^0.3, 
                        TruncQuantile = getdistrOption("TruncQuantile"), 
                        IQR.fac = 15, ..., diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,DiscreteDistribution'
TotalVarDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,AbscontDistribution'
TotalVarDist(e1,e2, ...)
## S4 method for signature 'DiscreteDistribution,DiscreteDistribution'
TotalVarDist(e1,e2, ...)
## S4 method for signature 'numeric,DiscreteDistribution'
TotalVarDist(e1, e2, ...)
## S4 method for signature 'DiscreteDistribution,numeric'
TotalVarDist(e1, e2, ...)
## S4 method for signature 'numeric,AbscontDistribution'
TotalVarDist(e1, e2, asis.smooth.discretize = "discretize", 
            n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e2),
            up.discr = getUp(e2), h.smooth = getdistrExOption("hSmooth"),
            rel.tol = .Machine$double.eps^0.3, 
            TruncQuantile = getdistrOption("TruncQuantile"), IQR.fac = 15, ...,
            diagnostic = FALSE)
## S4 method for signature 'AbscontDistribution,numeric'
TotalVarDist(e1, e2, asis.smooth.discretize = "discretize", 
            n.discr = getdistrExOption("nDiscretize"), low.discr = getLow(e1),
            up.discr = getUp(e1), h.smooth = getdistrExOption("hSmooth"),
            rel.tol = .Machine$double.eps^0.3, 
            TruncQuantile = getdistrOption("TruncQuantile"), IQR.fac = 15, ...,
            diagnostic = FALSE)
## S4 method for signature 'AcDcLcDistribution,AcDcLcDistribution'
TotalVarDist(e1, e2,                         
                        rel.tol = .Machine$double.eps^0.3, 
                        TruncQuantile = getdistrOption("TruncQuantile"), 
                        IQR.fac = 15, ..., diagnostic = FALSE)

Arguments

e1	object of class "Distribution" or "numeric"
e2	object of class "Distribution" or "numeric"
asis.smooth.discretize	possible methods are "asis", "smooth" and "discretize". Default is "discretize".
n.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the number of lattice points used to discretize the abs. cont. distribution.
low.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the lower end point of the lattice used to discretize the abs. cont. distribution.
up.discr	if asis.smooth.discretize is equal to "discretize" one has to specify the upper end point of the lattice used to discretize the abs. cont. distribution.
h.smooth	if asis.smooth.discretize is equal to "smooth" – i.e., the empirical distribution of the provided data should be smoothed – one has to specify this parameter.
rel.tol	relative accuracy requested in integration
TruncQuantile	Quantile the quantile based integration bounds (see details)
IQR.fac	Factor for the scale based integration bounds (see details)
...	further arguments to be used in particular methods – (in package distrEx: just used for distributions with a.c. parts, where it is used to pass on arguments to distrExIntegrate).
diagnostic	logical; if TRUE, the return value obtains an attribute "diagnostic" with diagnostic information on the integration, i.e., a list with entries method ("integrate" or "GLIntegrate"), call, result (the complete return value of the method), args (the args with which the method was called), and time (the time to compute the integral).

Details

For distances between absolutely continuous distributions, we use numerical integration; to determine sensible bounds we proceed as follows: by means of min(getLow(e1,eps=TruncQuantile),getLow(e2,eps=TruncQuantile)), max(getUp(e1,eps=TruncQuantile),getUp(e2,eps=TruncQuantile)) we determine quantile based bounds c(low.0,up.0), and by means of s1 <- max(IQR(e1),IQR(e2)); m1<- median(e1); m2 <- median(e2) and low.1 <- min(m1,m2)-s1*IQR.fac, up.1 <- max(m1,m2)+s1*IQR.fac we determine scale based bounds; these are combined by low <- max(low.0,low.1), up <- max(up.0,up1).

In case we want to compute the total variation distance between (empirical) data and an abs. cont. distribution, we can specify the parameter asis.smooth.discretize to avoid trivial distances (distance = 1).

Using asis.smooth.discretize = "discretize", which is the default, leads to a discretization of the provided abs. cont. distribution and the distance is computed between the provided data and the discretized distribution.

Using asis.smooth.discretize = "smooth" causes smoothing of the empirical distribution of the provided data. This is, the empirical data is convoluted with the normal distribution Norm(mean = 0, sd = h.smooth) which leads to an abs. cont. distribution. Afterwards the distance between the smoothed empirical distribution and the provided abs. cont. distribution is computed.

Diagnostics on the involved integrations are available if argument diagnostic is TRUE. Then there is attribute diagnostic attached to the return value, which may be inspected and accessed through showDiagnostic and getDiagnostic.

Value

Total variation distance of e1 and e2

Methods

e1 = "AbscontDistribution", e2 = "AbscontDistribution":

total variation distance of two absolutely continuous univariate distributions which is computed using distrExIntegrate.

e1 = "AbscontDistribution", e2 = "DiscreteDistribution":

total variation distance of absolutely continuous and discrete univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "DiscreteDistribution", e2 = "DiscreteDistribution":

total variation distance of two discrete univariate distributions which is computed using support and sum.

e1 = "DiscreteDistribution", e2 = "AbscontDistribution":

total variation distance of discrete and absolutely continuous univariate distributions (are mutually singular; i.e., have distance =1).

e1 = "numeric", e2 = "DiscreteDistribution":

Total variation distance between (empirical) data and a discrete distribution.

e1 = "DiscreteDistribution", e2 = "numeric":

Total variation distance between (empirical) data and a discrete distribution.

e1 = "numeric", e2 = "AbscontDistribution":

Total variation distance between (empirical) data and an abs. cont. distribution.

e1 = "AbscontDistribution", e1 = "numeric":

Total variation distance between (empirical) data and an abs. cont. distribution.

e1 = "AcDcLcDistribution", e2 = "AcDcLcDistribution":

Total variation distance of mixed discrete and absolutely continuous univariate distributions.

Author(s)

Matthias Kohl [email protected],
Peter Ruckdeschel [email protected]

References

Huber, P.J. (1981) Robust Statistics. New York: Wiley.

Rieder, H. (1994) Robust Asymptotic Statistics. New York: Springer.

See Also

TotalVarDist-methods, ContaminationSize, KolmogorovDist, HellingerDist, Distribution-class

Examples

TotalVarDist(Norm(), UnivarMixingDistribution(Norm(1,2),Norm(0.5,3),
                 mixCoeff=c(0.2,0.8)))
TotalVarDist(Norm(), Td(10))
TotalVarDist(Norm(mean = 50, sd = sqrt(25)), Binom(size = 100)) # mutually singular
TotalVarDist(Pois(10), Binom(size = 20)) 

x <- rnorm(100)
TotalVarDist(Norm(), x)
TotalVarDist(x, Norm(), asis.smooth.discretize = "smooth")

y <- (rbinom(50, size = 20, prob = 0.5)-10)/sqrt(5)
TotalVarDist(y, Norm())
TotalVarDist(y, Norm(), asis.smooth.discretize = "smooth")

TotalVarDist(rbinom(50, size = 20, prob = 0.5), Binom(size = 20, prob = 0.5))

---