Package 'GeneralizedHyperbolic'

Soil Electrical Conductivity

Description

Electrical conductivity of soil paste extracts from the Lower Arkansas River Valley, at sites upstream and downstream of the John Martin Reservoir.

Usage

data(ArkansasRiver)

Format

The format is: List of 2 $ upstream : num [1:823] 2.37 3.53 3.06 3.35 3.07 ... $ downstream: num [1:435] 8.75 6.59 5.09 6.03 5.64 ...

Details

Electrical conductivity is a measure of soil water salinity.

Source

This data set was supplied by Eric Morway ([email protected]).

References

Eric D. Morway and Timothy K. Gates (2011) Regional assessment of soil water salinity across an extensively irrigated river valley. Journal of Irrigation and Drainage Engineering, doi:10.1061/(ASCE)IR.1943-4774.0000411

Examples

data(ArkansasRiver)
lapply(ArkansasRiver, summary)
upstream <- ArkansasRiver[[1]]
downstream <- ArkansasRiver[[2]]
## Fit normal inverse Gaussian
## Hyperbolic can also be fitted but fit is not as good
fitUpstream <- nigFit(upstream)
summary(fitUpstream)
par(mfrow = c(2,2))
plot(fitUpstream)
fitDownstream <- nigFit(downstream)
summary(fitDownstream)
plot(fitDownstream)
par(mfrow = c(1,1))
## Combined plot to compare
## Reproduces Figure 3 from Morway and Gates (2011)
hist(upstream, col = "grey", xlab = "", ylab = "", cex.axis = 1.25,
     main = "", breaks = seq(0,20, by = 1), xlim = c(0,15), las = 1,
     ylim = c(0,0.5), freq = FALSE)
param <- coef(fitUpstream)
nigDens <- function(x) dnig(x, param = param)
curve(nigDens, 0, 15, n = 201, add = TRUE,
      ylab = NULL, col = "red", lty = 1, lwd = 1.7)

hist(downstream, add = TRUE, col = "black", angle = 45, density = 15,
     breaks = seq(0,20, by = 1), freq = FALSE)
param <- coef(fitDownstream)
nigDens <- function(x) dnig(x, param = param)
curve(nigDens, 0, 15, n = 201, add = TRUE,
      ylab = NULL, col = "red", lty = 1, lwd = 1.7)

mtext(expression(EC[e]), side = 1, line = 3, cex = 1.25)
mtext("Frequency", side = 2, line = 3, cex = 1.25)
legend(x = 7.5, y = 0.250, c("Upstream Region","Downstream Region"),
       col = c("black","black"), density = c(NA,25),
       fill = c("grey","black"), angle = c(NA,45),
       cex = 1.25, bty = "n", xpd = TRUE)

---

Functions for Calculating Moments

Description

Functions used to calculate the mean, variance, skewness and kurtosis of a hyperbolic distribution. Not expected to be called directly by users.

Usage

RLambda(zeta, lambda = 1)
SLambda(zeta, lambda = 1)
MLambda(zeta, lambda = 1)
WLambda1(zeta, lambda = 1)
WLambda2(zeta, lambda = 1)
WLambda3(zeta, lambda = 1)
WLambda4(zeta, lambda = 1)
gammaLambda1(hyperbPi, zeta, lambda = 1)
gammaLambda1(hyperbPi, zeta, lambda = 1)

Arguments

hyperbPi	Value of the parameter $\pi$ of the hyperbolic distribution.
zeta	Value of the parameter $\zeta$ of the hyperbolic distribution.
lambda	Parameter related to order of Bessel functions.

Value

The functions RLambda and SLambda are used in the calculation of the mean and variance. They are functions of the Bessel functions of the third kind, implemented in R as besselK. The other functions are used in calculation of higher moments. See Barndorff-Nielsen, O. and Blæsild, P. (1981) for details of the calculations.

The parameterization of the hyperbolic distribution used for this and other components of the HyperbolicDist package is the $(\pi,\zeta)$ one. See hyperbChangePars to transfer between parameterizations.

Author(s)

David Scott [email protected], Richard Trendall, Thomas Tran

References

Barndorff-Nielsen, O. and Blæsild, P (1981). Hyperbolic distributions and ramifications: contributions to theory and application. In Statistical Distributions in Scientific Work, eds., Taillie, C., Patil, G. P., and Baldessari, B. A., Vol. 4, pp. 19–44. Dordrecht: Reidel.

Barndorff-Nielsen, O. and Blæsild, P (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700–707. New York: Wiley.

See Also

dhyperb, hyperbMean,hyperbChangePars, besselK

---

Generalized Inverse Gaussian Distribution

Description

Density function, cumulative distribution function, quantile function and random number generation for the generalized inverse Gaussian distribution with parameter vector param. Utility routines are included for the derivative of the density function and to find suitable break points for use in determining the distribution function.

Usage

dgig(x, chi = 1, psi = 1, lambda = 1,
     param = c(chi, psi, lambda), KOmega = NULL)
pgig(q, chi = 1, psi = 1, lambda = 1,
     param = c(chi,psi,lambda), lower.tail = TRUE,
     ibfTol = .Machine$double.eps^(0.85), nmax = 200)
qgig(p, chi = 1, psi = 1, lambda = 1,
     param = c(chi, psi, lambda), lower.tail = TRUE,
     method = c("spline", "integrate"),
     nInterpol = 501, uniTol = 10^(-7),
     ibfTol = .Machine$double.eps^(0.85), nmax =200, ...)
rgig(n, chi = 1, psi = 1, lambda = 1,
     param = c(chi, psi, lambda))
rgig1(n, chi = 1, psi = 1, param = c(chi, psi))
ddgig(x, chi = 1, psi = 1, lambda = 1,
      param = c(chi, psi, lambda), KOmega = NULL)

Arguments

x, q	Vector of quantiles.
p	Vector of probabilities.
n	Number of observations to be generated.
chi	A shape parameter that by default holds a value of 1.
psi	Another shape parameter that is set to 1 by default.
lambda	Shape parameter of the GIG distribution. Common to all forms of parameterization. By default this is set to 1.
param	Parameter vector taking the form c(chi, psi, lambda) for rgig, or c(chi, psi) for rgig1.
method	Character. If "spline" quantiles are found from a spline approximation to the distribution function. If "integrate", the distribution function used is always obtained by integration.
lower.tail	Logical. If TRUE, probabilities are $P(X\leq x)$, otherwise as $P(X>x)$.
KOmega	Sets the value of the Bessel function in the density or derivative of the density. See Details.
ibfTol	Value of tolerance to be passed to incompleteBesselK by pgig.
nmax	Value of maximum order of the approximating series to be passed to incompleteBesselK by pgig.
nInterpol	The number of points used in qgig for cubic spline interpolation (see splinefun) of the distribution function.
uniTol	Value of tol in calls to uniroot. See uniroot.
...	Passes arguments to uniroot. See Details.

Details

The generalized inverse Gaussian distribution has density

$f(x)=\frac{(\psi/\chi)^{\frac{\lambda}{2}}} {2K_\lambda(\sqrt{\psi\chi})}x^{\lambda-1} e^{-\frac{1}{2}\left(\chi x^{-1}+\psi x\right)}$

for $x>0$, where $K_\lambda()$ is the modified Bessel function of the third kind with order $\lambda$.

The generalized inverse Gaussian distribution is investigated in detail in Jörgensen (1982).

Use gigChangePars to convert from the $(\delta,\gamma)$, $(\alpha,\beta)$, or $(\omega,\eta)$ parameterizations to the $(\chi,\psi)$, parameterization used above.

pgig calls the function incompleteBesselK from the package DistributionUtils to integrate the density function dgig. This can be expected to be accurate to about 13 decimal places on a 32-bit computer, often more accurate. The algorithm used is due to Slavinsky and Safouhi (2010).

Calculation of quantiles using qgig permits the use of two different methods. Both methods use uniroot to find the value of $x$ for which a given $q$ is equal $F(x)$ where $F$ denotes the cumulative distribution function. The difference is in how the numerical approximation to $F$ is obtained. The obvious and more accurate method is to calculate the value of $F(x)$ whenever it is required using a call to pghyp. This is what is done if the method is specified as "integrate". It is clear that the time required for this approach is roughly linear in the number of quantiles being calculated. A Q-Q plot of a large data set will clearly take some time. The alternative (and default) method is that for the major part of the distribution a spline approximation to $F(x)$ is calculated and quantiles found using uniroot with this approximation. For extreme values (for which the tail probability is less than $10^{-7}$), the integration method is still used even when the method specifed is "spline".

If accurate probabilities or quantiles are required, tolerances (intTol and uniTol) should be set to small values, say $10^{-10}$ or $10^{-12}$ with method = "integrate". Generally then accuracy might be expected to be at least $10^{-9}$. If the default values of the functions are used, accuracy can only be expected to be around $10^{-4}$. Note that on 32-bit systems .Machine$double.eps^0.25 = 0.0001220703 is a typical value.

Generalized inverse Gaussian observations are obtained via the algorithm of Dagpunar (1989).

Value

dgig gives the density, pgig gives the distribution function, qgig gives the quantile function, and rgig generates random variates. rgig1 generates random variates in the special case where $\lambda = 1$.

ddgig gives the derivative of dgig.

Author(s)

David Scott [email protected], Richard Trendall, and Melanie Luen.

