Package 'distr'

Description

distr provides a conceptual treatment of distributions by means of S4 classes. A mother class Distribution is introduced with slots for a parameter and —most important— for the four constitutive methods r, d, p, and q for simulation respectively for evaluation of density / c.d.f.\ and quantile function of the corresponding distribution. Most distributions of package stats (like normal, Poisson, etc.) are implemented as subclasses of either AbscontDistribution or DiscreteDistribution, which themselves are again subclasses of Distribution. Up to arguments referring to a parameter of the distribution (like mean for the normal distribution), these function slots have the same arguments as those of package stats, i.e.; for a distribution object X we may call these functions as

• r(X)(n)

• d(X)(x, log = FALSE)

• p(X)(q, lower.tail = TRUE, log.p = FALSE)

• q(X)(p, lower.tail = TRUE, log.p = FALSE)

For the arguments of these function slots see e.g. rnorm. Note that, as usual, slots d, p, and q are vectorized in their first argument, but are not on the subsequent ones. In the environments of RStudio, see https://posit.co and Jupyter IRKernel, see https://github.com/IRkernel/IRkernel, calls to q are caught away from standard R evaluation and are treated in a non-standard way. This non-standard evaluation in particular throws errors at calls to our accessor methods q to slot q of the respective distribution object. To amend this, we provide function q.l as alias to our accessors q, so that our packages also become available in these environments. Arithmetics and unary mathematical transformations for distributions are available: For Distribution objects X and Y expressions like 3*X+sin(exp(-Y/4+3)) have their natural interpretation as corresponding image distributions.

Details

Classes

Distribution classes have a slot param the class of which is is specialized for the particualar distributions. The parameter classes for the particular distributions have slots with names according to the corresponding [rdpq]<name> functions of package base. From version 1.9 on, AbscontDistribution and descendants have a slot gaps for gaps in the support. DiscreteDistribution and descendants have an additional slot support, which is again specialized to be a lattice for class LatticeDistribution.
For saved objects from earlier versions, we provide the methods isOldVersion, and conv2NewVersion to check whether the object was generated by an older version of this package and to convert such an object to the new format, respectively. This applies to objects of subclasses of AbscontDistribution lacking a gap-slot as well as to to objects of subclasses of LatticeDistribution lacking a lattice-slot.
To enhance accuracy, from version 1.9 on, we also provide subclasses AffLinAbscontDistribution, AffLinDiscreteDistribution, and AffLinLatticeDistribution, as well as the class union AffLinDistribution, so that in particular functionals like E from package distrEx can recur to exact formula more frequently: These classes have additional slots a, b, and X0 to reflect the fact, that a distribution object of theses classes has the same distribution as a*X0+b.
For all particular distributions, as well as for classes AbscontDistribution, DiscreteDistribution, LatticeDistribution, UnivarDistrList and DistrList generating functions are provided, e.g. X <- Norm(mean = 3, sd = 2). The same goes for the space classes. All slots should be inspected / modified by means of corresponding accessor- /replacement functions; e.g. mean(X) <- 3 Again to enhance accuracy, from version 2.0 on, we also provide subclasses UnivarMixingDistribution to support mixing distributions, UnivarLebDecDistribution, to support Lebesgue decomposed distributions (with a discrete and an a.c. part) as well as AffLinUnivarLebDecDistribution, for corresponding affine linear transformations. Class UnivarLebDecDistribution is closed under arithmetical operations + /, *, ^ for pairs of independent variables + +, - for pairs of independent variables + affine linear transformations + truncation, huberization, min/max which are all now available analytically.
(see Parameter classes).

[*]: there is a generating function with the same name
##########################
Distribution classes
##########################
slots: [<name>(<class>)]
img(rSpace), param(OptionalParameter),
r(function), d(OptionalFunction), p(OptionalFunction), q(OptionalFunction),
.withSim(logical), .withArith(logical), .logExact(logical), .lowerExact(logical),
Symmetry(DistributionSymmetry)
"Distribution"
|>"UnivariateDistribution"
|>|>"UnivarMixingDistribution"            [*]
|>|>|>"UnivarLebDecDistribution"          [*]
|>|>|>|>"AffLinUnivarLebDecDistribution"
|>|>|>"CompoundDistribution"              [*]
|>|>"AbscontDistribution"                 [*]
|>|>|>"AffLinAbscontDistribution"
|>|>|>"Arcsine"                           [*]
|>|>|>"Beta"                              [*]
|>|>|>"Cauchy"                            [*]
|>|>|>"ExpOrGammaOrChisq" (VIRTUAL)
|>|>|>|>"Exp"                             [*]
|>|>|>|>"Gammad"                          [*]
|>|>|>|>"Chisq"                           [*]
|>|>|>"Fd"                                [*]
|>|>|>"Lnorm"                             [*]
|>|>|>"Logis"                             [*]
|>|>|>"Norm"                              [*]
|>|>|>"Td"                                [*]
|>|>|>"Unif"                              [*]
|>|>|>"Weibull"                           [*]
|>|>|"DiscreteDistribution"               [*]
|>|>|>"AffLinDiscreteDistribution"
|>|>|>"LatticeDistribution"               [*]
|>|>|>|>"AffLinLatticeDistribution"
|>|>|>|>"Binom"                           [*]
|>|>|>|>"Dirac"                           [*]
|>|>|>|>"Hyper"                           [*]
|>|>|>|>"NBinom"                          [*]
|>|>|>|>|>"Geom"                          [*]
|>|>|>|>"Pois"                            [*]
"AffLinDistribution" = union ( "AffLinAbscontDistribution",
                               "AffLinDiscreteDistribution",
                               "AffLinUnivarLebDecDistribution" )
"DistrList"
|>"UnivarDistrList"                       [*]
"AcDcLc" = union ( "AbscontDistribution",
                   "DiscreteDistribution",
                   "UnivarLebDecDistribution" )
##########################
Parameter classes
##########################
"OptionalParameter"
|>"Parameter"
|>|>"BetaParameter"
|>|>"BinomParameter"
|>|>"CauchyParameter"
|>|>"ChisqParameter"
|>|>"DiracParameter"
|>|>"ExpParameter"
|>|>"FParameter"
|>|>"GammaParameter"
|>|>"GeomParameter"
|>|>"HyperParameter"
|>|>"LnormParameter"
|>|>"LogisParameter"
|>|>"NbinomParameter"
|>|>"NormParameter"
|>|>"UniNormParameter"
|>|>|>"PoisParameter"
|>|>"TParameter"
|>|>"UnifParameter"
|>|>"WeibullParameter"
##########################
Space classes
##########################
"rSpace"
|>"EuclideanSpace"
|>|>"Reals"
|>"Lattice"
|>"Naturals"
##########################
Symmetry classes
##########################
slots:
type(character), SymmCenter(ANY)
"Symmetry"
|>"NoSymmetry"          [*]
|>"EllipticalSymmetry"  [*]
|>|>"SphericalSymmetry" [*]
|>"DistributionSymmetry"
|>"FunctionSymmetry"
|>|>"NonSymmetric"      [*]
|>|>"EvenSymmetric"     [*]
|>|>"OddSymmetric"      [*]
list thereof
"DistrSymmList"         [*]
"FunSymmList"           [*]
##########################
Matrix classes
##########################
slots:
none
"PosSemDefSymmMatrix" [*] is subclass of class "matrix" of package "base".
|>"PosDefSymmMatrix"  [*]
##########################
Class unions
##########################
"OptionalNumeric" = union("numeric", "NULL")
"OptionalMatrix" = union("matrix","NULL")

Methods

The group Math of unary (see Math) as well as convolution are made available for distributions, see operators-methods ;in particular for convolution powers, we have method convpow. Besides, there are plot and print-methods for distributions. For the space classes, we have liesIn, for the DicreteDistribution class, we have liesInSupport, as well as a generating function. The "history" of distributions obtained by chaining operations may be shortened using simplifyr.

Functions

RtoDPQ                  Default procedure to fill slots d,p,q given r
                        for a.c. distributions
RtoDPQ.d                Default procedure to fill slots d,p,q given r
                        for discrete distributions
RtoDPQ.LC               Default procedure to fill slots d,p,q given r
                        for Lebesgue decomposed distributions
decomposePM             decomposes a distribution into positive and negative
                        part and, if discrete, into part '0'
simplifyD               tries to reduce/simplify mixing distribution using
                        that certain weights are 0
flat.LCD                makes a single UnivarLebDecDistribution out of
                        a list of UnivarLebDecDistribution with corresp. weights
flat.mix                makes a single UnivarLebDecDistribution out of
                        a list of a UnivarMixingDistribution
distroptions            Functions to change the global variables of the
                        package 'distr'
standardMethods         Utility to automatically generate accessor and
                        replacement functions

Extension Packages in distrXXX family

Please note that there are extension packages of this packages available on CRAN,

distrDoc

a documentation package providing joint documentation for all packages of the distrXXX family of packages in the form of vignette 'distr'; try require(distrDoc); vignette("distr").

distrEx

provides functionals (like E, sd, mad) operating on distributions, as well as distances between distributions and basic support for multivariate and conditional distributions.

distrSim

for the standardized treatment of simulations, also under contaminations.

distrTEst

with classes and methods for evaluations of statistical procedures on simulations generated by distrSim.

distrTeach

embodies illustrations for basic stats courses using our distribution classes.

distrMod

provides classes for parametric models and hence covers, in an object orientated way, estimation in statistical models.

distrEllipse

provides classes for elliptically symmetric distributions.

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.

Acknowledgement

We thank Martin Maechler, Josef Leydold, John Chambers, Duncan Murdoch, Gregory Warnes, Paul Gilbert, Kurt Hornik, Uwe Ligges, Torsten Hothorn, and Seth Falcon for their help in preparing this package.

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.

Demos

Demos are available — see demo(package="distr")

Note

Arithmetics on distribution objects are understood as operations on corresponding (independent) r.v.'s and not on distribution functions or densities.
See also distrARITH().
Some functions of package stats have intentionally been masked, but completely retain their functionality — see distrMASK().
Accuracy of these arithmetics is controlled by global options which may be inspected / set by distroptions() and getdistrOption(), confer distroptions .

