The Package 'SkewHyperbolic': Summary Information

Description

This package provides a collection of functions for working with the skew hyperbolic Student t-distribution.

Functions are provided for the density function (dskewhyp), distribution function (pskewhyp), quantiles (qskewhyp) and random number generation (rskewhyp). There are functions that fit the distribution to data (skewhypFit). The mean, variance, skewness, kurtosis and mode can be found using the functions skewhypMean, skewhypVar, skewhypSkew, skewhypKurt and skewhypMode respectively, and there is also a function to calculate moments of any order skewhypMom. To assess goodness of fit, there are functions to generate a Q-Q plot (qqskewhyp) and a P-P plot (ppskewhyp). S3 methods print, plot and summary are provided for the output of skwewhypFit.

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

References

Aas, K. and Haff, I. H. (2006). The Generalised Hyperbolic Skew Student's t-distribution, Journal of Financial Econometrics, 4, 275–309.

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

Scott, D. J., Würtz, D. and Tran, T. T. (2008) Moments of the Generalized Hyperbolic Distribution. Preprint.

See Also

dskewhyp, skewhypMean,skewhypMom, skewhypFit, skewhypFitStart, qqskewhyp, GeneralizedHyperbolicDistribution.

Internal Skew Hyperbolic Functions

Description

Internal functions for the package SkewHyperbolic

Details

Functions which are either not intended to be called by the user or are waiting to be documented.

Skewed Hyperbolic Student t-Distribution

Description

Density function, distribution function, quantiles and random number generation for the skew hyperbolic Student t-distribution, with parameters \beta (skewness), \delta (scale), \mu (location) and \nu (shape). Also a function for the derivative of the density function.

Usage

dskewhyp(x, mu = 0, delta = 1, beta = 1, nu = 1,
         param = c(mu,delta,beta,nu), log = FALSE,
         tolerance = .Machine$double.eps^0.5)
pskewhyp(q, mu = 0, delta = 1, beta = 1, nu = 1,
         param = c(mu, delta, beta, nu), log.p = FALSE,
         lower.tail = TRUE, subdivisions = 100,
         intTol = .Machine$double.eps^0.25, valueOnly = TRUE, ...)
qskewhyp(p, mu = 0, delta = 1, beta = 1, nu = 1,
         param = c(mu,delta, beta, nu),
         lower.tail = TRUE, log.p = FALSE,
         method = c("spline","integrate"),
         nInterpol = 501, uniTol = .Machine$double.eps^0.25,
         subdivisions = 100, intTol = uniTol, ...)
rskewhyp(n, mu = 0, delta = 1, beta = 1, nu = 1,
         param = c(mu,delta,beta,nu), log = FALSE)
ddskewhyp(x, mu = 0, delta = 1, beta = 1, nu = 1,
          param = c(mu,delta,beta,nu),log = FALSE,
          tolerance = .Machine$double.eps^0.5)

Arguments

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 skewhypCheckPars to see if they are valid.

The density function is

f(x)=\frac{2^{(1-\nu)/2}\delta^\nu%latex |\beta|^{(\nu+1)/2}% K_{(\nu+1)/2}\sqrt{(\beta^2(\delta^2+(x-\mu)^2))}% \exp(\beta(x-\mu))}{\Gamma(\nu/2)\sqrt{(\pi)}% \sqrt{(\delta^2+(x-\mu)^2)^{(\nu+1)/2}}}

when \beta \ne 0, and

f(x)=\frac{\Gamma((\nu+1)/2)}{\sqrt{\pi}\delta% \Gamma(\nu/2)}\left(1+\frac{(x-\mu)^2}{\delta^2}\right)^% {-(\nu+1)/2}

when \beta = 0, where K_{\nu}(.) is the modified Bessel function of the third kind with order \nu, and \Gamma(.) is the gamma function.

pskewhyp 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 qhyperb 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 phyperb. 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.

Note that when small values of \nu are used, and the density is skewed, there are often some extreme values generated by rskewhyp. These look like outliers, but are caused by the heaviness of the skewed tail, see Examples.

The extreme skewness of the distribution when \beta is large in absolute value and \nu is small make this distribution very challenging numerically.

Value

dskewhyp gives the density function, pskewhyp gives the distribution function, qskewhyp gives the quantile function and rskewhyp 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 pskewyhp which returns a list with components value and error.

ddskewhyp gives the derivative of dskewhyp.