References

Dagpunar, J.S. (1989). An easily implemented generalised inverse Gaussian generator. Commun. Statist. -Simula., 18, 703–710.

Jörgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics, Vol. 9, Springer-Verlag, New York.

Slevinsky, Richard M., and Safouhi, Hassan (2010) A recursive algorithm for the G transformation and accurate computation of incomplete Bessel functions. Appl. Numer. Math., In press.

See Also

safeIntegrate, integrate for its shortfalls, splinefun, uniroot and gigChangePars for changing parameters to the $(\chi,\psi)$ parameterization, dghyp for the generalized hyperbolic distribution.

Examples

param <- c(2, 3, 1)
gigRange <- gigCalcRange(param = param, tol = 10^(-3))
par(mfrow = c(1, 2))
curve(dgig(x, param = param), from = gigRange[1], to = gigRange[2],
      n = 1000)
title("Density of the\n Generalized Inverse Gaussian")
curve(pgig(x, param = param), from = gigRange[1], to = gigRange[2],
      n = 1000)
title("Distribution Function of the\n Generalized Inverse Gaussian")
dataVector <- rgig(500, param = param)
curve(dgig(x, param = param), range(dataVector)[1], range(dataVector)[2],
      n = 500)
hist(dataVector, freq = FALSE, add = TRUE)
title("Density and Histogram\n of the Generalized Inverse Gaussian")
DistributionUtils::logHist(dataVector, main =
        "Log-Density and Log-Histogram\n of the Generalized Inverse Gaussian")
curve(log(dgig(x, param = param)), add = TRUE,
      range(dataVector)[1], range(dataVector)[2], n = 500)
par(mfrow = c(2, 1))
curve(dgig(x, param = param), from = gigRange[1], to = gigRange[2],
      n = 1000)
title("Density of the\n Generalized Inverse Gaussian")
curve(ddgig(x, param = param), from = gigRange[1], to = gigRange[2],
      n = 1000)
title("Derivative of the Density\n of the Generalized Inverse Gaussian")

---

The Package ‘GeneralizedHyperbolic’: Summary Information

Description

This package provides a collection of functions for working with the generalized hyperbolic and related distributions.

For the hyperbolic distribution functions are provided for the density function, distribution function, quantiles, random number generation and fitting the hyperbolic distribution to data (hyperbFit). The function hyperbChangePars will interchange parameter values between different parameterizations. The mean, variance, skewness, kurtosis and mode of a given hyperbolic distribution are given by hyperbMean, hyperbVar, hyperbSkew, hyperbKurt, and hyperbMode respectively. For assessing the fit of the hyperbolic distribution to a set of data, the log-histogram is useful. See logHist from package DistributionUtils. Q-Q and P-P plots are also provided for assessing the fit of a hyperbolic distribution. A Cramér-von~Mises test of the goodness of fit of data to a hyperbolic distribution is given by hyperbCvMTest. S3 print, plot and summary methods are provided for the output of hyperbFit.

For the generalized hyperbolic distribution functions are provided for the density function, distribution function, quantiles, and for random number generation. The function ghypChangePars will interchange parameter values between different parameterizations. The mean, variance, and mode of a given generalized hyperbolic distribution are given by ghypMean, ghypVar, ghypSkew, ghypKurt, and ghypMode respectively. Q-Q and P-P plots are also provided for assessing the fit of a generalized hyperbolic distribution.

For the generalized inverse Gaussian distribution functions are provided for the density function, distribution function, quantiles, and for random number generation. The function gigChangePars will interchange parameter values between different parameterizations. The mean, variance, skewness, kurtosis and mode of a given generalized inverse Gaussian distribution are given by gigMean, gigVar, gigSkew, gigKurt, and gigMode respectively. Q-Q and P-P plots are also provided for assessing the fit of a generalized inverse Gaussian distribution.

For the skew-Laplace distribution functions are provided for the density function, distribution function, quantiles, and for random number generation. Q-Q and P-P plots are also provided for assessing the fit of a skew-Laplace distribution.

Acknowledgements

A number of students have worked on the package: Ai-Wei Lee, Jennifer Tso, Richard Trendall, Thomas Tran, Simon Potter and David Cusack.

Thanks to Ross Ihaka and Paul Murrell for their willingness to answer my questions, and to all the core group for the development of R.

Special thanks also to Diethelm Würtz without whose advice, this package would be far inferior.

LICENCE

This package and its documentation are usable under the terms of the "GNU General Public License", a copy of which is distributed with the package.

Author(s)

David Scott [email protected]

References

Barndorff-Nielsen, O. (1977) Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond., A353, 401–419.

Barndorff-Nielsen, O. and Blæsild, P (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700–707. New York: Wiley.

Fieller, N. J., Flenley, E. C. and Olbricht, W. (1992) Statistics of particle size data. Appl. Statist., 41, 127–146.

Jörgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics, Vol. 9, Springer-Verlag, New York.

Prause, K. (1999) The generalized hyperbolic models: Estimation, financial derivatives and risk measurement. PhD Thesis, Mathematics Faculty, University of Freiburg.

---

Generalized Hyperbolic Distribution

Description

Density function, distribution function, quantiles and random number generation for the generalized hyperbolic distribution, with parameters $\alpha$ (tail), $\beta$ (skewness), $\delta$ (peakness), $\mu$ (location) and $\lambda$ (shape).

Usage

dghyp(x, mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
      param = c(mu, delta, alpha, beta, lambda))
pghyp(q, mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
      param = c(mu, delta, alpha, beta, lambda),
      lower.tail = TRUE, subdivisions = 100,
      intTol = .Machine$double.eps^0.25, valueOnly = TRUE, ...)
qghyp(p, mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
      param = c(mu, delta, alpha, beta, lambda),
      lower.tail = TRUE, method = c("spline","integrate"),
      nInterpol = 501, uniTol = .Machine$double.eps^0.25,
      subdivisions = 100, intTol = uniTol, ...)
rghyp(n, mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
      param = c(mu, delta, alpha, beta, lambda))
ddghyp(x, mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
       param = c(mu, delta, alpha, beta, lambda))

Arguments

x, q	Vector of quantiles.
p	Vector of probabilities.
n	Number of random variates to be generated.
mu	Location parameter $\mu$, default is 0.
delta	Scale parameter $\delta$, default is 1.
alpha	Tail parameter $\alpha$, default is 1.
beta	Skewness parameter $\beta$, default is 0.
lambda	Shape parameter $\lambda$, default is 1.
param	Specifying the parameters as a vector of the form c(mu,delta,alpha,beta,lambda).
method	Character. If "spline" quantiles are found from a spline approximation to the distribution function. If "integrate", the distribution function used is always obtained by integration.
lower.tail	Logical. If TRUE, probabilities are $P(X\leq x)$, otherwise they are $P(X>x)$.
subdivisions	The maximum number of subdivisions used to integrate the density and determine the accuracy of the distribution function calculation.
intTol	Value of rel.tol and hence abs.tol in calls to integrate. See integrate.
valueOnly	Logical. If valueOnly = TRUE calls to pghyp only return the value obtained for the integral. If valueOnly = FALSE an estimate of the accuracy of the numerical integration is also returned.
nInterpol	Number of points used in qghyp for cubic spline interpolation of the distribution function.
uniTol	Value of tol in calls to uniroot. See uniroot.
...	Passes additional arguments to integrate in pghyp and qghyp, and to uniroot in qghyp.

Details

Users may either specify the values of the parameters individually or as a vector. If both forms are specified, then the values specified by the vector param will overwrite the other ones. In addition the parameter values are examined by calling the function ghypCheckPars to see if they are valid.

The density function is

$f(x)=c(\lambda,\alpha,\beta,\delta)\times \frac{K_{\lambda-1/2}(\alpha\sqrt{\delta^2+(x-\mu)^2})} {(\frac{\sqrt{\delta^2+(x-\mu)^2}}{\alpha})^{1/2-\lambda}} e^{\beta(x-\mu)}$

where $K_\nu()$ is the modified Bessel function of the third kind with order $\nu$, and

$c(\lambda,\alpha,\beta,\delta)= \frac{(\frac{\sqrt{\alpha^2-\beta^2}}{\delta})^\lambda} {\sqrt{2\pi}K_\lambda(\delta\sqrt{\alpha^2-\beta^2})}$

Use ghypChangePars to convert from the $(\rho, \zeta)$, $(\xi, \chi)$, $(\bar\alpha, \bar\beta)$, or $(\pi, \zeta)$ parameterizations to the $(\alpha, \beta)$ parameterization used above.

pghyp uses the function integrate to numerically integrate the density function. The integration is from -Inf to x if x is to the left of the mode, and from x to Inf if x is to the right of the mode. The probability calculated this way is subtracted from 1 if required. Integration in this manner appears to make calculation of the quantile function more stable in extreme cases.

Calculation of quantiles using qghyp permits the use of two different methods. Both methods use uniroot to find the value of $x$ for which a given $q$ is equal $F(x)$ where $F$ denotes the cumulative distribution function. The difference is in how the numerical approximation to $F$ is obtained. The obvious and more accurate method is to calculate the value of $F(x)$ whenever it is required using a call to pghyp. This is what is done if the method is specified as "integrate". It is clear that the time required for this approach is roughly linear in the number of quantiles being calculated. A Q-Q plot of a large data set will clearly take some time. The alternative (and default) method is that for the major part of the distribution a spline approximation to $F(x)$ is calculated and quantiles found using uniroot with this approximation. For extreme values (for which the tail probability is less than $10^{-7}$), the integration method is still used even when the method specifed is "spline".