Author(s)

Peter Ruckdeschel [email protected],
Thomas Stabla [email protected],
Florian Camphausen [email protected],
Matthias Kohl [email protected]
Maintainer: Peter Ruckdeschel [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 P. Ruckdeschel and M. Kohl (2014): General purpose convolution algorithm for distributions in S4-Classes by means of FFT. J. Statist. Softw. 59(4): 1-25. 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/

Examples

X <- Unif(2,3)
Y <- Pois(lambda = 3)
Z <- X+Y  # generates Law of corresponding independent variables
p(Z)(0.2)
r(Z)(1000)
plot(Z+sin(Norm()))

Description

Generates an object of class "AbscontDistribution"

Usage

AbscontDistribution(r = NULL, d = NULL, p = NULL, q = NULL,
                   gaps = NULL, param = NULL, img = new("Reals"),
                   .withSim = FALSE, .withArith = FALSE,
                    .lowerExact = FALSE, .logExact = FALSE,
                   withgaps = getdistrOption("withgaps"),
                   low1 = NULL, up1 = NULL, low = -Inf, up =Inf,
                   withStand = FALSE,
                   ngrid = getdistrOption("DefaultNrGridPoints"),
                   ep = getdistrOption("TruncQuantile"),
                   e = getdistrOption("RtoDPQ.e"),
                   Symmetry = NoSymmetry())

Arguments

Details

Typical usages are

  AbscontDistribution(r)
  AbscontDistribution(r = NULL, d)
  AbscontDistribution(r = NULL, d = NULL, p)
  AbscontDistribution(r = NULL, d = NULL, p = NULL, d)
  AbscontDistribution(r, d, p, q)
  

Minimally, only one of the slots r, d, p or q needs to be given as argument. The other non-given slots are then reconstructed according to the following scheme:

For this purpose, one may alternatively give arguments low1 and up1 (NULL each by default, and determined through slot q, resp. p, resp. d, resp. r in this order according to availability), for the (finite) range of values in the support of this distribution, as well as the possibly infinite theoretical range given by arguments low and up with default values -Inf, Inf, respectively. Of course all other slots may be specified as arguments.

Value

Object of class "AbscontDistribution"

Author(s)

Peter Ruckdeschel [email protected]

See Also

AbscontDistribution-class, DiscreteDistribution-class, RtoDPQ

Examples

plot(Norm())
plot(AbscontDistribution(r = rnorm))
plot(AbscontDistribution(d = dnorm))
plot(AbscontDistribution(p = pnorm))
plot(AbscontDistribution(q = qnorm))
plot(Ac <- AbscontDistribution(d = function(x, log = FALSE){
                                   d <- exp(-abs(x^3))
                                   ## unstandardized!!
                                   if(log) d <- log(d)
                                   return(d)}, 
                         withStand = TRUE))

Description

The AbscontDistribution-class is the mother-class of the classes Beta, Cauchy, Chisq, Exp, F, Gammad, Lnorm, Logis, Norm, T, Unif and Weibull. Further absolutely continuous distributions can be defined either by declaration of own random number generator, density, cumulative distribution and quantile functions, or as result of a convolution of two absolutely continuous distributions or by application of a mathematical operator to an absolutely continuous distribution.

Objects from the Class

Objects can be created by calls of the form new("AbscontDistribution", r, d, p, q). More comfortably, you may use the generating function AbscontDistribution. The result of these calls is an absolutely continuous distribution.

Slots

img

Object of class "Reals": the space of the image of this distribution which has dimension 1 and the name "Real Space"

param

Object of class "Parameter": the parameter of this distribution, having only the slot name "Parameter of an absolutely continuous distribution"

r

Object of class "function": generates random numbers

d

Object of class "function": density function

p

Object of class "function": cumulative distribution function

q

Object of class "function": quantile function

gaps

[from version 1.9 on] Object of class "OptionalMatrix", i.e.; an object which may either be NULL ora matrix. This slot, if non-NULL, contains left and right endpoints of intervals where the density of the object is 0. This slot may be inspected by the accessor gaps() and modified by a corresponding replacement method. It may also be filled automatically by setgaps(). For saved objects from earlier versions, we provide functions isOldVersion and conv2NewVersion.

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

Methods

initialize

signature(.Object = "AbscontDistribution"): initialize method

Math

signature(x = "AbscontDistribution"): application of a mathematical function, e.g. sin or exp (does not work with log, sign!), to this absolutely continouos distribution

• abs: signature(x = "AbscontDistribution"): exact image distribution of abs(x).

• exp: signature(x = "AbscontDistribution"): exact image distribution of exp(x).

• sign: signature(x = "AbscontDistribution"): exact image distribution of sign(x).

• sqrt: signature(x = "AbscontDistribution"): exact image distribution of sqrt(x).

• log: signature(x = "AbscontDistribution"): (with optional further argument base, defaulting to exp(1)) exact image distribution of log(x).

• log10: signature(x = "AbscontDistribution"): exact image distribution of log10(x).

• gamma: signature(x = "AbscontDistribution"): exact image distribution of gamma(x).

• lgamma: signature(x = "AbscontDistribution"): exact image distribution of lgamma(x).

• digamma: signature(x = "AbscontDistribution"): exact image distribution of digamma(x).

• sqrt: signature(x = "AbscontDistribution"): exact image distribution of sqrt(x).

-

signature(e1 = "AbscontDistribution"): application of ‘-’ to this absolutely continuous distribution.

*

signature(e1 = "AbscontDistribution", e2 = "numeric"): multiplication of this absolutely continuous distribution by an object of class "numeric"

/

signature(e1 = "AbscontDistribution", e2 = "numeric"): division of this absolutely continuous distribution by an object of class "numeric"

+

signature(e1 = "AbscontDistribution", e2 = "numeric"): addition of this absolutely continuous distribution to an object of class "numeric".

-

signature(e1 = "AbscontDistribution", e2 = "numeric"): subtraction of an object of class "numeric" from this absolutely continuous distribution.

*

signature(e1 = "numeric", e2 = "AbscontDistribution"): multiplication of this absolutely continuous distribution by an object of class "numeric".

+

signature(e1 = "numeric", e2 = "AbscontDistribution"): addition of this absolutely continuous distribution to an object of class "numeric".

-

signature(e1 = "numeric", e2 = "AbscontDistribution"): subtraction of this absolutely continuous distribution from an object of class "numeric".

+

signature(e1 = "AbscontDistribution", e2 = "AbscontDistribution"): Convolution of two absolutely continuous distributions. The slots p, d and q are approximated by grids.

-

signature(e1 = "AbscontDistribution", e2 = "AbscontDistribution"): Convolution of two absolutely continuous distributions. The slots p, d and q are approximated by grids.

plot

signature(object = "AbscontDistribution"): plots density, cumulative distribution and quantile function.

Internal subclass "AffLinAbscontDistribution"

To enhance accuracy of several functionals on distributions, mainly from package distrEx, from version 1.9 of this package on, there is an internally used (but exported) subclass "AffLinAbscontDistribution" which has extra slots a, b (both of class "numeric"), and X0 (of class "AbscontDistribution"), to capture the fact that the object has the same distribution as a * X0 + b. This is the class of the return value of methods

-

signature(e1 = "AbscontDistribution")

*

signature(e1 = "AbscontDistribution", e2 = "numeric")

/

signature(e1 = "AbscontDistribution", e2 = "numeric")

+

signature(e1 = "AbscontDistribution", e2 = "numeric")

-

signature(e1 = "AbscontDistribution", e2 = "numeric")

*

signature(e1 = "numeric", e2 = "AbscontDistribution")

+

signature(e1 = "numeric", e2 = "AbscontDistribution")

-

signature(e1 = "numeric", e2 = "AbscontDistribution")

-

signature(e1 = "AffLinAbscontDistribution")

*

signature(e1 = "AffLinAbscontDistribution", e2 = "numeric")

/

signature(e1 = "AffLinAbscontDistribution", e2 = "numeric")

+

signature(e1 = "AffLinAbscontDistribution", e2 = "numeric")

-

signature(e1 = "AffLinAbscontDistribution", e2 = "numeric")

*

signature(e1 = "numeric", e2 = "AffLinAbscontDistribution")

+

signature(e1 = "numeric", e2 = "AffLinAbscontDistribution")

-

signature(e1 = "numeric", e2 = "AffLinAbscontDistribution")

There also is a class union of "AffLinAbscontDistribution", "AffLinDiscreteDistribution", "AffLinUnivarLebDecDistribution" and called "AffLinDistribution" which is used for functionals.

Internal virtual superclass "AcDcLcDistribution"

As many operations should be valid no matter whether the operands are of class "AbscontDistribution", "DiscreteDistribution", or "UnivarLebDecDistribution", there is a class union of these classes called "AcDcLcDistribution"; in partiucalar methods for "*", "/", "^" (see operators-methods) and methods Minimum, Maximum, Truncate, and Huberize, and convpow are defined for this class union.

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

AbscontDistribution Parameter-class UnivariateDistribution-class Beta-class Cauchy-class Chisq-class Exp-class Fd-class Gammad-class Lnorm-class Logis-class Norm-class Td-class Unif-class Weibull-class DiscreteDistribution-class Reals-class RtoDPQ

Examples

N <-  Norm() # N is a normal distribution with mean=0 and sd=1.
E <-  Exp() # E is an exponential distribution with rate=1.
A1 <-  E+1 # a new absolutely continuous distributions with exact slots d, p, q
A2 <-  A1*3 # a new absolutely continuous distributions with exact slots d, p, q
A3 <- N*0.9 + E*0.1 # a new absolutely continuous distribution with approximated slots d, p, q
r(A3)(1) # one random number generated from this distribution, e.g. -0.7150937
d(A3)(0) # The (approximated) density for x=0 is 0.43799.
p(A3)(0) # The (approximated) probability that x <= 0 is 0.45620.
q(A3)(.1) # The (approximated) 10 percent quantile is -1.06015.
## in RStudio or Jupytier IRKernel, use q.l(.)(.) instead of q(.)(.)

Description

The Arcsine distribution has density

$f(x)=\frac{1}{\pi \sqrt{1-x^2} }$

for $-1 < x < 1$.