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

References

Aas, K. and Haff, I. H. (2006). The Generalised Hyperbolic Skew Student's t-distribution, Journal of Financial Econometrics, 4, 275–309.

See Also

safeIntegrate, integrate for its shortfalls, skewhypCheckPars, logHist. Also skewhypMean for information on moments and mode, and skewhypFit for fitting to data.

Examples

param <- c(0,1,40,10)
par(mfrow = c(1,2))
range <- skewhypCalcRange(param = param, tol = 10^(-2))

### curves of density and distribution
curve(dskewhyp(x, param = param), range[1], range[2], n = 1000)
title("Density of the \n Skew Hyperbolic Distribution")
curve(pskewhyp(x, param = param),
      range[1], range[2], n = 500)
title("Distribution Function of the \n Skew Hyperbolic Distribution")

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

##plots of density and derivative
par(mfrow = c(2,1))
curve(dskewhyp(x, param = param), range[1], range[2], n = 1000)
title("Density of the Skew\n Hyperbolic Distribution")
curve(ddskewhyp(x, param = param), range[1], range[2], n = 1000)
title("Derivative of the Density\n of the Skew Hyperbolic Distribution")

##  example of density and random numbers for beta large and nu small
par(mfrow = c(1,2))
param1 <- c(0,1,10,1)
data1 <- rskewhyp(1000, param = param1)
curve(dskewhyp(x, param = param1), range(data1)[1], range(data1)[2],
      n = 1000, col = 2, main = "Density and Histogram -- when nu is small")
hist(data1, freq = FALSE, add = TRUE)

DistributionUtils::logHist(data1, main =
      "Log-Density and Log-Histogram -- when nu is small")
curve(dskewhyp(x, param = param1, log = TRUE),
      from = min(data1), to = max(data1), n = 500, add = TRUE, col = 2)

Skew Hyperbolic Student t-Distribution Quantile-Quantile and Percent-Percent Plots

Description

qqskewhyp produces a skew hyperbolic t-distribution Q-Q plot of the values in y, ppskewhyp produces a skew hyperbolic t-distribution P-P (percent-percent) plot or probability plot of the values in y. Graphical parameters may be given as arguments to qqskewhyp and ppskewhyp.

Usage

qqskewhyp(y, mu = 0, delta = 1, beta = 1, nu = 1,
          param = c(mu, delta, beta, nu),
          main = "Skew Hyperbolic Student-t QQ Plot",
          xlab = "Theoretical Quantiles", ylab = "Sample Quantiles",
          plot.it = TRUE, line = TRUE, ...)
ppskewhyp(y, beta = NULL, delta = NULL, mu = NULL, nu = NULL,
          param = c(mu, delta, beta, nu),
          main = "Skew Hyperbolic Student-t P-P Plot",
          xlab = "Uniform Quantiles",
          ylab = "Probability-integral-transformed Data",
          plot.it = TRUE, line = TRUE, ...)

Arguments

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.

Value

For qqskewhyp and ppskewhyp, a list with components:

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

References

Aas, K. and Haff, I. H. (2006). The Generalised Hyperbolic Skew Student's t-distribution, Journal of Financial Econometrics, 4, 275–309.

See Also

ppoints, qqplot, dskewhyp

Examples

par(mfrow = c(1,2))
param <- c(0,1,0,10)
y <- rskewhyp(500, param = param)
qqskewhyp(y, param = param, main = "Skew Hyperbolic\n Q-Q Plot")
ppskewhyp(y, param = param, main = "Skew Hyperbolic\n P-P Plot")

Australian Institute of Sport data

Description

Data on 102 male and 100 female athletes collected at the Australian Institute of Sport, courtesy of Richard Telford and Ross Cunningham.

Usage

data(ais)

Format

A data frame with 202 observations on 13 variables.

Source

Cook and Weisberg (1994) via the package sn. This help file is a modification of the help file from the package sn.

References

Cook and Weisberg (1994), An Introduction to Regression Graphics. John Wiley & Sons, New York.