If accurate probabilities or quantiles are required, tolerances (intTol and uniTol) should be set to small values, say $10^{-10}$ or $10^{-12}$ with method = "integrate". Generally then accuracy might be expected to be at least $10^{-9}$. If the default values of the functions are used, accuracy can only be expected to be around $10^{-4}$. Note that on 32-bit systems .Machine$double.eps^0.25 = 0.0001220703 is a typical value.

Value

dghyp gives the density function, pghyp gives the distribution function, qghyp gives the quantile function and rghyp generates random variates.

An estimate of the accuracy of the approximation to the distribution function can be found by setting valueOnly = FALSE in the call to pghyp which returns a list with components value and error.

ddghyp gives the derivative of dghyp.

Author(s)

David Scott [email protected]

References

Barndorff-Nielsen, O., and Blæsild, P (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S., and Read, C. B., Vol. 3, pp. 700–707.-New York: Wiley.

Bibby, B. M., and Sörenson,M. (2003). Hyperbolic processes in finance. In Handbook of Heavy Tailed Distributions in Finance,ed., Rachev, S. T. pp. 212–248. Elsevier Science B.~V.

Dagpunar, J.S. (1989). An easily implemented generalised inverse Gaussian generator Commun. Statist.-Simula., 18, 703–710.

Prause, K. (1999) The generalized hyperbolic models: Estimation, financial derivatives and risk measurement. PhD Thesis, Mathematics Faculty, University of Freiburg.

See Also

dhyperb for the hyperbolic distribution, dgig for the generalized inverse Gaussian distribution, safeIntegrate, integrate for its shortfalls, also splinefun, uniroot and ghypChangePars for changing parameters to the $(\alpha,\beta)$ parameterization.

Examples

param <- c(0, 1, 3, 1, 1/2)
ghypRange <- ghypCalcRange(param = param, tol = 10^(-3))
par(mfrow = c(1, 2))

### curves of density and distribution
curve(dghyp(x, param = param), ghypRange[1], ghypRange[2], n = 1000)
title("Density of the \n Generalized  Hyperbolic Distribution")
curve(pghyp(x, param = param), ghypRange[1], ghypRange[2], n = 500)
title("Distribution Function of the \n Generalized Hyperbolic Distribution")

### curves of density and log density
par(mfrow = c(1, 2))
data <- rghyp(1000, param = param)
curve(dghyp(x, param = param), range(data)[1], range(data)[2],
      n = 1000, col = 2)
hist(data, freq = FALSE, add = TRUE)
title("Density and Histogram of the\n Generalized Hyperbolic Distribution")
DistributionUtils::logHist(data,
    main = "Log-Density and Log-Histogram of\n the Generalized Hyperbolic Distribution")
curve(log(dghyp(x, param = param)),
      range(data)[1], range(data)[2],
      n = 500, add = TRUE, col = 2)

### plots of density and derivative
par(mfrow = c(2, 1))
curve(dghyp(x, param = param), ghypRange[1], ghypRange[2], n = 1000)
title("Density of the\n Generalized  Hyperbolic Distribution")
curve(ddghyp(x, param = param), ghypRange[1], ghypRange[2], n = 1000)
title("Derivative of the Density of the\n Generalized Hyperbolic Distribution")

---

Generalized Hyperbolic Quantile-Quantile and Percent-Percent Plots

Description

qqghyp produces a generalized hyperbolic Q-Q plot of the values in y.

ppghyp produces a generalized hyperbolic P-P (percent-percent) or probability plot of the values in y.

Graphical parameters may be given as arguments to qqghyp, and ppghyp.

Usage

qqghyp(y, mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
       param = c(mu, delta, alpha, beta, lambda),
       main = "Generalized Hyperbolic Q-Q Plot",
       xlab = "Theoretical Quantiles",
       ylab = "Sample Quantiles",
       plot.it = TRUE, line = TRUE, ...)

ppghyp(y, mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
       param = c(mu, delta, alpha, beta, lambda),
       main = "Generalized Hyperbolic P-P Plot",
       xlab = "Uniform Quantiles",
       ylab = "Probability-integral-transformed Data",
       plot.it = TRUE, line = TRUE, ...)

Arguments

y	The data sample.
mu	$\mu$ is the location parameter. By default this is set to 0.
delta	$\delta$ is the scale parameter of the distribution. A default value of 1 has been set.
alpha	$\alpha$ is the tail parameter, with a default value of 1.
beta	$\beta$ is the skewness parameter, by default this is 0.
lambda	$\lambda$ is the shape parameter and dictates the shape that the distribution shall take. Default value is 1.
param	Parameters of the generalized hyperbolic distribution.
xlab, ylab, main	Plot labels.
plot.it	Logical. Should the result be plotted?
line	Add line through origin with unit slope.
...	Further graphical parameters.

Value

For qqghyp and ppghyp, a list with components:

x	The x coordinates of the points that are to be plotted.
y	The y coordinates of the points that are to be plotted.

References

Wilk, M. B. and Gnanadesikan, R. (1968) Probability plotting methods for the analysis of data. Biometrika. 55, 1–17.

See Also

ppoints, dghyp.

Examples

par(mfrow = c(1, 2))
y <- rghyp(200, param = c(2, 2, 2, 1, 2))
qqghyp(y, param = c(2, 2, 2, 1, 2), line = FALSE)
abline(0, 1, col = 2)
ppghyp(y, param = c(2, 2, 2, 1, 2))

---

Range of a Generalized Hyperbolic Distribution

Description

Given the parameter vector Theta of a generalized hyperbolic distribution, this function determines the range outside of which the density function is negligible, to a specified tolerance. The parameterization used is the $(\alpha, \beta)$ one (see dghyp). To use another parameterization, use ghypChangePars.

Usage

ghypCalcRange(mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
              param = c(mu, delta, alpha, beta, lambda),
              tol = 10^(-5), density = TRUE, ...)

Arguments

mu	$\mu$ is the location parameter. By default this is set to 0.
delta	$\delta$ is the scale parameter of the distribution. A default value of 1 has been set.
alpha	$\alpha$ is the tail parameter, with a default value of 1.
beta	$\beta$ is the skewness parameter, by default this is 0.
lambda	$\lambda$ is the shape parameter and dictates the shape that the distribution shall take. Default value is 1.
param	Value of parameter vector specifying the generalized hyperbolic distribution. This takes the form c(mu, delta,alpha, beta, lambda).
tol	Tolerance.
density	Logical. If TRUE, the bounds are for the density function. If FALSE, they should be for the probability distribution, but this has not yet been implemented.
...	Extra arguments for calls to uniroot.

Details

The particular generalized hyperbolic distribution being considered is specified by the value of the parameter value param.

If density = TRUE, the function gives a range, outside of which the density is less than the given tolerance. Useful for plotting the density. Also used in determining break points for the separate sections over which numerical integration is used to determine the distribution function. The points are found by using uniroot on the density function.

If density = FALSE, the function returns the message: "Distribution function bounds not yet implemented".

Value

A two-component vector giving the lower and upper ends of the range.

Author(s)

David Scott [email protected]

References

Barndorff-Nielsen, O. and Blæsild, P (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700–707. New York: Wiley.

See Also

dghyp, ghypChangePars

Examples

param <- c(0, 1, 5, 3, 1)
maxDens <- dghyp(ghypMode(param = param), param = param)
ghypRange <- ghypCalcRange(param = param, tol = 10^(-3) * maxDens)
ghypRange
curve(dghyp(x, param = param), ghypRange[1], ghypRange[2])
## Not run: ghypCalcRange(param = param, tol = 10^(-3), density = FALSE)

---

Change Parameterizations of the Generalized Hyperbolic Distribution

Description

This function interchanges between the following 5 parameterizations of the generalized hyperbolic distribution:

1. $\mu, \delta, \alpha, \beta, \lambda$

2. $\mu, \delta, \rho, \zeta, \lambda$

3. $\mu, \delta, \xi, \chi, \lambda$

4. $\mu, \delta, \bar\alpha, \bar\beta, \lambda$

5. $\mu, \delta, \pi, \zeta, \lambda$

The first four are the parameterizations given in Prause (1999). The final parameterization has proven useful in fitting.

Usage

ghypChangePars(from, to, param, noNames = FALSE)

Arguments

from	The set of parameters to change from.
to	The set of parameters to change to.
param	"from" parameter vector consisting of 5 numerical elements.
noNames	Logical. When TRUE, suppresses the parameter names in the output.

Details

In the 5 parameterizations, the following must be positive:

1. $\alpha, \delta$

2. $\zeta, \delta$

3. $\xi, \delta$

4. $\bar\alpha, \delta$

5. $\zeta, \delta$

Furthermore, note that in the first parameterization $\alpha$ must be greater than the absolute value of $\beta$; in the third parameterization, $\xi$ must be less than one, and the absolute value of $\chi$ must be less than $\xi$; and in the fourth parameterization, $\bar\alpha$ must be greater than the absolute value of $\bar\beta$.

Value

A numerical vector of length 5 representing param in the to parameterization.

Author(s)

David Scott [email protected], Jennifer Tso, Richard Trendall

References

Barndorff-Nielsen, O. and Blæsild, P. (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700–707. New York: Wiley.