Objects from the Class

Objects can be created by calls of the form Arcsine(). This object is an Arcsine distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

r

Object of class "function": generates random numbers (calls function rArcsine)

d

Object of class "function": density function (calls function dArcsine)

p

Object of class "function": cumulative function (calls function pArcsine)

q

Object of class "function": inverse of the cumulative function (calls function qArcsine)

.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 "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "Arcsine"): initialize method

Author(s)

Peter Ruckdeschel [email protected]

See Also

AbscontDistribution-class Reals-class

Examples

A <- Arcsine()
# A is a Arcsine distribution with shape1 = 1 and shape2 = 1.
r(A)(3) # three random number generated from this distribution, e.g. 0.6979795
d(A)(c(-2,-1,-0.2,0,0.2,1,2)) # Density at x=c(-1,-0.2,0,0.2,1).
p(A)(c(-2,-1,-0.2,0,0.2,1,2)) # cdf at q=c(-1,-0.2,0,0.2,1).
q(A)(c(0,0.2,1,2)) # quantile function at at x=c(0,0.2,1).
## in RStudio or Jupyter IRKernel, use q.l(A)(c(0,0.2,1,2)) instead

Description

The Beta distribution with parameters shape1 $= a$ and shape2 $= b$ has density

$f(x)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}{x}^{a-1} {(1-x)}^{b-1}$

for $a > 0$, $b > 0$ and $0 \le x \le 1$ where the boundary values at $x=0$ or $x=1$ are defined as by continuity (as limits).

Ad hoc methods

For R Version <2.3.0 ad hoc methods are provided for slots q, r if ncp!=0; for R Version >=2.3.0 the methods from package stats are used.

Objects from the Class

Objects can be created by calls of the form Beta(shape1, shape2). This object is a beta distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "BetaParameter": the parameter of this distribution (shape1 and shape2), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rbeta)

d

Object of class "function": density function (calls function dbeta)

p

Object of class "function": cumulative function (calls function pbeta)

q

Object of class "function": inverse of the cumulative function (calls function qbeta)

.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 "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "Beta"): initialize method

shape1

signature(object = "Beta"): returns the slot shape1 of the parameter of the distribution

shape1<-

signature(object = "Beta"): modifies the slot shape1 of the parameter of the distribution

shape2

signature(object = "Beta"): returns the slot shape2 of the parameter of the distribution

shape2<-

signature(object = "Beta"): modifies the slot shape2 of the parameter of the distribution

-

signature(e1 = "numeric", e2 = "Beta") if ncp(e2)==0 and e1 == 1, an exact (central) Beta(shape1 = shape2(e2), shape2 = shape1(e2)) is returned, else the default method is used; exact

Note

The non-central Beta distribution is defined (Johnson et al, 1995, pp. 502) as the distribution of $X/(X+Y)$ where $X \sim \chi^2_{2a}(\lambda)$ and $Y \sim \chi^2_{2b}$. C.f. rbeta

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

BetaParameter-class AbscontDistribution-class Reals-class rbeta

Examples

B <- Beta(shape1 = 1, shape2 = 1)
# B is a beta distribution with shape1 = 1 and shape2 = 1.
r(B)(1) # one random number generated from this distribution, e.g. 0.6979795
d(B)(1) # Density of this distribution is 1 for x=1.
p(B)(1) # Probability that x < 1 is 1.
q(B)(.1) # Probability that x < 0.1 is 0.1.
shape1(B) # shape1 of this distribution is 1.
shape1(B) <- 2 # shape1 of this distribution is now 2.
Bn <- Beta(shape1 = 1, shape2 = 3, ncp = 5) 
# Bn is a beta distribution with shape1 = 1 and shape2 = 3 and ncp = 5.
B0 <- Bn; ncp(B0) <- 0; 
# B0 is just the same beta distribution as Bn but with ncp = 0
q(B0)(0.1) ## 
q(Bn)(0.1) ## => from R 2.3.0 on ncp no longer ignored...
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)

Description

The parameter of a beta distribution, used by Beta-class

Objects from the Class

Objects can be created by calls of the form new("BetaParameter", shape1, shape2, ncp). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Beta is instantiated.

Slots

shape1

Object of class "numeric": the shape1 of a beta distribution

shape2

Object of class "numeric": the shape2 of a beta distribution

ncp

Object of class "numeric": the noncentrality parameter of a beta distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "BetaParameter"): initialize method

shape1

signature(object = "BetaParameter"): returns the slot shape1 of the parameter of the distribution

shape1<-

signature(object = "BetaParameter"): modifies the slot shape1 of the parameter of the distribution

shape2

signature(object = "BetaParameter"): returns the slot shape2 of the parameter of the distribution

shape2<-

signature(object = "BetaParameter"): modifies the slot shape2 of the parameter of the distribution

ncp

signature(object = "BetaParameter"): returns the slot ncp of the parameter of the distribution

ncp<-

signature(object = "BetaParameter"): modifies the slot ncp of the parameter of the distribution

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

Beta-class Parameter-class

Examples

W <- new("BetaParameter", shape1 = 1, shape2 = 1, ncp = 0)
shape2(W) # shape2 of this distribution is 1.
shape2(W) <- 2 # shape2 of this distribution is now 2.

Description

The binomial distribution with size $= n$, by default $=1$, and prob $= p$, by default $=0.5$, has density

$p(x) = {n \choose x} {p}^{x} {(1-p)}^{n-x}$

for $x = 0, \ldots, n$.

C.f.rbinom

Objects from the Class

Objects can be created by calls of the form Binom(prob, size). This object is a binomial distribution.

Slots

img

Object of class "Naturals": The space of the image of this distribution has got dimension 1 and the name "Natural Space".

param

Object of class "BinomParameter": the parameter of this distribution (prob, size), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rbinom)

d

Object of class "function": density function (calls function dbinom)

p

Object of class "function": cumulative function (calls function pbinom)

q

Object of class "function": inverse of the cumulative function (calls function qbinom). The quantile is defined as the smallest value x such that F(x) >= p, where F is the cumulative function.

support

Object of class "numeric": a (sorted) vector containing the support of the discrete density 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 "DiscreteDistribution", directly.
Class "UnivariateDistribution", by class "DiscreteDistribution".
Class "Distribution", by class "DiscreteDistribution".

Methods

+

signature(e1 = "Binom", e2 = "Binom"): For two binomial distributions with equal probabilities the exact convolution formula is implemented thereby improving the general numerical accuracy.

initialize

signature(.Object = "Binom"): initialize method

prob

signature(object = "Binom"): returns the slot prob of the parameter of the distribution

prob<-

signature(object = "Binom"): modifies the slot prob of the parameter of the distribution

size

signature(object = "Binom"): returns the slot size of the parameter of the distribution

size<-

signature(object = "Binom"): modifies the slot size of the parameter of the distribution

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

BinomParameter-class DiscreteDistribution-class Naturals-class rbinom

Examples

B <- Binom(prob=0.5,size=1) # B is a binomial distribution with prob=0.5 and size=1.
r(B)(1) # # one random number generated from this distribution, e.g. 1
d(B)(1) # Density of this distribution is  0.5 for x=1.
p(B)(0.4) # Probability that x<0.4 is 0.5.
q(B)(.1) # x=0 is the smallest value x such that p(B)(x)>=0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
size(B) # size of this distribution is 1.
size(B) <- 2 # size of this distribution is now 2.
C <- Binom(prob = 0.5, size = 1) # C is a binomial distribution with prob=0.5 and size=1.
D <- Binom(prob = 0.6, size = 1) # D is a binomial distribution with prob=0.6 and size=1.
E <- B + C # E is a binomial distribution with prob=0.5 and size=3.
F <- B + D # F is an object of class LatticeDistribution.
G <- B + as(D,"DiscreteDistribution") ## DiscreteDistribution

Description

The parameter of a binomial distribution, used by Binom-class

Objects from the Class

Objects can be created by calls of the form new("BinomParameter", prob, size). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Binom is instantiated.

Slots

prob

Object of class "numeric": the probability of a binomial distribution

size

Object of class "numeric": the size of a binomial distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "BinomParameter"): initialize method

prob

signature(object = "BinomParameter"): returns the slot prob of the parameter of the distribution

prob<-

signature(object = "BinomParameter"): modifies the slot prob of the parameter of the distribution

size

signature(object = "BinomParameter"): returns the slot size of the parameter of the distribution

size<-

signature(object = "BinomParameter"): modifies the slot size of the parameter of the distribution

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

Binom-class Parameter-class

Examples

W <- new("BinomParameter",prob=0.5,size=1)
size(W) # size of this distribution is 1.
size(W) <- 2 # size of this distribution is now 2.

Description

The Cauchy distribution with location $l$, by default $=0$, and scale $s$ , by default $=1$,has density

$f(x) = \frac{1}{\pi s} \left( 1 + \left(\frac{x - l}{s}\right)^2 \right)^{-1}$

for all $x$. C.f. rcauchy

Objects from the Class

Objects can be created by calls of the form Cauchy(location, scale). This object is a Cauchy distribution.

Slots

img

Object of class "Reals": The domain of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "CauchyParameter": the parameter of this distribution (location and scale), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rcauchy)

d

Object of class "function": density function (calls function dcauchy)

p

Object of class "function": cumulative function (calls function pcauchy)

q

Object of class "function": inverse of the cumulative function (calls function qcauchy)

.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 "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Is-Relations

By means of setIs, R “knows” that a distribution object obj of class "Cauchy" with location 0 and scale 1 also is a T distribution with parameters df = 1, ncp = 0.

Methods

initialize

signature(.Object = "Cauchy"): initialize method

location

signature(object = "Cauchy"): returns the slot location of the parameter of the distribution

location<-

signature(object = "Cauchy"): modifies the slot location of the parameter of the distribution

scale

signature(object = "Cauchy"): returns the slot scale of the parameter of the distribution

scale<-

signature(object = "Cauchy"): modifies the slot scale of the parameter of the distribution

+

signature(e1 = "Cauchy", e2 = "Cauchy"): For the Cauchy distribution the exact convolution formula is implemented thereby improving the general numerical approximation.

*

signature(e1 = "Cauchy", e2 = "numeric")

+

signature(e1 = "Cauchy", e2 = "numeric"): For the Cauchy location scale family we use its closedness under affine linear transformations.

further arithmetic methods see operators-methods

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

CauchyParameter-class AbscontDistribution-class Reals-class rcauchy

Examples

C <- Cauchy(location = 1, scale = 1) # C is a Cauchy distribution with location=1 and scale=1.
r(C)(1) # one random number generated from this distribution, e.g. 4.104603
d(C)(1) # Density of this distribution is 0.3183099 for x=1.
p(C)(1) # Probability that x<1 is 0.5.
q(C)(.1) # Probability that x<-2.077684 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
location(C) # location of this distribution is 1.
location(C) <- 2 # location of this distribution is now 2.
is(C,"Td") # no
C0 <- Cauchy() # standard, i.e. location = 0, scale = 1
is(C0,"Td") # yes
as(C0,"Td")

Description

The parameter of a Cauchy distribution, used by Cauchy-class

Objects from the Class

Objects can be created by calls of the form new("CauchyParameter", location, scale). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Cauchy is instantiated.