Examples

data(ais)
Fe <- ais$Fe
### Not enough data to find starting values
### Use default parameter values as starting values
FeFit <- skewhypFit(Fe, startValues = "US", paramStart = c(0,1,1,1))
### Ferritin must always be non-negative
### Probability of negative values is small for fitted distribution
pskewhyp(0, param = FeFit$param)

Dow Jones Log Return Data

Description

Log returns of daily closing value data from the dow jones index, from 04/JAN/1999 to 08/JUL/2003. The original data used to calculate these was the dji data set available in the QRMlib package.

Usage

data(lrdji)

Format

A vector of 1132 observations.

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

Source

library(QRMlib) data(dji)

References

McNeil, A. & Ulman, S. (2008). QRMlib http://cran.r-project.org/web/packages/QRMlib/index.html

Examples

data(lrdji)
##fit a skew hyperbolic student t-distribution to the data
fit <- skewhypFit(lrdji, plot = TRUE, print = TRUE)

Log Returns of the NOK/EUR Exchange Rate

Description

Log returns of daily closing value data of the NOK/EUR (Norwegian Kroner/Euro) exchange rate, from 04/JAN/1999 to 08/JUL/2003. The original data was downloaded from the oanda website. The data was selected to be as similar as possible to the data used in the Aas & Haff article (see References).

Usage

data(lrnokeur)

Format

A vector of 1647 observations.

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

Source

http://www.oanda.com

References

Aas, K. and Haff, I. H. (2006). The Generalised Hyperbolic Skew Student's t-distribution, Journal of Financial Econometrics, 4, 275–309.

Examples

##Fit the skew hyperbolic students-t distribution to the data
data(lrnokeur)
fit <- skewhypFit(lrnokeur, method = "nlm", plot = TRUE, print = TRUE)

Range of a Skew Hyperbolic Student t-Distribution

Description

Given the parameter vector param, or parameter values of a skew hyperbolic Student t-distribution, this function determines the range outside of which the density function or distribution function are negligible, to a specified tolerance.

Usage

skewhypCalcRange(mu = 0, delta = 1, beta = 1, nu = 1,
                 param = c(mu,delta,beta,nu), density = TRUE,
                 tol= 10^(-5), ...)
skewhypStepSize(dist, delta, beta, nu, side = c("right","left"))

Arguments

Details

The particular skew hyperbolic distribution being considered is specified by either the individual parameter values, or the parameter vector param. If both are specified, the values in param will overwriete the other ones. In addition the parameter values are examined by calling the function skewhypCheckPars to see if they are valid.

The function skewhypCalcRange returns the range outside of which the density function or distribution function are less than the given tolerance. The points are found by using uniroot on the density or distribution function.

The function skewhypStepSize is used for stepping to the right or the left to obtain an enclosing interval so uniroot can be used to search. When the tail is declining exponentially the step is just a linear function of the current distance from the mode. If the tail is declining only as a power of x, an exponential step is used.

skewhypStepSize is for internal use and is not expected to be called by users. It is documented here for completeness.

Value

The function skewhypCalcRange returns a two component vector giving the lower and upper limits of the range.

skewhypStepSize returns the size of the step.

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

References

Aas, K. and Haff, I. H. (2006). The Generalised Hyperbolic Skew Student's t-distribution, Journal of Financial Econometrics, 4, 275–309.

See Also

uniroot, dskewhyp, skewhypCheckPars

Examples

param <- c(0,1,10,10)
range <- skewhypCalcRange(param = param, tol = 10^(-2))
range
curve(dskewhyp(x, param = c(0,1,5,10), range[1], range[2]))

param <- c(0,1,20,1)
(range <- skewhypCalcRange(param = param))
round(integrate(dskewhyp, -Inf, range[1], param = param)$value,7)
round(integrate(dskewhyp, range[2], Inf, param = param)$value,7)

Check Parameters of the Skew Hyperbolic Student t-distribution

Description

Given a set of parameters for the skew hyperbolic Student t-distribution, the function checks that the parameters are in the correct range, and that the set has the correct length of 4.

Usage

skewhypCheckPars(param)

Arguments

Details

The vector param should be of the form c(mu,delta,beta,nu). If either delta or nu is not greater than zero an error message is returned. If the vector param does not have a length of 4 then an error message is returned.