Prause, K. (1999) The generalized hyperbolic models: Estimation, financial derivatives and risk measurement. PhD Thesis, Mathematics Faculty, University of Freiburg.

See Also

dghyp

Examples

param1 <- c(0, 3, 2, 1, 2)               # Parameterization 1
param2 <- ghypChangePars(1, 2, param1)   # Convert to parameterization 2
param2                                   # Parameterization 2
ghypChangePars(2, 1, param2)             # Back to parameterization 1

---

Check Parameters of the Generalized Hyperbolic Distribution

Description

Given a putative set of parameters for the generalized hyperbolic distribution, the functions checks if they are in the correct range, and if they correspond to the boundary cases.

Usage

ghypCheckPars(param)

Arguments

param

Numeric. Putative parameter values for a generalized hyperblic distribution.

Details

The vector param takes the form c(mu, delta, alpha, beta, lambda).

If alpha is negative, an error is returned.

If lambda is 0 then the absolute value of beta must be less than alpha and delta must be greater than zero. If either of these conditions are false, than a error is returned.

If lambda is greater than 0 the absolute value of beta must be less than alpha. delta must also be non-negative. When either one of these is not true, an error is returned.

If lambda is less than 0 then the absolute value of beta must be equal to alpha. delta must be greater than 0, if both conditions are not true, an error is returned.

Value

A list with components:

case	Either "" or "error".
errMessage	An appropriate error message if an error was found, the empty string "" otherwise.

Author(s)

David Scott [email protected]

References

Paolella, Marc S. (2007) Intermediate Probability: A Computational Approach, Chichester: Wiley

See Also

dghyp

Examples

ghypCheckPars(c(0, 2.5, -0.5, 1, 0))      # error
ghypCheckPars(c(0, 2.5, 0.5, 0, 0))       # normal
ghypCheckPars(c(0, 1, 1, -1, 0))          # error
ghypCheckPars(c(2, 0, 1, 0.5, 0))         # error
ghypCheckPars(c(0, 5, 2, 1.5, 0))         # normal
ghypCheckPars(c(0, -2.5, -0.5, 1, 1))     # error
ghypCheckPars(c(0, -1, 0.5, 1, 1))        # error
ghypCheckPars(c(0, 0, -0.5, -1, 1))       # error
ghypCheckPars(c(2, 0, 0.5, 0, -1))        # error
ghypCheckPars(c(2, 0, 1, 0.5, 1))         # skew laplace
ghypCheckPars(c(0, 1, 1, 1, -1))          # skew hyperbolic

---

Calculate Moments of the Generalized Hyperbolic Distribution

Description

Function to calculate raw moments, mu moments, central moments and moments about any other given location for the generalized hyperbolic distribution.

Usage

ghypMom(order, mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
        param = c(mu, delta, alpha, beta, lambda),
        momType = c("raw", "central", "mu"), about = 0)

Arguments

order	Numeric. The order of the moment to be calculated. Not permitted to be a vector. Must be a positive whole number except for moments about zero.
mu	$\mu$ is the location parameter. By default this is set to 0.
delta	$\delta$ is the scale parameter of the distribution. A default value of 1 has been set.
alpha	$\alpha$ is the tail parameter, with a default value of 1.
beta	$\beta$ is the skewness parameter, by default this is 0.
lambda	$\lambda$ is the shape parameter and dictates the shape that the distribution shall take. Default value is 1.
param	Numeric. The parameter vector specifying the generalized hyperbolic distribution. Of the form c(mu, delta, alpha, beta, lambda) (see dghyp).
momType	Common types of moments to be calculated, default is "raw". See Details.
about	Numeric. The point around which the moment is to be calculated.

Details

Checking whether order is a whole number is carried out using the function is.wholenumber.

momType can be either "raw" (moments about zero), "mu" (moments about mu), or "central" (moments about mean). If one of these moment types is specified, then there is no need to specify the about value. For moments about any other location, the about value must be specified. In the case that both momType and about are specified and contradicting, the function will always calculate the moments based on about rather than momType.

To calculate moments of the generalized hyperbolic distribution, the function firstly calculates mu moments by formula defined below and then transforms mu moments to central moments or raw moments or moments about any other locations as required by calling momChangeAbout.

The mu moments are obtained from the recursion formula given in Scott, Würtz and Tran (2011).

Value

The moment specified.

Author(s)

David Scott [email protected]

References

Scott, D. J., Würtz, D., Dong, C. and Tran, T. T. (2011) Moments of the generalized hyperbolic distribution. Comp. Statistics., 26, 459–476.

See Also

ghypChangePars and from package DistributionUtils: logHist, is.wholenumber, momChangeAbout, and momIntegrated.

Further, ghypMean, ghypVar, ghypSkew, ghypKurt.

Examples

param <- c(1, 2, 2, 1, 2)
mu <- param[1]
### mu moments
m1 <- ghypMean(param = param)
m1 - mu
ghypMom(1, param = param, momType = "mu")

## Comparison, using momIntegrated from pkg 'DistributionUtils':
momIntegrated <- DistributionUtils :: momIntegrated

momIntegrated("ghyp", order = 1, param = param, about = mu)
ghypMom(2, param = param, momType = "mu")
momIntegrated("ghyp", order = 2, param = param, about = mu)
ghypMom(10, param = param, momType = "mu")
momIntegrated("ghyp", order = 10, param = param, about = mu)

### raw moments
ghypMean(param = param)
ghypMom(1, param = param, momType = "raw")
momIntegrated("ghyp", order = 1, param = param, about = 0)
ghypMom(2, param = param, momType = "raw")
momIntegrated("ghyp", order = 2, param = param, about = 0)
ghypMom(10, param = param, momType = "raw")
momIntegrated("ghyp", order = 10, param = param, about = 0)

### central moments
ghypMom(1, param = param, momType = "central")
momIntegrated("ghyp", order = 1, param = param, about = m1)
ghypVar(param = param)
ghypMom(2, param = param, momType = "central")
momIntegrated("ghyp", order = 2, param = param, about = m1)
ghypMom(10, param = param, momType = "central")
momIntegrated("ghyp", order = 10, param = param, about = m1)

---

Parameter Sets for the Generalized Hyperbolic Distribution

Description

These objects store different parameter sets of the generalized hyperbolic distribution as matrices for testing or demonstration purposes.

The parameter sets ghypSmallShape and ghypLargeShape have a constant location parameter of $\mu$ = 0, and constant scale parameter $\delta$ = 1. In ghypSmallParam and ghypLargeParam the values of the location and scale parameters vary. In these parameter sets the location parameter $\mu$ = 0 takes values from {0, 1} and {-1, 0, 1, 2} respectively. For the scale parameter $\delta$, values are drawn from {1, 5} and {1, 2, 5, 10} respectively.

For the shape parameters $\alpha$ and $\beta$ the approach is more complex. The values for these shape parameters were chosen by choosing values of $\xi$ and $\chi$ which range over the shape triangle, then the function ghypChangePars was applied to convert them to the $\alpha, \beta$ parameterization. The resulting $\alpha, \beta$ values were then rounded to three decimal places. See the examples for the values of $\xi$ and $\chi$ for the large parameter sets.

The values of $\lambda$ are drawn from {-0.5, 0, 1} in ghypSmallShape and {-1, -0.5, 0, 0.5, 1, 2} in ghypLargeShape.

Usage

ghypSmallShape
  ghypLargeShape
  ghypSmallParam
  ghypLargeParam

Format

ghypSmallShape: a 22 by 5 matrix; ghypLargeShape: a 90 by 5 matrix; ghypSmallParam: a 84 by 5 matrix; ghypLargeParam: a 1440 by 5 matrix.

Author(s)

David Scott [email protected]

Examples

data(ghypParam)
plotShapeTriangle()
xis <- rep(c(0.1,0.3,0.5,0.7,0.9), 1:5)
chis <- c(0,-0.25,0.25,-0.45,0,0.45,-0.65,-0.3,0.3,0.65,
          -0.85,-0.4,0,0.4,0.85)
points(chis, xis, pch = 20, col = "red")


## Testing the accuracy of ghypMean
for (i in 1:nrow(ghypSmallParam)) {
  param <- ghypSmallParam[i, ]
  x <- rghyp(1000, param = param)
  sampleMean <- mean(x)
  funMean <- ghypMean(param = param)
  difference <- abs(sampleMean - funMean)
  print(difference)
}

---

Rescale a generalized hyperbolic distribution

Description

Given a specific mean and standard deviation will rescale any given generalized hyperbolic distribution to have the same shape but the specified mean and standard deviation. Can be used to standardize a generalized hyperbolic distribution to have mean zero and standard deviation one.

Usage

ghypScale(newMean, newSD,
          mu = 0, delta = 1, alpha = 1, beta = 0, lambda = 1,
          param = c(mu, delta, alpha, beta, lambda))

Arguments

newMean	Numeric. The required mean of the rescaled distribution.
newSD	Numeric. The required standard deviation of the rescaled distribution.
mu	Numeric. Location parameter $\mu$ of the starting distribution, default is0.
delta	Numeric. Scale parameter $\delta$ of the starting distribution, default is 1.
alpha	Numeric. Tail parameter $\alpha$ of the starting distribution, default is 1.
beta	Numeric. Skewness parameter $\beta$ of the starting distribution, default is 0.
lambda	Numeric. Shape parameter $\lambda$ of the starting distribution, default is 1.
param	Numeric. Specifying the parameters of the starting distribution as a vector of the form c(mu,delta,alpha,beta,lambda).