Slots

location:

Object of class "numeric": the location of a Cauchy distribution

scale

Object of class "numeric": the scale of a Cauchy distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "CauchyParameter"): initialize method

scale

signature(object = "CauchyParameter"): returns the slot scale of the parameter of the distribution

scale<-

signature(object = "CauchyParameter"): modifies the slot scale of the parameter of the distribution

location

signature(object = "CauchyParameter"): returns the slot location of the parameter of the distribution

location<-

signature(object = "CauchyParameter"): modifies the slot location of the parameter of the distribution

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

Cauchy-class Parameter-class

Examples

W <- new("CauchyParameter",location=1,scale=1)
location(W) # location of this distribution is 1.
location(W) <- 2 # location of this distribution is now 2.

Description

The chi-squared distribution with df$= n$ degrees of freedom has density

$f_n(x) = \frac{1}{{2}^{n/2} \Gamma (n/2)} {x}^{n/2-1} {e}^{-x/2}$

for $x > 0$. The mean and variance are $n$ and $2n$.

The non-central chi-squared distribution with df$= n$ degrees of freedom and non-centrality parameter ncp $= \lambda$ has density

$f(x) = e^{-\lambda / 2} \sum_{r=0}^\infty \frac{(\lambda/2)^r}{r!}\, f_{n + 2r}(x)$

for $x \ge 0$. For integer $n$, this is the distribution of the sum of squares of $n$ normals each with variance one, $\lambda$ being the sum of squares of the normal means.

C.f. rchisq

Objects from the Class

Objects can be created by calls of the form Chisq(df, ncp). This object is a chi-squared distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "ChisqParameter": the parameter of this distribution (df and ncp), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rchisq)

d

Object of class "function": density function (calls function dchisq)

p

Object of class "function": cumulative function (calls function pchisq)

q

Object of class "function": inverse of the cumulative function (calls function qchisq)

.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 "ExpOrGammaOrChisq", directly.
Class "AbscontDistribution", by class "ExpOrGammaOrChisq".
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "UnivariateDistribution".

Is-Relations

By means of setIs, R “knows” that a distribution object obj of class "Chisq" with non-centrality 0 also is a Gamma distribution with parameters shape = df(obj)/2, scale = 2.

Methods

initialize

signature(.Object = "Chisq"): initialize method

df

signature(object = "Chisq"): returns the slot df of the parameter of the distribution

df<-

signature(object = "Chisq"): modifies the slot df of the parameter of the distribution

ncp

signature(object = "Chisq"): returns the slot ncp of the parameter of the distribution

ncp<-

signature(object = "Chisq"): modifies the slot ncp of the parameter of the distribution

+

signature(e1 = "Chisq", e2 = "Chisq"): For the chi-squared distribution we use its closedness under convolutions.

Note

Warning: The code for pchisq and qchisq is unreliable for values of ncp above approximately 290.

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

ChisqParameter-class AbscontDistribution-class Reals-class rchisq

Examples

C <- Chisq(df = 1, ncp = 1) # C is a chi-squared distribution with df=1 and ncp=1.
r(C)(1) # one random number generated from this distribution, e.g. 0.2557184
d(C)(1) # Density of this distribution is 0.2264666 for x = 1.
p(C)(1) # Probability that x < 1 is 0.4772499.
q(C)(.1) # Probability that x < 0.04270125 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
df(C) # df of this distribution is 1.
df(C) <- 2 # df of this distribution is now 2.
is(C, "Gammad") # no
C0 <- Chisq() # default: Chisq(df=1,ncp=0)
is(C0, "Gammad") # yes
as(C0,"Gammad")

Description

The parameter of a chi-squared distribution, used by Chisq-class

Objects from the Class

Objects can be created by calls of the form new("ChisqParameter", ncp, df). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Chisq is instantiated.

Slots

ncp

Object of class "numeric": the ncp of a chi-squared distribution

df

Object of class "numeric": the df of a chi-squared distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "ChisqParameter"): initialize method

df

signature(object = "ChisqParameter"): returns the slot df of the parameter of the distribution

df<-

signature(object = "ChisqParameter"): modifies the slot df of the parameter of the distribution

ncp

signature(object = "ChisqParameter"): returns the slot ncp of the parameter of the distribution

ncp<-

signature(object = "ChisqParameter"): modifies the slot ncp of the parameter of the distribution

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

Chisq-class Parameter-class

Examples

W <- new("ChisqParameter",df=1,ncp=1)
ncp(W) # ncp of this distribution is 1.
ncp(W) <- 2 # ncp of this distribution is now 2.

Description

Generates an object of class "CompoundDistribution".

Usage

CompoundDistribution(NumbOfSummandsDistr, SummandsDistr, .withSim = FALSE,
                                 withSimplify = FALSE)

Arguments

Value

Object of class "CompoundDistribution", or if argument withSimplify is TRUE the result of simplifyD applied to the compound distribution, i.e. an object of class "UnivarLebDecDistribution", or if degenerate, of class "AbscontDistribution" or "DiscreteDistribution".

Author(s)

Peter Ruckdeschel [email protected]

See Also

CompoundDistribution-class, simplifyD

Examples

CP0 <- CompoundDistribution(Pois(), Norm())
CP0
CP1 <- CompoundDistribution(DiscreteDistribution(supp = c(1,5,9,11),
                            prob = dbinom(0:3, size = 3,prob = 0.3)),Norm())
CP1
UL <- UnivarDistrList(Norm(), Binom(10,0.3), Chisq(df=4), Norm(),
                      Binom(10,0.3), Chisq(df=4), Norm(), Binom(10,0.3),
                      Chisq(df=4), Td(5), Td(10))
CP2 <- CompoundDistribution(DiscreteDistribution(supp = c(1,5,9,11),
                      prob = dbinom(0:3, size = 3, prob = 0.3)),UL)
plot(CP2)

Description

CompoundDistribution-class is a class to formalize compound distributions; it is a subclass to class UnivarMixingDistribution.

Objects from the Class

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

Slots

NumbOfSummandsDistr

Object of class "DiscreteDistribution", the frequency distribution.

SummandsDistr

Object of class "UnivDistrListOrDistribution", that is, either of class "UnivarDistrList" (non i.i.d. case) or of class "UnivariateDistribution" (i.i.d. case); the summand distribution(s).

mixCoeff

Object of class "numeric": a vector of probabilities for the mixing components.

mixDistr

Object of class "UnivarDistrList": a list of univariate distributions containing the mixing components; must be of same length as mixCoeff.

img

Object of class "Reals": the space of the image of this distribution which has dimension 1 and the name "Real Space"

param

Object of class "Parameter": the parameter of this distribution, having only the slot name "Parameter of a discrete distribution"

r

Object of class "function": generates random numbers

d

fixed to NULL

p

Object of class "function": cumulative distribution function

q

Object of class "function": 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 "UnivarMixingDistribution" class "UnivarDistribution" by class "UnivarMixingDistribution", class "Distribution" by class "UnivariateDistribution".

Methods

show

signature(object = "CompoundDistribution") prints the object

SummandsDistr

signature(object = "CompoundDistribution") returns the corresponding slot

NumbOfSummandsDistr

signature(object = "CompoundDistribution") returns the corresponding slot

setAs relations

There is a coerce method to coerce objects of class "CompoundDistribution" to class UnivarLebDecDistribution; this is done by a simple call to simplifyD.

Author(s)

Peter Ruckdeschel [email protected]

See Also

Parameter-class, UnivariateDistribution-class, LatticeDistribution-class, AbscontDistribution-class, simplifyD, flat.mix

Examples

CP <- CompoundDistribution(Pois(),Norm())
CP
p(CP)(0.3)          
plot(CP)

Description

Method convpow determines the distribution of the sum of N univariate i.i.d r.v's by means of DFT

Usage

convpow(D1,...)
  ## S4 method for signature 'AbscontDistribution'
convpow(D1,N)
  ## S4 method for signature 'LatticeDistribution'
convpow(D1,N, 
                     ep = getdistrOption("TruncQuantile"))
  ## S4 method for signature 'DiscreteDistribution'
convpow(D1,N)
  ## S4 method for signature 'AcDcLcDistribution'
convpow(D1,N, 
                     ep = getdistrOption("TruncQuantile"))

Arguments

Details

in the methods implemented a second argument N is obligatory; the general methods use a general purpose convolution algorithm for distributions by means of D/FFT. In case of an argument of class "UnivarLebDecDistribution", the result will in generally be again of class "UnivarLebDecDistribution". However, if acWeight(D1) is positive, discreteWeight(convpow(D1,N)) will decay exponentially in N, hence from some (small) $N_0$ on, the result will be of class "AbscontDistribution". This is used algorithmically, too, as then only the a.c. part needs to be convolved. In case of an argument D1 of class "DiscreteDistribution", for N equal to 0,1 we return the obvious solutions, and for N==2 the return value is D1+D1. For N>2, we split up N into N=N1+N2, N1=floor(N/2) and recursively return convpow(D1,N1)+convpow(D1,N2).

Value

Object of class "AbscontDistribution", "DiscreteDistribution", "LatticeDistribution" resp. "AcDcLcDistribution"

further S4-Methods

There are particular methods for the following classes, using explicit convolution formulae:

signature(D1="Norm")

returns class "Norm"

signature(D1="Nbinom")

returns class "Nbinom"

signature(D1="Binom")

returns class "Binom"

signature(D1="Cauchy")

returns class "Cauchy"

signature(D1="ExpOrGammaOrChisq")

returns class "Gammad" —if D1 may be coerced to Gammad

signature(D1="Pois")

returns class "Pois"

signature(D1="Dirac")

returns class "Dirac"

Author(s)

Peter Ruckdeschel [email protected]
Matthias Kohl [email protected] Thomas Stabla [email protected]

References

Kohl, M., Ruckdeschel, P., (2014): General purpose convolution algorithm for distributions in S4-Classes by means of FFT. J. Statist. Softw. 59(4): 1-25.