Value

A list with components:

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

See Also

dskewhyp

Examples

skewhypCheckPars(c(0,1,1,1))     #normal
skewhypCheckPars(c(0,0,1,1))     #error
skewhypCheckPars(c(0,1,1,-1))    #error
skewhypCheckPars(c(0,1,1))       #error

Fit the Skew Hyperbolic Student t-Distribution to Data

Description

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

Usage

skewhypFit(x, freq = NULL, breaks = NULL, startValues = "LA",
           paramStart = NULL, method = "Nelder-Mead", hessian = TRUE,
           plots = FALSE, printOut = TRUE, controlBFGS = list(maxit = 200),
           controlNM = list(maxit = 1000), maxitNLM = 1500, ...)
## S3 method for class 'skewhypFit'
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 'skewhypFit'
print(x,digits = max(3, getOption("digits") - 3),...)

Arguments

Details

startValues can be either "US"(User-supplied) or "LA" (Linear approximation) If startValues = "US" then a value for paramStart must be supplied. For the details concerning the use of startValues and paramStart see skewhypFitStart.

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 the details of how to pass control information using optim and nlm, see optim and nlm.

Value

skewhypFit returns a list with components:

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

References

Aas, K. and Haff, I. H. (2006). The Generalised Hyperbolic Skew Student's t-distribution, Journal of Financial Econometrics, 4, 275–309.

See Also

optim, nlm, par, hist, density, logHist, qqskewhyp, ppskewhyp, dskewhyp and skewhypFitStart.

Examples

## See how well skewhypFit works
param <- c(0, 1, 4, 10)
data <- rskewhyp(500, param = param)
fit <- skewhypFit(data)
## Use data set NOK/EUR as per Aas&Haff
data(lrnokeur)
nkfit <- skewhypFit(lrnokeur, method = "nlm")
## Use data set DJI
data(lrdji)
djfit <- skewhypFit(lrdji)

Find Starting Values for Fitting a Skew Hyperbolic Student t-Distribution

Description

Finds starting values for input to a maximum likelihood routine for fitting a skew hyperbolic t-distribution to data.

Usage

skewhypFitStart(x, breaks = NULL, startValues = "LA", paramStart = NULL,
...)
skewhypFitStartLA(x, breaks = NULL)
skewhypFitStartMM(x, nuTol = 1e-6, nuStart = 5, ...)

Arguments

Details

startValues can be either "US"(User-supplied), "LA"(Linear approximation), or "MM"(Method of moments).

If startValues = "US" then a value for paramStart must be supplied. The parameters are checked for validity by the function skewhypCheckPars.

If startValues = "LA" a linear approximation is made to the log-density in each of the tails, from which the estimates for \nu and \beta are found. The remaining two parameters, \delta and \mu are found by solving the moment equations for mean and variance. Since the variance does not exist for values of \nu \le 4, the estimate of \nu will be at least 4.1. Note that if the distribution is too skewed, there are not enough points in the lighter tail to fit the required linear model, and the method will stop and return a warning. User supplied values will have to be used in this case.

If startValues = "MM", start values are found using the method of moments as outlined in Aas and Haff (2006).

Value

skewhypFitStart returns a list with components:

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

References

Aas, K. and Haff, I. H. (2006). The Generalised Hyperbolic Skew Student's t-distribution, Journal of Financial Econometrics, 4, 275–309.

See Also

hist, density, dskewhyp, skewhypFit, skewhypCheckPars

Examples

## find starting values to feed to skewhypFit
data(lrnokeur)
skewhypFitStart(lrnokeur, startValues="LA")$paramStart
## user supplied values
skewhypFitStart(lrnokeur, startValues="US",
                paramStart=c(0,0.01,0,5))$paramStart

Moments and Mode of the Skew Hyperbolic Student t-Distribution.

Description

Functions to calculate the mean, variance, skewness, kurtosis and mode of a specified skew hyperbolic t-distribution.

Usage

skewhypMean(mu = 0, delta = 1, beta = 1, nu = 1,
            param = c(mu,delta,beta,nu))
skewhypVar(mu = 0, delta = 1, beta = 1, nu = 1,
           param = c(mu,delta,beta,nu))
skewhypSkew(mu = 0, delta = 1, beta = 1, nu = 1,
            param = c(mu,delta,beta,nu))
skewhypKurt(mu = 0, delta = 1, beta = 1, nu = 1,
            param = c(mu,delta,beta,nu))
skewhypMode(mu = 0, delta = 1, beta = 1, nu = 1,
            param = c(mu,delta,beta,nu),
            tolerance = .Machine$double.eps ^ 0.5)

Arguments

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 skewhypCheckPars to see if they are valid.