Value

A numerical vector of length 5 giving the value of the parameters in the rescaled generalized hyperbolic distribution in the usual ($\alpha, \beta$) parameterization.

Author(s)

David Scott [email protected]

Examples

param <- c(2,10,0.1,0.07,-0.5) # a normal inverse Gaussian
ghypMean(param = param)
ghypVar(param = param)
## convert to standardized parameters
(newParam <- ghypScale(0, 1, param = param))
ghypMean(param = newParam)
ghypVar(param = newParam)

## try some other mean and sd
(newParam <- ghypScale(1, 1, param = param))
ghypMean(param = newParam)
sqrt(ghypVar(param = newParam))
(newParam <- ghypScale(10, 2, param = param))
ghypMean(param = newParam)
sqrt(ghypVar(param = newParam))

---

Range of a Generalized Inverse Gaussian Distribution

Description

Given the parameter vector param of a generalized inverse Gaussian distribution, this function determines the range outside of which the density function is negligible, to a specified tolerance. The parameterization used is the $(\chi, \psi)$ one (see dgig). To use another parameterization, use gigChangePars.

Usage

gigCalcRange(chi = 1, psi = 1, lambda = 1,
             param = c(chi, psi, lambda),
             tol = 10^(-5), density = TRUE, ...)

Arguments

chi	A shape parameter that by default holds a value of 1.
psi	Another shape parameter that is set to 1 by default.
lambda	Shape parameter of the GIG distribution. Common to all forms of parameterization. By default this is set to 1.
param	Value of parameter vector specifying the generalized inverse Gaussian distribution.
tol	Tolerance.
density	Logical. If TRUE, the bounds are for the density function. If FALSE, they should be for the probability distribution, but this has not yet been implemented.
...	Extra arguments for calls to uniroot.

Details

The particular generalized inverse Gaussian distribution being considered is specified by the value of the parameter value param.

If density = TRUE, the function gives a range, outside of which the density is less than the given tolerance. Useful for plotting the density. Also used in determining break points for the separate sections over which numerical integration is used to determine the distribution function. The points are found by using uniroot on the density function.

If density = FALSE, the function returns the message: "Distribution function bounds not yet implemented".

Value

A two-component vector giving the lower and upper ends of the range.

Author(s)

David Scott [email protected]

References

Jörgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics, Vol. 9, Springer-Verlag, New York.

See Also

dgig, gigChangePars

Examples

param <- c(2.5, 0.5, 5)
maxDens <- dgig(gigMode(param = param), param = param)
gigRange <- gigCalcRange(param = param, tol = 10^(-3) * maxDens)
gigRange
curve(dgig(x, param = param), gigRange[1], gigRange[2])
## Not run: gigCalcRange(param = param, tol = 10^(-3), density = FALSE)

---

Change Parameterizations of the Generalized Inverse Gaussian Distribution

Description

This function interchanges between the following 4 parameterizations of the generalized inverse Gaussian distribution:

1. $(\chi, \psi, \lambda)$

2. $(\delta, \gamma, \lambda)$

3. $(\alpha, \beta, \lambda)$

4. $(\omega, \eta, \lambda)$

See Jörgensen (1982) and Dagpunar (1989)

Usage

gigChangePars(from, to, param, noNames = FALSE)

Arguments

from	The set of parameters to change from.
to	The set of parameters to change to.
param	“from” parameter vector consisting of 3 numerical elements.
noNames	Logical. When TRUE, suppresses the parameter names in the output.

Details

The range of $\lambda$ is the whole real line. In each parameterization, the other two parameters must take positive values.

Value

A numerical vector of length 3 representing param in the “to” parameterization.

Author(s)

David Scott [email protected]

References

Jörgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics, Vol. 9, Springer-Verlag, New York.

Dagpunar, J. S. (1989). An easily implemented generalised inverse Gaussian generator, Commun. Statist.—Simula., 18, 703–710.

See Also

dgig

Examples

param1 <- c(2.5, 0.5, 5)                # Parameterisation 1
param2 <- gigChangePars(1, 2, param1)   # Convert to parameterization 2
param2                                  # Parameterization 2
gigChangePars(2, 1, as.numeric(param2)) # Convert back to parameterization 1

---

Check Parameters of the Generalized Inverse Gaussian Distribution

Description

Given a putative set of parameters for the generalized inverse Gaussian distribution, the functions checks if they are in the correct range, and if they correspond to the boundary cases.

Usage

gigCheckPars(param, ...)

Arguments

param	Numeric. Putative parameter values for a generalized inverse Gaussian distribution.
...	Further arguments for calls to all.equal.

Details

The vector param takes the form c(chi, psi, lambda).

If either chi or psi is negative, an error is returned.

If chi is 0 (to within tolerance allowed by all.equal) then psi and lambda must be positive or an error is returned. If these conditions are satisfied, the distribution is identified as a gamma distribution.

If psi is 0 (to within tolerance allowed by all.equal) then chi must be positive and lambda must be negative or an error is returned. If these conditions are satisfied, the distribution is identified as an inverse gamma distribution.

If both chi and psi are positive, then the distribution is identified as a normal generalized inverse Gaussian distribution.

Value

A list with components:

case	Whichever of "error", "gamma", invgamma, or "normal" is identified by the function.
errMessage	An appropriate error message if an error was found, the empty string "" otherwise.

Author(s)

David Scott [email protected]

References

Paolella, Marc S. (2007) Intermediate Probability: A Computational Approach, Chichester: Wiley

See Also

dgig

Examples

gigCheckPars(c(5, 2.5, -0.5))      # normal
gigCheckPars(c(-5, 2.5, 0.5))      # error
gigCheckPars(c(5, -2.5, 0.5))      # error
gigCheckPars(c(-5, -2.5, 0.5))     # error
gigCheckPars(c(0, 2.5, 0.5))       # gamma
gigCheckPars(c(0, 2.5, -0.5))      # error
gigCheckPars(c(0, 0, 0.5))         # error
gigCheckPars(c(0, 0, -0.5))        # error
gigCheckPars(c(5, 0, 0.5))         # error
gigCheckPars(c(5, 0, -0.5))        # invgamma

---

Fit the Generalized Inverse Gausssian Distribution to Data

Description

Fits a generalized inverse Gaussian distribution to data. Displays the histogram, log-histogram (both with fitted densities), Q-Q plot and P-P plot for the fit which has the maximum likelihood.

Usage

gigFit(x, freq = NULL, paramStart = NULL,
         startMethod = c("Nelder-Mead","BFGS"),
         startValues = c("LM","GammaIG","MoM","Symb","US"),
         method = c("Nelder-Mead","BFGS","nlm"),
         stand = TRUE, plots = FALSE, printOut = FALSE,
         controlBFGS = list(maxit = 200),
         controlNM = list(maxit = 1000),
         maxitNLM = 1500, ...)


  ## S3 method for class 'gigFit'
print(x,
        digits = max(3, getOption("digits") - 3), ...)

  ## S3 method for class 'gigFit'
plot(x, which = 1:4,
       plotTitles = paste(c("Histogram of ", "Log-Histogram of ",
                          "Q-Q Plot of ", "P-P Plot of "),
                          x$obsName, sep = ""),
       ask = prod(par("mfcol")) < length(which) & dev.interactive(), ...)

  ## S3 method for class 'gigFit'
coef(object, ...)

  ## S3 method for class 'gigFit'
vcov(object, ...)

Arguments

x	Data vector for gigFit. Object of class "gigFit" for print.gigFit and plot.gigFit.
freq	A vector of weights with length equal to length(x).
paramStart	A user specified starting parameter vector param taking the form c(chi, psi, lambda).
startMethod	Method used by gigFitStartMoM in calls to optim.
startValues	Code giving the method of determining starting values for finding the maximum likelihood estimate of param.
method	Different optimisation methods to consider. See Details.
stand	Logical. If TRUE, the data is first standardized by dividing by the sample standard deviation.
plots	Logical. If FALSE suppresses printing of the histogram, log-histogram, Q-Q plot and P-P plot.
printOut	Logical. If FALSE suppresses printing of results of fitting.
controlBFGS	A list of control parameters for optim when using the "BFGS" optimisation.
controlNM	A list of control parameters for optim when using the "Nelder-Mead" optimisation.
maxitNLM	A positive integer specifying the maximum number of iterations when using the "nlm" optimisation.
digits	Desired number of digits when the object is printed.
which	If a subset of the plots is required, specify a subset of the numbers 1:4.
plotTitles	Titles to appear above the plots.
ask	Logical. If TRUE, the user is asked before each plot, see par(ask = .).
...	Passes arguments to optim, par, hist, logHist, qqgig and ppgig.
object	Object of class "gigFit" for coef.gigFit and for vcov.gigFit.

Details

Possible values of the argument startValues are the following:

"LM"

Based on fitting linear models to the upper tails of the data x and the inverse of the data 1/x.

"GammaIG"

Based on fitting gamma and inverse gamma distributions.

"MoM"

Method of moments.

"Symb"

Not yet implemented.

"US"

User-supplied.

If startValues = "US" then a value must be supplied for paramStart.

For the details concerning the use of paramStart, startMethod, and startValues, see gigFitStart.

The three optimisation methods currently available are:

"BFGS"

Uses the quasi-Newton method "BFGS" as documented in optim.