See Also

operators, distrARITH()

Examples

convpow(Exp()+Pois(),4)

Description

d-methods

Methods

d

signature(object = "Distribution"): returns the density function

See Also

Distribution-class

Description

decomposePM-methods

Usage

decomposePM(object)

Arguments

Details

There are particular return types for the following classes

"AbscontDistribution"

a list with components "neg" and "pos" for the respective negative and positive part; each of these parts in its turn is a list with components D for the distribution (in this case of class "AbscontDistribution" again) and w for the weight of the respective part; if the weight of the negative part is 0, the corresponding distribution is set to -abs(Norm()), and respectively, if the weight of the positive part is 0, the corresponding distribution is set to abs(Norm()).

"DiscreteDistribution"

a list with components "neg", "pos" and "0" for the respective negative, positive and zero part; each of these parts in its turn is a list with components D for the distribution (in this case of class "DiscreteDistribution" again) and w for the weight of the respective part; while the distribution of the zero part is always Dirac(0), if the weight of the negative part is 0, the corresponding distribution is set to Dirac(-1), and respectively, if the weight of the positive part is 0, the corresponding distribution is set to Dirac(1).

"UnivarLebDecDistribution"

a list with components "neg", "pos" and "0" for the respective negative, positive and zero part; each of these parts in its turn is a list with components D for the distribution (in case of components "neg", "pos" of class "UnivarLebDecDistribution" again, while the distribution of the zero part is always Dirac(0)) and w for the weight of the respective part; it is build up by calling decomposePM for acPart(object) and discretePart(object) separately, hence if weights of some parts are zero the corresponding procedure mentionned for these methods applies.

Method decomposePM is used by our multiplication, division and exponentiation ("*", "/" "^") - methods.

Value

the positive and negative part of the distribution together with corresponding weights as a list.

See Also

AbscontDistribution-class, DiscreteDistribution-class, UnivarLebDecDistribution-class, operators-methods

Examples

decomposePM(Norm())
decomposePM(Binom(2,0.3)-Binom(5,.4))
decomposePM(UnivarLebDecDistribution(Norm(),Binom(2,0.3)-Binom(5,.4), 
            acWeight = 0.3))

Description

The double exponential or Laplace distribution with rate $\lambda$ has density

$f(x) = \frac{1}{2}\lambda {e}^{- \lambda |x|}$

C.f. Exp-class, rexp

Objects from the Class

Objects can be created by calls of the form DExp(rate). This object is a double exponential (or Laplace) distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "ExpParameter": the parameter of this distribution (rate), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rexp)

d

Object of class "function": density function (calls function dexp)

p

Object of class "function": cumulative function (calls function pexp)

q

Object of class "function": inverse of the cumulative function (calls function qexp)

.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 "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution". Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "DExp"): initialize method

rate

signature(object = "DExp"): returns the slot rate of the parameter of the distribution

rate<-

signature(object = "DExp"): modifies the slot rate of the parameter of the distribution

*

signature(e1 = "DExp", e2 = "numeric"): For the Laplace distribution we use its closedness under scaling transformations.

Author(s)

Peter Ruckdeschel [email protected]

See Also

Exp-class ExpParameter-class AbscontDistribution-class Reals-class rexp

Examples

D <- DExp(rate = 1) # D is a Laplace distribution with rate = 1.
r(D)(1) # one random number generated from this distribution, e.g. 0.4190765
d(D)(1) # Density of this distribution is 0.1839397 for x = 1.
p(D)(1) # Probability that x < 1 is 0.8160603.
q(D)(.1) # Probability that x < -1.609438 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
rate(D) # rate of this distribution is 1.
rate(D) <- 2 # rate of this distribution is now 2.
3*D ###  still a DExp -distribution

Description

df-methods

Methods

df

signature(object = "TParameter"): returns the slot df of the parameter of the distribution

df<-

signature(object = "TParameter"): modifies the slot df of the parameter of the distribution

df

signature(object = "Td"): returns the slot df of the parameter of the distribution

df<-

signature(object = "Td"): modifies the slot df of the parameter of the distribution

df

signature(object = "ChisqParameter"): returns the slot df of the parameter of the distribution

df<-

signature(object = "ChisqParameter"): modifies the slot df of the parameter of the distribution

df

signature(object = "Chisq"): returns the slot df of the parameter of the distribution

df<-

signature(object = "Chisq"): modifies the slot df of the parameter of the distribution

Description

df-methods

Methods

df1

signature(object = "FParameter"): returns the slot df1 of the parameter of an F-distribution

df1<-

signature(object = "FParameter"): modifies the slot df1 of the parameter of an F-distribution

df1

signature(object = "Fd"): returns the slot df1 of the slot param of the distribution

df1<-

signature(object = "Fd"): modifies the slot df1 of the slot param of the distribution

Description

df-methods

Methods

df2

signature(object = "FParameter"): returns the slot df2 of the parameter of an F-distribution

df2<-

signature(object = "FParameter"): modifies the slot df2 of the parameter of an F-distribution

df2

signature(object = "Fd"): returns the slot df2 of the slot param of the distribution

df2<-

signature(object = "Fd"): modifies the slot df2 of the slot param of the distribution

Description

dim-methods

Methods

dim

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

See Also

UnivariateDistribution-class

Description

dimension-methods

Methods

dimension

signature(object = "EuclideanSpace"): returns the dimension of the space

dimension<-

signature(object = "EuclideanSpace"): modifies the dimension of the space

Description

The Dirac distribution with location $l$, by default $=0$, has density $d(x) = 1$ for $x = l$, $0$ else.

Objects from the Class

Objects can be created by calls of the form Dirac(location). This object is a Dirac distribution.

Slots

img

Object of class "Naturals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "DiracParameter": the parameter of this distribution (location), declared at its instantiation

r

Object of class "function": generates random numbers

d

Object of class "function": density function

p

Object of class "function": cumulative function

q

Object of class "function": inverse of the cumulative function

support

Object of class "numeric": a (sorted) vector containing the support of the discrete density 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 "DiscreteDistribution", directly.
Class "UnivariateDistribution", by class "DiscreteDistribution".
Class "Distribution", by class "DiscreteDistribution".

Methods

-

signature(e1 = "Dirac", e2 = "Dirac")

+

signature(e1 = "Dirac", e2 = "Dirac")

*

signature(e1 = "Dirac", e2 = "Dirac")

/

signature(e1 = "Dirac", e2 = "Dirac"): For the Dirac distribution these operations are trivial.

initialize

signature(.Object = "Dirac"): initialize method

location

signature(object = "Dirac"): returns the slot location of the parameter of the distribution

location<-

signature(object = "Dirac"): modifies the slot location of the parameter of the distribution

log

signature(object = "Dirac"): returns an object of class "Dirac" distribution with log-transformed location parameter.

Math

signature(object = "Dirac"): given a "Math" group generic fun an object of class "Dirac" distribution with fun-transformed location parameter is returned.

further arithmetic methods see operators-methods

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

DiracParameter-class DiscreteDistribution-class Naturals-class

Examples

D <- Dirac(location = 0) # D is a Dirac distribution with location=0.
r(D)(1)
# r(D)(1) generates a pseudo-random-number according to a Dirac
# distribution with location = 0,
# which of course will take 0 as value almost surely.
d(D)(0) # Density of this distribution is 1 for x = 0.
p(D)(1) # Probability that x < 1 is 1.
q(D)(.1) # q(D)(x) is always 0 (= location).
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
location(D) # location of this distribution is 0.
location(D) <- 2 # location of this distribution is now 2.

Description

The parameter of a Dirac distribution, used by Dirac-class

Objects from the Class

Objects can be created by calls of the form new("DiracParameter", location). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Dirac is instantiated.

Slots

location

Object of class "numeric": the location of a Dirac distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "DiracParameter"): initialize method

location

signature(object = "DiracParameter"): returns the slot location of the parameter of the distribution

location<-

signature(object = "DiracParameter"): modifies the slot location of the parameter of the distribution

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

Dirac-class Parameter-class

Examples

W <- new("DiracParameter",location=1)
location(W) # location of this distribution is 1.
location(W) <- 2 # location of this distribution is now 2.

Description

Generates an object of class "DiscreteDistribution"

Usage

DiscreteDistribution(supp, prob, .withArith=FALSE, .withSim=FALSE, 
                       .lowerExact = TRUE, .logExact = FALSE,
             .DistrCollapse = getdistrOption("DistrCollapse"),
             .DistrCollapse.Unique.Warn = 
                  getdistrOption("DistrCollapse.Unique.Warn"),
             .DistrResolution = getdistrOption("DistrResolution"),
             Symmetry = NoSymmetry())

Arguments

Details

If prob is missing, all elements in supp are equally weighted.

Typical usages are

    DiscreteDistribution(supp, prob)
    DiscreteDistribution(supp)
  

Value

Object of class "DiscreteDistribution"

Note

Working with a computer, we use a finite interval as support which carries at least mass 1-getdistrOption("TruncQuantile").

Also, we require that support points have distance at least .DistrResoltion, if this condition fails, upon a suggestion by Jacob van Etten, [email protected], we use the global option .DistrCollapse to decide whether we use collapsing or not. If we do so, we collapse support points if they are too close to each other, taking the (left most) median among them as new support point which accumulates all the mass of the collapsed points. With .DistrCollapse==FALSE, we at least collapse points according to the result of unique(), and if after this collapsing, the minimal distance is less than .DistrResoltion, we throw an error. By .DistrCollapse.Unique.Warn, we control, whether we throw a warning upon collapsing or not.

Author(s)

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

See Also

DiscreteDistribution-class AbscontDistribution-class RtoDPQ.d

Examples

# Dirac-measure at 0
D1 <- DiscreteDistribution(supp = 0)
D1
# simple discrete distribution
D2 <- DiscreteDistribution(supp = c(1:5), prob = c(0.1, 0.2, 0.3, 0.2, 0.2))
D2

plot(D2)

Description

The DiscreteDistribution-class is the mother-class of the class LatticeDistribution.

Objects from the Class

Objects can be created by calls to new("DiscreteDistribution", ...), but more easily is the use of the generating function "DiscreteDistribution". This generating function, from version 1.9 on, has been moved to this package from package distrEx.