The moments are calculated as per formulae in Aas&Haff(2006) and the mode is calculated by numerical optimisation of the density function using optim.

Note that the mean does not exist when \nu = 2, the variance does not exist for \nu \le 4, the skewness does not exist for \nu \le 6, and the kurtosis does not exist for \nu \le 8.

Value

skewhypMean gives the mean of the skew hyperbolic t-distribution, skewhypVar the variance, skewhypSkew the skewness, skewhypKurt the kurtosis and skewhypMode the mode.

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

References

Aas, K. and Haff, I. H. (2006). The Generalised Hyperbolic Skew Student's t-distribution, Journal of Financial Econometrics, 4, 275–309.

See Also

dskewhyp, optim, skewhypCheckPars, skewhypMom

Examples

param <- c(10,1,5,9)
skewhypMean(param = param)
skewhypVar(param = param)
skewhypSkew(param = param)
skewhypKurt(param = param)
skewhypMode(param = param)
range <- skewhypCalcRange(param = param)
curve(dskewhyp(x, param = param), range[1], range[2])
abline(v = skewhypMode(param = param), col = "red")
abline(v = skewhypMean(param = param), col = "blue")

Calculate Moments of the Skew Hyperbolic Student t-Distribution.

Description

This function can be used to calculate the raw moments, mu moments, central moments, and moments about any other given location for the skew hyperbolic t-distribution.

Usage

skewhypMom(order, mu = 0, delta = 1, beta = 1, nu = 1,
           param = c(mu,delta,beta,nu), momType = "raw", about = 0)

Arguments

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 skewhypCheckPars to see if they are valid.

order is also checked by calling the function is.wholenumber in the DistributionUtils package to see whether a whole number is given.

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

To calculate the moments of the skew hyperbolic t-distribution, the function first calculates the mu moments by the formula defined below, and then transforms them to any of the other types of moment by calling momChangeAbout in the DistributionUtils package.

The mu moments of the skew hyperbolic t-distribution are given by:

\bar{M}_k = \sum\limits_{\ell = \lfloor(k+1)/2\rfloor}^{k} a_{k, \ell} \beta^{2\ell - k} \left[\frac{\delta^{2\ell} \Gamma(\nu/2 - \ell)}{ \Gamma(\nu/2) 2^{\ell} }\right]

where k = \code{order} and k > 0 and a_{k, \ell} is the recursive coefficient (see momRecursion for details).

This formula is given in Scott, Würtz and Tran (2008). Note that the [.] part of this formula is actually equivalent to the formula for the raw moments of the inverse gamma distribution, so the function calls gammaRawMom in the GeneralizedHyperbolic package when implementing the computations.

Value

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

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

References

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

Scott, D. J., Würtz, D. and Tran, T. T. (2008) Moments of the Generalized Hyperbolic Distribution. Preprint.

See Also

skewhypCheckPars, skewhypMean, is.wholenumber, momRecursion, momChangeAbout and gigMom.

Examples

param = c(1,2,3,10)
##Raw moments of the skew hyperbolic t distribution
skewhypMom(3, param = param, momType = "raw")
##Mu moments
skewhypMom(3, param = param, momType = "mu")
##Central moments
skewhypMom(3, param = param, momType = "central")
##Moments about any location
skewhypMom(3, param = param, about = 5)

Parameter Sets for the Skew Hyperbolic t-Distribution

Description

These objects store different parameter sets of the skew hyperbolic t distribution for testing or demonstrating purpose as matrices.

Specifically, the parameter sets skewhypSmallShape and skewhypLargeShape have constant location parameter of \mu = 0 and scale parameter of \delta = 1.

The skewness parameter \beta takes values from {0,2} in skewhypSmallShape and skewhypSmallParam, and from {-5,0,1,2,5} in skewhypLargeShape and skewhypLargeParam.

The shape parameter \nu takes values from {1,5} in skewhypSmallShape and skewhypSmallParam, and from {1,2,5,10,20} in skewhypLargeShape and skewhypLargeParam.