"Nelder-Mead"

Uses an implementation of the Nelder and Mead method as documented in optim.

"nlm"

Uses the nlm function in R.

For details of how to pass control information for optimisation using optim and nlm, see optim and nlm.

When method = "nlm" is used, warnings may be produced. These do not appear to be a problem.

Value

gigFit returns a list with components:

param	A vector giving the maximum likelihood estimate of param, as c(chi, psi, lambda).
maxLik	The value of the maximised log-likelihood.
method	Optimisation method used.
conv	Convergence code. See the relevant documentation (either optim or nlm) for details on convergence.
iter	Number of iterations of optimisation routine.
obs	The data used to fit the generalized inverse Gaussian distribution.
obsName	A character string with the actual x argument name.
paramStart	Starting value of param returned by call to gigFitStart.
svName	Descriptive name for the method finding start values.
startValues	Acronym for the method of finding start values.
breaks	The cell boundaries found by a call to hist.
midpoints	The cell midpoints found by a call to hist.
empDens	The estimated density found by a call to hist.

Author(s)

David Scott [email protected], David Cusack

References

Jörgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics, Vol. 9, Springer-Verlag, New York.

See Also

optim, par, hist, logHist (pkg DistributionUtils), qqgig, ppgig, and gigFitStart.

Examples

param <- c(1, 1, 1)
dataVector <- rgig(500, param = param)
## See how well gigFit works
gigFit(dataVector)
##gigFit(dataVector, plots = TRUE)

## See how well gigFit works in the limiting cases
## Gamma case
dataVector2 <- rgamma(500, shape = 1, rate = 1)
gigFit(dataVector2)

## Inverse gamma
require(actuar)
dataVector3 <- rinvgamma(500, shape = 1, rate = 1)
gigFit(dataVector3)

## Use nlm instead of default
gigFit(dataVector, method = "nlm")

---

Find Starting Values for Fitting a Generalized Inverse Gaussian Distribution

Description

Finds starting values for input to a maximum likelihood routine for fitting the generalized inverse Gaussian distribution to data.

Usage

gigFitStart(x, startValues = c("LM","GammaIG","MoM","Symb","US"),
              paramStart = NULL,
              startMethodMoM = c("Nelder-Mead","BFGS"), ...)
  gigFitStartMoM(x, paramStart = NULL,
                 startMethodMoM = "Nelder-Mead", ...)
  gigFitStartLM(x, ...)

Arguments

x	Data vector.
startValues	Acronym indicating the method to use for obtaining starting values to be used as input to gigFit. See Details.
paramStart	Starting values for param if startValues = "US".
startMethodMoM	Method used by call to optim in finding method of moments estimates.
...	Passes arguments to optim and calls to hist.

Details

Possible values of the argument startValues are the following:

"LM"

Based on fitting linear models to the upper tails of the data x and the inverse of the data 1/x.

"GammaIG"

Based on fitting gamma and inverse gamma distributions.

"MoM"

Method of moments.

"Symb"

Not yet implemented.

"US"

User-supplied.

If startValues = "US" then a value must be supplied for paramStart.

When startValues = "MoM" an initial optimisation is needed to find the starting values. This optimisations starts from arbitrary values, c(1,1,1) for the parameters $(\chi,\psi,\lambda)$ and calls optim with the method given by startMethodMoM. Other starting values for the method of moments can be used by supplying a value for paramStart.

The default method of finding starting values is "LM". Testing indicates this is quite fast and finds good starting values. In addition, it does not require any starting values itself.

gigFitStartMoM is called by gigFitStart and implements the method of moments approach.

gigFitStartLM is called by gigFitStart and implements the linear models approach.

Value

gigFitStart returns a list with components:

paramStart	A vector with elements chi, psi, and lambda giving the starting value of param.
breaks	The cell boundaries found by a call to hist.
midpoints	The cell midpoints found by a call to hist.
empDens	The estimated density found by a call to hist.

gigFitStartMoM and gigFitStartLM each return paramStart, the starting value of param, to the calling function gigFitStart

Author(s)

David Scott [email protected], David Cusack

See Also

dgig, gigFit.

Examples

param <- c(1, 1, 1)
dataVector <- rgig(500, param = param)
gigFitStart(dataVector)

---

Calculate Two-Sided Hessian for the Generalized Inverse Gaussian Distribution

Description

Calculates the Hessian of a function, either exactly or approximately. Used to obtaining the information matrix for maximum likelihood estimation.

Usage

gigHessian(x, param, hessianMethod = "tsHessian",
           whichParam = 1)

Arguments

x	Data vector.
param	The maximum likelihood estimates parameter vector of the generalized inverse Gaussian distribution. There are five different sets of parameterazations can be used in this function, the first four sets are listed in gigChangePars and the last set is the log scale of the first set of the parameterization, i.e., mu,log(delta),Pi,log(zeta).
hessianMethod	Only the approximate method ("tsHessian") has actually been implemented so far.
whichParam	Numeric. A number between indicating which parameterization the argument param relates to. Only parameterization 1 is available so far.

Details

The approximate Hessian is obtained via a call to tsHessian from the package DistributionUtils. summary.gigFit calls the function gigHessian to calculate the Hessian matrix when the argument hessian = TRUE.

Value

gigHessian gives the approximate Hessian matrix for the data vector x and the estimated parameter vector param.

Author(s)

David Scott [email protected], David Cusack

Examples

### Calculate the approximate Hessian using gigHessian:
param <- c(1,1,1)
dataVector <- rgig(500, param = param)
fit <- gigFit(dataVector)
coef <- coef(fit)
gigHessian(x = dataVector, param = coef, hessianMethod = "tsHessian",
              whichParam = 1)

### Or calculate the approximate Hessian using summary.gigFit method:
summary(fit, hessian = TRUE)

---

Calculate Moments of the Generalized Inverse Gaussian Distribution

Description

Functions to calculate raw moments and moments about a given location for the generalized inverse Gaussian (GIG) distribution, including the gamma and inverse gamma distributions as special cases.

Usage

gigRawMom(order, chi = 1, psi = 1, lambda = 1,
          param = c(chi, psi, lambda))
gigMom(order, chi = 1, psi = 1, lambda = 1,
       param = c(chi, psi, lambda), about = 0)
gammaRawMom(order, shape = 1, rate = 1, scale = 1/rate)

Arguments

order	Numeric. The order of the moment to be calculated. Not permitted to be a vector. Must be a positive whole number except for moments about zero.
chi	A shape parameter that by default holds a value of 1.
psi	Another shape parameter that is set to 1 by default.
lambda	Shape parameter of the GIG distribution. Common to all forms of parameterization. By default this is set to 1.
param	Numeric. The parameter vector specifying the GIG distribution. Of the form c(chi, psi, lambda) (see dgig).
about	Numeric. The point around which the moment is to be calculated.
shape	Numeric. The shape parameter, must be non-negative, not permitted to be a vector.
scale	Numeric. The scale parameter, must be positive, not permitted to be a vector.
rate	Numeric. The rate parameter, an alternative way to specify the scale.

Details

The vector param of parameters is examined using gigCheckPars to see if the parameters are valid for the GIG distribution and if they correspond to the special cases which are the gamma and inverse gamma distributions. Checking of special cases and valid parameter vector values is carried out using the function gigCheckPars. Checking whether order is a whole number is carried out using the function is.wholenumber.

Raw moments (moments about zero) are calculated using the functions gigRawMom or gammaRawMom. For moments not about zero, the function momChangeAbout is used to derive moments about another point from raw moments. Note that raw moments of the inverse gamma distribution can be obtained from the raw moments of the gamma distribution because of the relationship between the two distributions. An alternative implementation of raw moments of the gamma and inverse gamma distributions may be found in the package actuar and these may be faster since they are written in C.

To calculate the raw moments of the GIG distribution it is convenient to use the alternative parameterization of the GIG in terms of $\omega$ and $\eta$, given as parameterization 3 in gigChangePars. Then the raw moment of the GIG distribution of order $k$ is given by

$\eta^k K_{\lambda+k}(\omega)/K_{\lambda}(\omega)$

where $K_\lambda()$ is the modified Bessel function of the third kind of order $\lambda$.

The raw moment of the gamma distribution of order $k$ with shape parameter $\alpha$ and rate parameter $\beta$ is given by

$\beta^{-k}\Gamma(\alpha+k)/\Gamma(\alpha)$

The raw moment of order $k$ of the inverse gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$ is the raw moment of order $-k$ of the gamma distribution with shape parameter $\alpha$ and rate parameter $1/\beta$.

Value

The moment specified. In the case of raw moments, Inf is returned if the moment is infinite.

Author(s)

David Scott [email protected]

References

Paolella, Marc S. (2007) Intermediate Probability: A Computational Approach, Chichester: Wiley

See Also

gigCheckPars, gigChangePars and from package DistributionUtils: is.wholenumber, momChangeAbout, momIntegrated

Further, gigMean, gigVar, gigSkew, gigKurt.