Slots

img

Object of class "Reals": the space of the image of this distribution which has dimension 1 and the name "Real Space"

param

Object of class "Parameter": the parameter of this distribution, having only the slot name "Parameter of a discrete distribution"

r

Object of class "function": generates random numbers

d

Object of class "function": density/probability function

p

Object of class "function": cumulative distribution function

q

Object of class "function": 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

.finSupport

logical: used internally to check whether the true support is finite; in case img is one-dimensional, it is of length 2 (left and right end).

Symmetry

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

Extends

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

Methods

initialize

signature(.Object = "DiscreteDistribution"): initialize method

coerce

signature(from = "DiscreteDistribution", to = "LatticeDistribution"): coerce method to class "LatticeDistribution" (checks if support is a lattice)

Math

signature(x = "DiscreteDistribution"): application of a mathematical function, e.g. sin or tan to this discrete distribution

• abs: signature(x = "DiscreteDistribution"): exact image distribution of abs(x).

• exp: signature(x = "DiscreteDistribution"): exact image distribution of exp(x).

• sign: signature(x = "DiscreteDistribution"): exact image distribution of sign(x).

• sqrt: signature(x = "DiscreteDistribution"): exact image distribution of sqrt(x).

• log: signature(x = "DiscreteDistribution"): (with optional further argument base, defaulting to exp(1)) exact image distribution of log(x).

• log10: signature(x = "DiscreteDistribution"): exact image distribution of log10(x).

• gamma: signature(x = "DiscreteDistribution"): exact image distribution of gamma(x).

• lgamma: signature(x = "DiscreteDistribution"): exact image distribution of lgamma(x).

• digamma: signature(x = "DiscreteDistribution"): exact image distribution of digamma(x).

-

signature(e1 = "DiscreteDistribution"): application of ‘-’ to this discrete distribution

*

signature(e1 = "DiscreteDistribution", e2 = "numeric"): multiplication of this discrete distribution by an object of class ‘numeric’

/

signature(e1 = "DiscreteDistribution", e2 = "numeric"): division of this discrete distribution by an object of class ‘numeric’

+

signature(e1 = "DiscreteDistribution", e2 = "numeric"): addition of this discrete distribution to an object of class ‘numeric’

-

signature(e1 = "DiscreteDistribution", e2 = "numeric"): subtraction of an object of class ‘numeric’ from this discrete distribution

*

signature(e1 = "numeric", e2 = "DiscreteDistribution"): multiplication of this discrete distribution by an object of class ‘numeric’

+

signature(e1 = "numeric", e2 = "DiscreteDistribution"): addition of this discrete distribution to an object of class ‘numeric’

-

signature(e1 = "numeric", e2 = "DiscreteDistribution"): subtraction of this discrete distribution from an object of class ‘numeric’

+

signature(e1 = "DiscreteDistribution", e2 = "DiscreteDistribution"): Convolution of two discrete distributions. The slots p, d and q are approximated on a common grid.

-

signature(e1 = "DiscreteDistribution", e2 = "DiscreteDistribution"): Convolution of two discrete distributions. The slots p, d and q are approximated on a common grid.

support

signature(object = "DiscreteDistribution"): returns the support

p.l

signature(object = "DiscreteDistribution"): returns the left continuous cumulative distribution function, i.e.; $p.l(t) = P(object < t)$

q.r

signature(object = "DiscreteDistribution"): returns the right-continuous quantile function, i.e.; ${\rm q.r}(s)=\sup\{t \,\big|\, P({\tt object}\ge t)\leq s\}$

plot

signature(object = "DiscreteDistribution"): plots density, cumulative distribution and quantile function

Internal subclass "AffLinDiscreteDistribution"

To enhance accuracy of several functionals on distributions, mainly from package distrEx, from version 1.9 of this package on, there is an internally used (but exported) subclass "AffLinDiscreteDistribution" which has extra slots a, b (both of class "numeric"), and X0 (of class "DiscreteDistribution"), to capture the fact that the object has the same distribution as a * X0 + b. This is the class of the return value of methods

-

signature(e1 = "DiscreteDistribution")

*

signature(e1 = "DiscreteDistribution", e2 = "numeric")

/

signature(e1 = "DiscreteDistribution", e2 = "numeric")

+

signature(e1 = "DiscreteDistribution", e2 = "numeric")

-

signature(e1 = "DiscreteDistribution", e2 = "numeric")

*

signature(e1 = "numeric", e2 = "DiscreteDistribution")

+

signature(e1 = "numeric", e2 = "DiscreteDistribution")

-

signature(e1 = "numeric", e2 = "DiscreteDistribution")

-

signature(e1 = "AffLinDiscreteDistribution")

*

signature(e1 = "AffLinDiscreteDistribution", e2 = "numeric")

/

signature(e1 = "AffLinDiscreteDistribution", e2 = "numeric")

+

signature(e1 = "AffLinDiscreteDistribution", e2 = "numeric")

-

signature(e1 = "AffLinDiscreteDistribution", e2 = "numeric")

*

signature(e1 = "numeric", e2 = "AffLinDiscreteDistribution")

+

signature(e1 = "numeric", e2 = "AffLinDiscreteDistribution")

-

signature(e1 = "numeric", e2 = "AffLinDiscreteDistribution")

There also is a class union of "AffLinAbscontDistribution", "AffLinDiscreteDistribution", "AffLinUnivarLebDecDistribution" and called "AffLinDistribution" which is used for functionals.

Internal virtual superclass "AcDcLcDistribution"

As many operations should be valid no matter whether the operands are of class "AbscontDistribution", "DiscreteDistribution", or "UnivarLebDecDistribution", there is a class union of these classes called "AcDcLcDistribution"; in partiucalar methods for "*", "/", "^" (see operators-methods) and methods Minimum, Maximum, Truncate, and Huberize, and convpow are defined for this class union.

Note

Working with a computer, we use a finite interval as support which carries at least mass 1-getdistrOption("TruncQuantile").

Also, we require that support points have distance at least getdistrOption("DistrResoltion"), if this condition fails, upon a suggestion by Jacob van Etten, [email protected], we use the global option getdistrOption("DistrCollapse") to decide whether we use collapsing or not. If we do so, we collapse support points if they are too close to each other, taking the (left most) median among them as new support point which accumulates all the mass of the collapsed points. With getdistrOption("DistrCollapse")==FALSE, we at least collapse points according to the result of unique(), and if after this collapsing, the minimal distance is less than getdistrOption("DistrResoltion"), we throw an error. By getdistrOption("DistrCollapse.Unique.Warn"), we control, whether we throw a warning upon collapsing or not.

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

Parameter-class UnivariateDistribution-class LatticeDistribution-class AbscontDistribution-class Reals-class RtoDPQ.d

Examples

# Dirac-measure at 0
D1 <- DiscreteDistribution(supp = 0)
support(D1)

# simple discrete distribution
D2 <- DiscreteDistribution(supp = c(1:5), prob = c(0.1, 0.2, 0.3, 0.2, 0.2))
plot(D2)
(pp <- p(D2)(support(D2)))
p(D2)(support(D2)-1e-5)
p(D2)(support(D2)+1e-5)
p.l(D2)(support(D2))
p.l(D2)(support(D2)-1e-5)
p.l(D2)(support(D2)+1e-5)
q(D2)(pp)
q(D2)(pp-1e-5)
q(D2)(pp+1e-5)
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
q.r(D2)(pp)
q.r(D2)(pp-1e-5)
q.r(D2)(pp+1e-5)

Description

The parameter of a geometric distribution, used by Geom-class

Objects from the Class

Objects were created by calls of the form new("GeomParameter", prob). Usually an object of this class was not needed on its own, it was generated automatically when an object of the class Geom is instantiated.

Slots

prob

Object of class "numeric": the probability of a geometric distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "GeomParameter"): initialize method

prob

signature(object = "GeomParameter"): returns the slot prob of the parameter of the distribution

prob<-

signature(object = "GeomParameter"): modifies the slot prob of the parameter of the distribution

Defunct

The use of class GeomParameter is defunct as of version 2.8.0; it is to be replaced by a corresponding use of class NbinomParameter with slot size = 1 which may be generated, e.g. by new("NbinomParameter", prob, size = 1, name = "Parameter of a Geometric distribution")

Author(s)

Peter Ruckdeschel [email protected]

See Also

Defunct

Description

Provides information on the interpretation of arithmetics operating on Distributions in package distr

Usage

distrARITH(library = NULL)

Arguments

Value

no value is returned

Author(s)

Peter Ruckdeschel [email protected]

Examples

## IGNORE_RDIFF_BEGIN
distrARITH()
## IGNORE_RDIFF_END

Description

The Distribution-class is the mother-class of class UnivariateDistribution.

Objects from the Class

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

Slots

img

Object of class "rSpace": the space of the image

param

Object of class "OptionalParameter": the parameter

r

Object of class "function": generates random numbers

d

Object of class "OptionalFunction": density function

p

Object of class "OptionalFunction": cumulative distribution function

q

Object of class "OptionalFunction": 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.

Methods

img

signature(object = "Distribution"): returns the space of the image

param

signature(object = "Distribution"): returns the parameter

r

signature(object = "Distribution"): returns the random number generator

d

signature(object = "Distribution"): returns the density function

p

signature(object = "Distribution"): returns the cumulative distribution function

q

signature(object = "Distribution"): returns the quantile function

.logExact

signature(object = "Distribution"): returns slot .logExact if existing; else tries to convert the object to a newer version of its class by conv2NewVersion and returns the corresponding slot of the converted object.

.lowerExact

signature(object = "Distribution"): returns slot .lowerExact if existing; else tries to convert the object to a newer version of its class by conv2NewVersion and returns the corresponding slot of the converted object.

Symmetry:

returns slot Symmetry if existing; else tries to convert the object to a newer version of its class by conv2NewVersion and returns the corresponding slot of the converted object.

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

UnivariateDistribution-class Parameter-class

Description

Class of symmetries for distributions.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

type

Object of class "character": discribes type of symmetry.

SymmCenter

Object of class "OptionalNumeric": center of symmetry.

Extends

Class "Symmetry", directly.

Author(s)

Matthias Kohl [email protected]

See Also

Symmetry-class, Distribution-class, OptionalNumeric-class

Description

Generates an object of class "DistrList".