Examples

## Computations, using momIntegrated from pkg 'DistributionUtils':
momIntegrated <- DistributionUtils :: momIntegrated

### Raw moments of the generalized inverse Gaussian distribution
param <- c(5, 2.5, -0.5)
gigRawMom(1, param = param)
momIntegrated("gig", order = 1, param = param, about = 0)
gigRawMom(2, param = param)
momIntegrated("gig", order = 2, param = param, about = 0)
gigRawMom(10, param = param)
momIntegrated("gig", order = 10, param = param, about = 0)
gigRawMom(2.5, param = param)

### Moments of the generalized inverse Gaussian distribution
param <- c(5, 2.5, -0.5)
(m1 <- gigRawMom(1, param = param))
gigMom(1, param = param)
gigMom(2, param = param, about = m1)
(m2 <- momIntegrated("gig", order = 2, param = param, about = m1))
gigMom(1, param = param, about = m1)
gigMom(3, param = param, about = m1)
momIntegrated("gig", order = 3, param = param, about = m1)

### Raw moments of the gamma distribution
shape <- 2
rate <- 3
param <- c(shape, rate)
gammaRawMom(1, shape, rate)
momIntegrated("gamma", order = 1, shape = shape, rate = rate, about = 0)
gammaRawMom(2, shape, rate)
momIntegrated("gamma", order = 2, shape = shape, rate = rate, about = 0)
gammaRawMom(10, shape, rate)
momIntegrated("gamma", order = 10, shape = shape, rate = rate, about = 0)

### Moments of the inverse gamma distribution
param <- c(5, 0, -0.5)
gigRawMom(2, param = param)             # Inf
gigRawMom(-2, param = param)
momIntegrated("invgamma", order = -2, shape = -param[3],
              rate = param[1]/2, about = 0)

### An example where the moment is infinite: inverse gamma
param <- c(5, 0, -0.5)
gigMom(1, param = param)
gigMom(2, param = param)

---

Parameter Sets for the Generalized Inverse Gaussian Distribution

Description

These objects store different parameter sets of the generalized inverse Gaussian distribution as matrices for testing or demonstration purposes.

The parameter sets gigSmallParam and gigLargeParam give combinations of values of the parameters $\chi$, $\psi$ and $\lambda$. For gigSmallParam, the values of $\chi$ and $\psi$ are chosen from {0.1, 0.5, 2, 5, 20, 50}, and the values of $\lambda$ from {-0.5, 0, 0.5, 1, 5}. For gigLargeParam, the values of $\chi$ and $\psi$ are chosen from {0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100}, and the values of $\lambda$ from {-2, -1, -0.5, 0, 0.1, 0.2, 0.5, 1, 2, 5, 10}.

Usage

gigSmallParam
  gigLargeParam

Format

gigSmallParam: a 125 by 3 matrix; gigLargeParam: a 1100 by 3 matrix.

Author(s)

David Scott [email protected]

Examples

data(gigParam)
## Check values of chi and psi
plot(gigLargeParam[, 1], gigLargeParam[, 2])
### Check all three parameters
pairs(gigLargeParam,
  labels = c(expression(chi),expression(psi),expression(lambda)))

## Testing the accuracy of gigMean
for (i in 1:nrow(gigSmallParam)) {
  param <- gigSmallParam[i, ]
  x <- rgig(1000, param = param)
  sampleMean <- mean(x)
  funMean <- gigMean(param = param)
  difference <- abs(sampleMean - funMean)
  print(difference)
}

---

Generalized Inverse Gaussian Quantile-Quantile and Percent-Percent Plots

Description

qqgig produces a generalized inverse Gaussian QQ plot of the values in y.

ppgig produces a generalized inverse Gaussian PP (percent-percent) or probability plot of the values in y.

If line = TRUE, a line with zero intercept and unit slope is added to the plot.

Graphical parameters may be given as arguments to qqgig, and ppgig.

Usage

qqgig(y, chi = 1, psi = 1, lambda = 1,
      param = c(chi, psi, lambda),
      main = "GIG Q-Q Plot",
      xlab = "Theoretical Quantiles",
      ylab = "Sample Quantiles",
      plot.it = TRUE, line = TRUE, ...)

ppgig(y, chi = 1, psi = 1, lambda = 1,
      param = c(chi, psi, lambda),
      main = "GIG P-P Plot",
      xlab = "Uniform Quantiles",
      ylab = "Probability-integral-transformed Data",
      plot.it = TRUE, line = TRUE, ...)

Arguments

y	The data sample.
chi	A shape parameter that by default holds a value of 1.
psi	Another shape parameter that is set to 1 by default.
lambda	Shape parameter of the GIG distribution. Common to all forms of parameterization. By default this is set to 1.
param	Parameters of the generalized inverse Gaussian distribution.
xlab, ylab, main	Plot labels.
plot.it	Logical. TRUE denotes the results should be plotted.
line	Logical. If TRUE, a line with zero intercept and unit slope is added to the plot.
...	Further graphical parameters.

Value

For qqgig and ppgig, a list with components:

x	The x coordinates of the points that are be plotted.
y	The y coordinates of the points that are be plotted.

References

Wilk, M. B. and Gnanadesikan, R. (1968) Probability plotting methods for the analysis of data. Biometrika. 55, 1–17.

See Also

ppoints, dgig.

Examples

par(mfrow = c(1, 2))
y <- rgig(1000, param = c(2, 3, 1))
qqgig(y, param = c(2, 3, 1), line = FALSE)
abline(0, 1, col = 2)
ppgig(y, param = c(2, 3, 1))

---

Range of a Hyperbolic Distribution

Description

Given the parameter vector param of a hyperbolic distribution, this function calculates the range outside of which the distribution has negligible probability, or the density function is negligible, to a specified tolerance. The parameterization used is the $(\alpha, \beta)$ one (see dhyperb). To use another parameterization, use hyperbChangePars.

Usage

hyperbCalcRange(mu = 0, delta = 1, alpha = 1, beta = 0,
                param = c(mu, delta, alpha, beta),
                tol = 10^(-5), density = TRUE, ...)

Arguments

mu	$\mu$ is the location parameter. By default this is set to 0.
delta	$\delta$ is the scale parameter of the distribution. A default value of 1 has been set.
alpha	$\alpha$ is the tail parameter, with a default value of 1.
beta	$\beta$ is the skewness parameter, by default this is 0.
param	Value of parameter vector specifying the hyperbolic distribution. This takes the form c(mu, delta, alpha, beta).
tol	Tolerance.
density	Logical. If FALSE, the bounds are for the probability distribution. If TRUE, they are for the density function.
...	Extra arguments for calls to uniroot.

Details

The particular hyperbolic distribution being considered is specified by the value of the parameter value param.

If density = FALSE, the function calculates the effective range of the distribution, which is used in calculating the distribution function and quantiles, and may be used in determining the range when plotting the distribution. By effective range is meant that the probability of an observation being greater than the upper end is less than the specified tolerance tol. Likewise for being smaller than the lower end of the range. Note that this has not been implemented yet.

If density = TRUE, the function gives a range, outside of which the density is less than the given tolerance. Useful for plotting the density.

Value

A two-component vector giving the lower and upper ends of the range.

Author(s)

David Scott [email protected], Jennifer Tso, Richard Trendall

References

Barndorff-Nielsen, O. and Blæsild, P (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700–707. New York: Wiley.

See Also

dhyperb, hyperbChangePars

Examples

par(mfrow = c(1, 2))
param <- c(0, 1, 3, 1)
hyperbRange <- hyperbCalcRange(param = param, tol = 10^(-3))
hyperbRange
curve(phyperb(x, param = param), hyperbRange[1], hyperbRange[2])
maxDens <- dhyperb(hyperbMode(param = param), param = param)
hyperbRange <- hyperbCalcRange(param = param, tol = 10^(-3) * maxDens, density = TRUE)
hyperbRange
curve(dhyperb(x, param = param), hyperbRange[1], hyperbRange[2])

---

Change Parameterizations of the Hyperbolic Distribution

Description

This function interchanges between the following 4 parameterizations of the hyperbolic distribution:

1. $\mu, \delta, \pi, \zeta$

2. $\mu, \delta, \alpha, \beta$

3. $\mu, \delta, \phi, \gamma$

4. $\mu, \delta, \xi, \chi$

The first three are given in Barndorff-Nielsen and Blæsild (1983), and the fourth in Prause (1999)

Usage

hyperbChangePars(from, to, param, noNames = FALSE)

Arguments

from	The set of parameters to change from.
to	The set of parameters to change to.
param	"from" parameter vector consisting of 4 numerical elements.
noNames	Logical. When TRUE, suppresses the parameter names in the output.

Details

In the 4 parameterizations, the following must be positive:

1. $\zeta, \delta$

2. $\alpha, \delta$

3. $\phi, \gamma, \delta$

4. $\xi, \delta$

Furthermore, note that in the second parameterization $\alpha$ must be greater than the absolute value of $\beta$, while in the fourth parameterization, $\xi$ must be less than one, and the absolute value of $\chi$ must be less than $\xi$.

Value

A numerical vector of length 4 representing param in the to parameterization.

Author(s)

David Scott [email protected], Jennifer Tso, Richard Trendall

References

Barndorff-Nielsen, O. and Blæsild, P. (1983). Hyperbolic distributions. In Encyclopedia of Statistical Sciences, eds., Johnson, N. L., Kotz, S. and Read, C. B., Vol. 3, pp. 700–707. New York: Wiley.

Prause, K. (1999) The generalized hyperbolic models: Estimation, financial derivatives and risk measurement. PhD Thesis, Mathematics Faculty, University of Freiburg.