Usage

DistrList(..., Dlist)

Arguments

Value

Object of class "DistrList"

Author(s)

Matthias Kohl [email protected]

See Also

DistrList-class, UnivarDistrList-class, UnivarDistrList

Examples

(DL <- DistrList(Norm(), Exp(), Pois()))
plot(DL)
as(Norm(), "DistrList")

## The function is currently defined as
function(...){ 
    new("DistrList", list(...)) 
}

Description

Create a list of distributions

Objects from the Class

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

Slots

.Data

Object of class "list". A list of distributions.

Extends

Class "list", from data part.
Class "vector", by class "list".

Methods

show

signature(object = "DistrList")

plot

signature(object = "DistrList")

coerce

signature(from = "Distribution", to = "DistrList"): create a "DistrList" object from a "Distribution" object

Author(s)

Matthias Kohl [email protected]

See Also

DistrList, Distribution-class

Examples

(DL <- new("DistrList", list(Norm(), Exp())))
plot(DL)
as(Norm(), "DistrList")

Description

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

Usage

distrMASK(library = NULL)

Arguments

Value

no value is returned

Author(s)

Peter Ruckdeschel [email protected]

Examples

## IGNORE_RDIFF_BEGIN
distrMASK()
## IGNORE_RDIFF_END

Description

With distroptions and getdistrOption you may inspect and change the global variables used by package distr.

Usage

distroptions(...)
getdistrOption(x)

Arguments

Details

Invoking distroptions() with no arguments returns a list with the current values of the options. To access the value of a single option, one should use getdistrOption("WarningSim"), e.g., rather than distroptions("WarningSim") which is a list of length one.

Value

distroptions() returns a list of the global options of distr.
distroptions("RtoDPQ.e") returns the global option RtoDPQ.e as a list of length 1.
distroptions("RtoDPQ.e" = 3) sets the value of the global option RtoDPQ.e to 3. getdistrOption("RtoDPQ.e") the current value set for option RtoDPQ.e.

Currently available options

DefaultNrGridPoints

default number of grid points in integration, default value: 2^12

DistrResolution

minimal spacing between two mass points in a discrete distribution, default value: 1e-6

DistrCollapse

logical; in discrete distributions, shall support points with distance smaller than DistrResolution be collapsed; default value: TRUE

TruncQuantile

argument for q-slot at which to truncate; also, for discrete distributions, support is restricted to [q(TruncQuantile),q(1-TruncQuantile)], default value: 1e-5

DefaultNrFFTGridPointsExponent

by default, for e = DefaultNrFFTGridPointsExponent, FFT uses $2^e$ gridpoints; default value: 12

RtoDPQ.e

by default, for reconstructing the d-,p-,q-slots out of simulations by slot r, RtoDPQ resp. RtoDPQ.d use $10^e$ simulations, where e = RtoDPQ.e, default value: 5

WarningSim

if WarningSim==TRUE, print/show issue a warning as to the precision of d-,p-,q-slots when these are obtained by RtoDPQ resp. RtoDPQ.d, default value: TRUE

WarningArith

if WarningArith==TRUE, print/show issue a warning as to the interpretation of arithmetics operating on distributions, when the corresponding distribution to be plotted/shown is obtained by such an operation; keep in mind that arithmetics in fact operate on random variables distributed according to the given distributions and not on corresponding cdf's or densities; default value: TRUE

withSweave

is code run in Sweave (then no new graphic devices are opened), default value: FALSE

withgaps

controls whether in the return value of arithmetic operations the slot gaps of an the AbscontDistribution part is filled automatically based on empirical evaluations via setgaps —default TRUE

simplifyD

controls whether in the return value of arithmetic operations there is a call to simplifyD or not —default TRUE

use.generalized.inverse.by.default

logical; decides whether by default (i.e., if argument generalized of solve is not explicitely set), solve is to use generalized inverses if the original solve-method from package base fails; if the option is FALSE, in case of failure, and unless argument generalized is not explicitely set to TRUE, solve will throw an error as is the base-method behavior. The default value is TRUE.

DistrCollapse.Unique.Warn

controls whether there is a warning whenever collapsing occurs or when two points are collapsed by a call to unique() (default behaviour if DistrCollapse is FALSE); —default FALSE

warn.makeDNew

controls whether a warning is issued once in internal utility .makeDNew standard integration with integrate throws an error—default TRUE

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

options, getOption

Examples

distroptions("RtoDPQ.e") # returns the value of RtoDPQ.e, by default = 5
currentDistrOptions <- distroptions()
distroptions(RtoDPQ.e = 6)
distroptions("RtoDPQ.e") 
getdistrOption("RtoDPQ.e") 
distroptions(c("WarningSim","WarningArith"))   
getdistrOption("WarningSim")   
distroptions("WarningSim" = FALSE)   
         # switches off warnings as to (In)accuracy due to simulations
distroptions("WarningArith" = FALSE) 
         # switches off warnings as to arithmetics
distroptions(currentDistrOptions)

Description

Generates an object of class "DistrSymmList".

Usage

DistrSymmList(...)

Arguments

Value

Object of class "DistrSymmList"

Author(s)

Matthias Kohl [email protected]

See Also

DistrSymmList-class

Examples

DistrSymmList(NoSymmetry(), SphericalSymmetry(SymmCenter = 1), 
              EllipticalSymmetry(SymmCenter = 2))

## The function is currently defined as
function (...){
    new("DistrSymmList", list(...))
}

Description

Create a list of symmetries for a list of distributions

Objects from the Class

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

Slots

.Data

Object of class "list". A list of objects of class "DistributionSymmetry".

Extends

Class "list", from data part.
Class "vector", by class "list".

Author(s)

Matthias Kohl [email protected]

See Also

DistributionSymmetry-class

Examples

new("DistrSymmList", list(NoSymmetry(), SphericalSymmetry(SymmCenter = 1), 
                          EllipticalSymmetry(SymmCenter = 2)))

Description

Generates an object of class "EllipticalSymmetry".

Usage

EllipticalSymmetry(SymmCenter = 0)

Arguments

Value

Object of class "EllipticalSymmetry"

Author(s)

Matthias Kohl [email protected]

See Also

EllipticalSymmetry-class, DistributionSymmetry-class

Examples

EllipticalSymmetry()

## The function is currently defined as
function(SymmCenter = 0){ 
    new("EllipticalSymmetry", SymmCenter = SymmCenter) 
}

Description

Class for elliptically symmetric distributions.

Objects from the Class

Objects can be created by calls of the form new("EllipticalSymmetry"). More frequently they are created via the generating function EllipticalSymmetry. Elliptical symmetry for instance leads to a simplification for the computation of optimally robust influence curves.

Slots

type

Object of class "character": contains “elliptical symmetric distribution”

SymmCenter

Object of class "numeric": center of symmetry

Extends

Class "DistributionSymmetry", directly.
Class "Symmetry", by class "DistributionSymmetry".

Author(s)

Matthias Kohl [email protected]

See Also

EllipticalSymmetry, DistributionSymmetry-class

Examples

new("EllipticalSymmetry")

Description

Generates an object of class "DiscreteDistribution"

Usage

EmpiricalDistribution(data, .withArith=FALSE, .withSim=FALSE, 
                        .lowerExact = TRUE, .logExact = FALSE,
                        .DistrCollapse = getdistrOption("DistrCollapse"),
                        .DistrCollapse.Unique.Warn = 
                             getdistrOption("DistrCollapse.Unique.Warn"),
                        .DistrResolution = getdistrOption("DistrResolution"),
                        Symmetry = NoSymmetry())

Arguments

Details

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

Typical usage is

    EmpiricalDistribution(data)
  

Value

Object of class "DiscreteDistribution"

Author(s)

Matthias Kohl [email protected]

See Also

DiscreteDistribution DiscreteDistribution-class

Examples

x <- rnorm(20)
D1 <- EmpiricalDistribution(data = x)
D1

plot(D1)

Description

The distribution-classes contain a slot where the sample space is stored. One typical sample space is the Euclidean Space in dimension k.

Usage

EuclideanSpace(dimension = 1)

Arguments

Objects from the Class

Objects could theoretically be created by calls of the form new("EuclideanSpace", dimension, name). Usually an object of this class is not needed on its own. EuclideanSpace is the mother-class of the class Reals, which is generated automatically when a univariate absolutly continuous distribution is instantiated.

Slots

dimension

Object of class "numeric": the dimension of the space, by default = 1

name

Object of class "character": the name of the space, by default = "Euclidean Space"

Extends

Class "rSpace", directly.

Methods

initialize

signature(.Object = "EuclideanSpace"): initialize method

liesIn

signature(object = "EuclideanSpace", x = "numeric"): Does a particular vector lie in this space or not?

dimension

signature(object = "EuclideanSpace"): returns the dimension of the space

dimension<-

signature(object = "EuclideanSpace"): modifies the dimension of the space

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

rSpace-class Reals-class Distribution-class liesIn-methods

Examples

E <- EuclideanSpace(dimension = 2) 
dimension(E) # The dimension of this space is 2.
dimension(E) <- 3 # The dimension of this space is now 3.
liesIn(E,c(0,0,0)) # TRUE
liesIn(E,c(0,0)) # FALSE

Description

The exponential distribution with rate $\lambda$ has density

$f(x) = \lambda {e}^{- \lambda x}$

for $x \ge 0$.

C.f. rexp

Objects from the Class

Objects can be created by calls of the form Exp(rate). This object is an exponential distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "ExpParameter": the parameter of this distribution (rate), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rexp)

d

Object of class "function": density function (calls function dexp)

p

Object of class "function": cumulative function (calls function pexp)

q

Object of class "function": inverse of the cumulative function (calls function qexp)

.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 "ExpOrGammaOrChisq", directly.
Class "AbscontDistribution", by class "ExpOrGammaOrChisq".
Class "UnivariateDistribution", by class "AbscontDistribution". Class "Distribution", by class "AbscontDistribution".

Is-Relations

By means of setIs, R “knows” that a distribution object obj of class "Exp" also is a Gamma distribution with parameters shape = 1, scale = 1/rate(obj) and a Weibull distribution with parameters shape = 1, scale = 1/rate(obj)

Methods

initialize

signature(.Object = "Exp"): initialize method

rate

signature(object = "Exp"): returns the slot rate of the parameter of the distribution

rate<-

signature(object = "Exp"): modifies the slot rate of the parameter of the distribution

*

signature(e1 = "Exp", e2 = "numeric"): For the exponential distribution we use its closedness under positive scaling transformations.