See Also

dhyperb

Examples

param1 <- c(2, 1, 3, 1)                    # Parameterization 1
param2 <- hyperbChangePars(1, 2, param1)   # Convert to parameterization 2
param2                                     # Parameterization 2
hyperbChangePars(2, 1, param2)             # Back to parameterization 1

---

Cramer-von~Mises Test of a Hyperbolic Distribution

Description

Carry out a Cramér-von~Mises test of a hyperbolic distribution where the parameters of the distribution are estimated, or calculate the p-value for such a test.

Usage

hyperbCvMTest(x, mu = 0, delta = 1, alpha = 1, beta = 0,
              param = c(mu, delta, alpha, beta),
              conf.level = 0.95, ...)
hyperbCvMTestPValue(delta = 1, alpha = 1, beta = 0, Wsq, digits = 3)
## S3 method for class 'hyperbCvMTest'
print(x, prefix = "\t", ...)

Arguments

x	A numeric vector of data values for hyperbCvMTest, or object of class "hyperbCvMTest" for print.hyperbCvMTest.
mu	$\mu$ is the location parameter. By default this is set to 0.
delta	$\delta$ is the scale parameter of the distribution. A default value of 1 has been set.
alpha	$\alpha$ is the tail parameter, with a default value of 1.
beta	$\beta$ is the skewness parameter, by default this is 0.
param	Parameters of the hyperbolic distribution taking the form c(mu, delta, alpha, beta).
conf.level	Confidence level of the the confidence interval.
...	Further arguments to be passed to or from methods.
Wsq	Value of the test statistic in the Cramér-von~Mises test of the hyperbolic distribution.
digits	Number of decimal places for p-value.
prefix	Character(s) to be printed before the description of the test.

Details

hyperbCvMTest carries out a Cramér-von~Mises goodness-of-fit test of the hyperbolic distribution. The parameter param must be given in the $(\alpha, \beta)$ parameterization.

hyperbCvMTestPValue calculates the p-value of the test, and is not expected to be called by the user. The method used is interpolation in Table 5 given in Puig & Stephens (2001), which assumes all the parameters of the distribution are unknown. Since the table used is limited, large p-values are simply given as “>~0.25” and very small ones as “<~0.01”. The table is created as the matrix wsqTable when the package GeneralizedHyperbolic is invoked.

print.hyperbCvMTest prints the output from the Cramér-von~Mises goodness-of-fit test for the hyperbolic distribution in very similar format to that provided by print.htest. The only reason for having a special print method is that p-values can be given as less than some value or greater than some value, such as “<\ ~0.01”, or “>\ ~0.25”.

Value

hyperbCvMTest returns a list with class hyperbCvMTest containing the following components:

statistic	The value of the test statistic.
method	A character string with the value “Cramér-von~Mises test of hyperbolic distribution”.
data.name	A character string giving the name(s) of the data.
parameter	The value of the parameter param
p.value	The p-value of the test.
warn	A warning if the parameter values are outside the limits of the table given in Puig & Stephens (2001).

hyperbCvMTestPValue returns a list with the elements p.value and warn only.

Author(s)

David Scott, Thomas Tran

References

Puig, Pedro and Stephens, Michael A. (2001), Goodness-of-fit tests for the hyperbolic distribution. The Canadian Journal of Statistics/La Revue Canadienne de Statistique, 29, 309–320.

Examples

param <- c(2, 2, 2, 1.5)
dataVector <- rhyperb(500, param = param)
fittedparam <- hyperbFit(dataVector)$param
hyperbCvMTest(dataVector, param = fittedparam)
dataVector <- rnorm(1000)
fittedparam <- hyperbFit(dataVector, startValues = "FN")$param
hyperbCvMTest(dataVector, param = fittedparam)

---

Fit the Hyperbolic Distribution to Data

Description

Fits a hyperbolic distribution to data. Displays the histogram, log-histogram (both with fitted densities), Q-Q plot and P-P plot for the fit which has the maximum likelihood.

Usage

hyperbFit(x, freq = NULL, paramStart = NULL,
            startMethod = c("Nelder-Mead","BFGS"),
            startValues = c("BN","US","FN","SL","MoM"),
            criterion = "MLE",
            method = c("Nelder-Mead","BFGS","nlm",
                       "L-BFGS-B","nlminb","constrOptim"),
            plots = FALSE, printOut = FALSE,
            controlBFGS = list(maxit = 200),
            controlNM = list(maxit = 1000), maxitNLM = 1500,
            controlLBFGSB = list(maxit = 200),
            controlNLMINB = list(),
            controlCO = list(), ...)

  ## S3 method for class 'hyperbFit'
print(x,
        digits = max(3, getOption("digits") - 3), ...)

  ## S3 method for class 'hyperbFit'
plot(x, which = 1:4,
     plotTitles = paste(c("Histogram of ","Log-Histogram of ",
                        "Q-Q Plot of ","P-P Plot of "), x$obsName,
                        sep = ""),
     ask = prod(par("mfcol")) < length(which) & dev.interactive(), ...)

  ## S3 method for class 'hyperbFit'
coef(object, ...)

  ## S3 method for class 'hyperbFit'
vcov(object, ...)

Arguments

x	Data vector for hyperbFit. Object of class "hyperbFit" for print.hyperbFit and plot.hyperbFit.
freq	A vector of weights with length equal to length(x).
paramStart	A user specified starting parameter vector param taking the form c(mu, delta, alpha, beta).
startMethod	Method used by hyperbFitStart in calls to optim.
startValues	Code giving the method of determining starting values for finding the maximum likelihood estimate of param.
criterion	Currently only "MLE" is implemented.
method	Different optimisation methods to consider. See Details.
plots	Logical. If FALSE suppresses printing of the histogram, log-histogram, Q-Q plot and P-P plot.
printOut	Logical. If FALSE suppresses printing of results of fitting.
controlBFGS	A list of control parameters for optim when using the "BFGS" optimisation.
controlNM	A list of control parameters for optim when using the "Nelder-Mead" optimisation.
maxitNLM	A positive integer specifying the maximum number of iterations when using the "nlm" optimisation.
controlLBFGSB	A list of control parameters for optim when using the "L-BFGS-B" optimisation.
controlNLMINB	A list of control parameters for nlminb when using the "nlminb" optimisation.
controlCO	A list of control parameters for constrOptim when using the "constrOptim" optimisation.
digits	Desired number of digits when the object is printed.
which	If a subset of the plots is required, specify a subset of the numbers 1:4.
plotTitles	Titles to appear above the plots.
ask	Logical. If TRUE, the user is asked before each plot, see par(ask = .).
...	Passes arguments to par, hist, logHist, qqhyperb and pphyperb.
object	Object of class "hyperbFit" for coef.hyperbFit and for vcov.hyperbFit.

Details

startMethod can be either "BFGS" or "Nelder-Mead".

startValues can be one of the following:

"US"

User-supplied.

"BN"

Based on Barndorff-Nielsen (1977).

"FN"

A fitted normal distribution.

"SL"

Based on a fitted skew-Laplace distribution.

"MoM"

Method of moments.

For the details concerning the use of paramStart, startMethod, and startValues, see hyperbFitStart.

The six optimisation methods currently available are:

"BFGS"

Uses the quasi-Newton method "BFGS" as documented in optim.

"Nelder-Mead"

Uses an implementation of the Nelder and Mead method as documented in optim.

"nlm"

Uses the nlm function in R.

"L-BFGS-B"

Uses the quasi-Newton method with box constraints "L-BFGS-B" as documented in optim.

"nlminb"

Uses the nlminb function in R.

"constrOptim"

Uses the constrOptim function in R.

For details of how to pass control information for optimisation using optim, nlm, nlminb and constrOptim, see optim, nlm, nlminb and constrOptim.

When method = "nlm" is used, warnings may be produced. These do not appear to be a problem.

Value

hyperbFit returns a list with components:

param	A vector giving the maximum likelihood estimate of param, as c(mu, delta, alpha, beta).
maxLik	The value of the maximised log-likelihood.
method	Optimisation method used.
conv	Convergence code. See the relevant documentation (either optim or nlm) for details on convergence.
iter	Number of iterations of optimisation routine.
obs	The data used to fit the hyperbolic distribution.
obsName	A character string with the actual x argument name.
paramStart	Starting value of param returned by call to hyperbFitStart.
svName	Descriptive name for the method finding start values.
startValues	Acronym for the method of finding start values.
breaks	The cell boundaries found by a call to hist.
midpoints	The cell midpoints found by a call to hist.
empDens	The estimated density found by a call to hist.

Author(s)

David Scott [email protected], Ai-Wei Lee, Jennifer Tso, Richard Trendall, Thomas Tran, Christine Yang Dong

References

Barndorff-Nielsen, O. (1977) Exponentially decreasing distributions for the logarithm of particle size, Proc. Roy. Soc. Lond. A353, 401–419.

Fieller, N. J., Flenley, E. C. and Olbricht, W. (1992) Statistics of particle size data. Appl. Statist. 41, 127–146.

See Also

optim, nlm, nlminb, constrOptim, par, hist, logHist (pkg DistributionUtils), qqhyperb, pphyperb, dskewlap and hyperbFitStart.

Examples

param <- c(2, 2, 2, 1)
dataVector <- rhyperb(500, param = param)
## See how well hyperbFit works
hyperbFit(dataVector)
hyperbFit(dataVector, plots = TRUE)
fit <- hyperbFit(dataVector)
par(mfrow = c(1, 2))
plot(fit, which = c(1, 3))

## Use nlm instead of default
hyperbFit(dataVector, method = "nlm")

---