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

ExpParameter-class AbscontDistribution-class Reals-class rexp

Examples

E <- Exp(rate = 1) # E is a exp distribution with rate = 1.
r(E)(1) # one random number generated from this distribution, e.g. 0.4190765
d(E)(1) # Density of this distribution is 0.3678794 for x = 1.
p(E)(1) # Probability that x < 1 is 0.6321206.
q(E)(.1) # Probability that x < 0.1053605 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
rate(E) # rate of this distribution is 1.
rate(E) <- 2 # rate of this distribution is now 2.
is(E, "Gammad") # yes
as(E,"Gammad")
is(E, "Weibull") 
E+E+E ###  a Gammad -distribution
2*E+Gammad(scale=1)

Description

The parameter of an exponential distribution, used by Exp-class and DExp-class

Objects from the Class

Objects can be created by calls of the form new("ExpParameter", rate). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Exp is instantiated.

Slots

rate

Object of class "numeric": the rate of an exponential distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "ExpParameter"): initialize method

rate

signature(object = "ExpParameter"): returns the slot rate of the parameter of the distribution

rate<-

signature(object = "ExpParameter"): modifies the slot rate of the parameter of the distribution

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

Exp-class DExp-class Parameter-class

Examples

W <- new("ExpParameter", rate = 1)
rate(W) # rate of this distribution is 1.
rate(W) <- 2 # rate of this distribution is now 2.

Description

The F distribution with df1 = $n_1$, by default = 1, and df2 = $n_2$, by default = 1, degrees of freedom has density

$d(x) = \frac{\Gamma(n_1/2 + n_2/2)}{\Gamma(n_1/2)\Gamma(n_2/2)} \left(\frac{n_1}{n_2}\right)^{n_1/2} x^{n_1/2 -1} \left(1 + \frac{n_1 x}{n_2}\right)^{-(n_1 + n_2) / 2}$

for $x > 0$.

C.f. rf

Objects from the Class

Objects can be created by calls of the form Fd(df1, df2). This object is a F distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "FParameter": the parameter of this distribution (df1 and df2), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rf)

d

Object of class "function": density function (calls function df)

p

Object of class "function": cumulative function (calls function pf)

q

Object of class "function": inverse of the cumulative function (calls function qf)

.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 "AbscontDistribution", directly.
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "AbscontDistribution".

Methods

initialize

signature(.Object = "Fd"): initialize method

df1

signature(object = "Fd"): returns the slot df1 of the parameter of the distribution

df1<-

signature(object = "Fd"): modifies the slot df1 of the parameter of the distribution

df2

signature(object = "Fd"): returns the slot df2 of the parameter of the distribution

df2<-

signature(object = "Fd"): modifies the slot df2 of the parameter of the distribution

Ad hoc methods

• An ad hoc method is provided for slot d if ncp!=0.

• For R Version <2.3.0 ad hoc methods are provided for slots q, r if ncp!=0; for R Version >=2.3.0 the methods from package stats are used.

Note

It is the distribution of the ratio of the mean squares of n1 and n2 independent standard normals, and hence of the ratio of two independent chi-squared variates each divided by its degrees of freedom. Since the ratio of a normal and the root mean-square of m independent normals has a Student's $t_m$ distribution, the square of a $t_m$ variate has a F distribution on 1 and m degrees of freedom.

The non-central F distribution is again the ratio of mean squares of independent normals of unit variance, but those in the numerator are allowed to have non-zero means and ncp is the sum of squares of the means.

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

FParameter-class AbscontDistribution-class Reals-class rf

Examples

F <- Fd(df1 = 1, df2 = 1) # F is a F distribution with df=1 and df2=1.
r(F)(1) # one random number generated from this distribution, e.g. 29.37863
d(F)(1) # Density of this distribution is 0.1591549 for x=1 .
p(F)(1) # Probability that x<1 is 0.5.
q(F)(.1) # Probability that x<0.02508563 is 0.1.
## in RStudio or Jupyter IRKernel, use q.l(.)(.) instead of q(.)(.)
df1(F) # df1 of this distribution is 1.
df1(F) <- 2 # df1 of this distribution is now 2.
Fn <- Fd(df1 = 1, df2 = 1, ncp = 0.5) 
  # Fn is a F distribution with df=1, df2=1 and ncp =0.5.
d(Fn)(1) ## from R 2.3.0 on ncp no longer ignored...

Description

flattens a list of Lebesgue decomposed distributions endowed with weights to give one Lebesgue decomposed distribution

Usage

flat.LCD(..., mixCoeff = NULL, withgaps = getdistrOption("withgaps"))

Arguments

Details

flat.LCD flattens a list of Lebesgue decomposed distributions given through ..., i.e., it takes all list elements and mixing coefficients and builds up the mixed distribution (forgetting about the components); the result will be one distribution of class UnivarLebDecDistribution. If mixCoeff is missing, all list elements are equally weighted. It is used internally in our methods for "*", "/", "^" (see operators-methods), Minimum, and convpow, as well in method flat.mix.

Value

flat.LCD returns an object of class UnivarLebDecDistribution.

Author(s)

Peter Ruckdeschel [email protected]

See Also

UnivarLebDecDistribution-class, operators-methods

Examples

D1 <- as(Norm(),"UnivarLebDecDistribution")
D2 <- as(Pois(1),"UnivarLebDecDistribution")
D3 <- as(Binom(1,.4),"UnivarLebDecDistribution")
flat.LCD(D1,D2,D3, mixCoeff = c(0.4,0.5,0.1))

Description

function to do get empirical density, cumulative distribution and quantile function from random numbers

Usage

flat.mix(object)

Arguments

Details

flat.mix generates $10^e$ random numbers, by default

$e = RtoDPQ.e$

. Replicates are assumed to be part of the discrete part, unique values to be part of the a.c. part of the distribution. For the replicated ones, we generate a discrete distribution by a call to DiscreteDistribution. The a.c. density is formed on the basis of $n$ points using approxfun and density (applied to the unique values), by default

$n = DefaultNrGridPoints$

. The cumulative distribution function is based on all random variables, and, as well as the quantile function, is also created on the basis of $n$ points using approxfun and ecdf. Of course, the results are usually not exact as they rely on random numbers.

Value

flat.mix returns an object of class UnivarLebDecDistribution.

Note

Use RtoDPQ for absolutely continuous and RtoDPQ.d for discrete distributions.

Author(s)

Peter Ruckdeschel [email protected]

See Also

UnivariateDistribution-class, density, approxfun, ecdf

Examples

D1 <- Norm()
D2 <- Pois(1)
D3 <- Binom(1,.4)
D4 <- UnivarMixingDistribution(D1,D2,D3, mixCoeff = c(0.4,0.5,0.1), 
      withSimplify = FALSE)
D <- UnivarMixingDistribution(D1,D4,D1,D2, mixCoeff = c(0.4,0.3,0.1,0.2), 
      withSimplify = FALSE)
D
D0<-flat.mix(D)
D0
plot(D0)

Description

The parameter of a F distribution, used by Fd-class

Objects from the Class

Objects can be created by calls of the form new("FParameter", df1, df2, ncp). Usually an object of this class is not needed on its own, it is generated automatically when an object of the class Fd is instantiated.

Slots

df1

Object of class "numeric": the degrees of freedom of the nominator of an F distribution

df2

Object of class "numeric": the degrees of freedom of the denominator of an F distribution

ncp

Object of class "numeric": the noncentrality parameter of an F distribution

name

Object of class "character": a name / comment for the parameters

Extends

Class "Parameter", directly.

Methods

initialize

signature(.Object = "FParameter"): initialize method

df1

signature(object = "FParameter"): returns the slot df1 of the parameter of the distribution

df1<-

signature(object = "FParameter"): modifies the slot df1 of the parameter of the distribution

df2

signature(object = "FParameter"): returns the slot df2 of the parameter of the distribution

df2<-

signature(object = "FParameter"): modifies the slot df2 of the parameter of the distribution

ncp

signature(object = "FParameter"): returns the slot ncp of the parameter of the distribution

ncp<-

signature(object = "FParameter"): modifies the slot ncp of the parameter of the distribution

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]

See Also

Fd-class Parameter-class

Examples

W <- new("FParameter", df1 = 1, df2 = 1, ncp = 0)
df2(W) # df2 of this distribution is 1.
df2(W) <- 2 # df2 of this distribution is now 2.

Description

The Gammad distribution with parameters shape $=\alpha$, by default = 1, and scale $=\sigma$, by default = 1, has density

$d(x)= \frac{1}{{\sigma}^{\alpha}\Gamma(\alpha)} {x}^{\alpha-1} e^{-x/\sigma}$

for $x > 0$, $\alpha > 0$ and $\sigma > 0$. The mean and variance are $E(X) = \alpha\sigma$ and $Var(X) = \alpha\sigma^2$. C.f. rgamma

Objects from the Class

Objects can be created by calls of the form Gammad(scale, shape). This object is a gamma distribution.

Slots

img

Object of class "Reals": The space of the image of this distribution has got dimension 1 and the name "Real Space".

param

Object of class "GammaParameter": the parameter of this distribution (scale and shape), declared at its instantiation

r

Object of class "function": generates random numbers (calls function rgamma)

d

Object of class "function": density function (calls function dgamma)

p

Object of class "function": cumulative function (calls function pgamma)

q

Object of class "function": inverse of the cumulative function (calls function qgamma)

.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 "ExpOrGammaOrChisq", directly.
Class "AbscontDistribution", by class "ExpOrGammaOrChisq".
Class "UnivariateDistribution", by class "AbscontDistribution".
Class "Distribution", by class "UnivariateDistribution".

Methods

initialize

signature(.Object = "Gammad"): initialize method

scale

signature(object = "Gammad"): returns the slot scale of the parameter of the distribution

scale<-

signature(object = "Gammad"): modifies the slot scale of the parameter of the distribution

shape

signature(object = "Gammad"): returns the slot shape of the parameter of the distribution

shape<-

signature(object = "Gammad"): modifies the slot shape of the parameter of the distribution

+

signature(e1 = "Gammad", e2 = "Gammad"): For the Gamma distribution we use its closedness under convolutions.

*

signature(e1 = "Gammad", e2 = "numeric"): For the Gamma distribution we use its closedness under positive scaling transformations.

Author(s)

Thomas Stabla [email protected],
Florian Camphausen [email protected],
Peter Ruckdeschel [email protected],
Matthias Kohl [email protected]