Usage

  data(skewhypParam)

Format

skewhypSmallShape: a 4 by 4 matrix;

skewhypLargeShape: a 25 by 4 matrix;

skewhypSmallParam: a 16 by 4 matrix;

skewhypLargeParam: a 400 by 4 matrix.

Author(s)

David Scott d.scott@auckland.ac.nz

Examples

data(skewhypParam)
### Testing the accuracy of skewhypMean
for (i in 1:nrow(skewhypSmallParam)) {
  param <- skewhypSmallParam[i, ]
  x <- rskewhyp(1000, param = param)
  sampleMean <- mean(x)
  distMean <- skewhypMean(param = param)
  difference <- abs(sampleMean - distMean)
  print(difference)
}

Tail Plot Line

Description

Adds skew hyperbolic t-distribution line to a tail plot

Usage

skewhypTailPlotLine(x, mu = 0, delta = 1, beta = 1, nu = 1,
                    param = c(mu,delta,beta,nu),
                    side = c("right", "left"), ...)

Arguments

Details

The function tailPlot from DistributionUtils can be used to draw either a left-hand or right-hand tail plot of the data x. See for example Resnick (2007), p.105. The left-hand tail plot plots the empirical distribution of the data against the order statistics, for order statistic values below the median. The right-hand tail plot plots one minus the empirical distribution of the data against the order statistics, for order statistic values above the median. The default is for the y-axis to be plotted on a log scale.

skewhypTailPlotLine adds the line derived from the given skew hyperbolic t-distribution to an already drawn tail plot.

Value

Returns NULL invisibly.

Author(s)

David Scott d.scott@auckland.ac.nz

References

Aas, Kjersti and Hobæk Haff, Ingrid (2006) The generalised hyperbolic skew Student's t-distribution. Journal of Financial Econometrics, 4, 275–309.

Resnick, S. (2007) Heavy-Tail Phenomena, New York: Springer.

See Also

tailPlot and skewhypFit.

Examples

### Draw tail plot of some data
tailPlot <- DistributionUtils :: tailPlot  ## for convenience below
param <- c(0,1,1,10)
x <- rskewhyp(200, param = param)
tailPlot(x)
### Add skew hyperbolic t-distribution line
skewhypTailPlotLine(x, param = param)
### Parameters from fit may look better
paramFit <- skewhypFit(x, plots = FALSE)$param
tailPlot(x)
skewhypTailPlotLine(x, param = param)
skewhypTailPlotLine(x, param = paramFit, col = "steelblue")

### Left tail example
tailPlot(x, side = "l")
### Add skew hyperbolic t-distribution line
skewhypTailPlotLine(x, param = paramFit, side = "l")
### Log scale on both axes
tailPlot(x, side = "r", log = "xy")
### Add skew hyperbolic t-distribution line
skewhypTailPlotLine(x, param = paramFit, side = "r")

Summarising the Skew Hyperbolic Student t-Distribution Fit

Description

summary Method for class "skewhypFit".

Usage

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

Arguments

Details

summary.skewhypFit calculates standard errors for errors for the estimates of \mu, \delta, \beta and \nu of the skew hyperbolic Student t-distribution parameter vector param, if the Hessian from the call to optim or nlm is available. Because the parameters in the call to the optimiser are \mu, \log(\delta), \beta and \log(\nu) the delta method is used to obtain standard errors for \delta and \nu

Value

If the Hessian is available summary.skewhyhpFit computes standard errors of \mu, \delta, \beta and \nu, and adds them to object as object$sds. Otherwise, no calculations are performed and the composition object is unaltered.

summary.skewhypFit invisibly returns x with class changed to summary.skewhypFit.

See skewhypFit for the composition of an object of class skewhypFit.

print.summary.skewhypFit prints a summary in the same format as print.skewhypFit when the Hessian is not available from the fit. When the Hessian is available, the standard errors for the parameter estimates are printed in parentheses beneath the parameter estimates, in the manner of fitdistr in the package MASS.

Author(s)

David Scott d.scott@auckland.ac.nz, Fiona Grimson

References

Aas, K. and Haff, I. H. (2006). The Generalised Hyperbolic Skew Student's t-distribution, Journal of Financial Econometrics, 4, 275–309.

See Also

skewhypFit, dskewhyp, summary

Examples

## Continuing the skewhypFit(.) example:
data(lrdji)
djfit <- skewhypFit(lrdji, print = FALSE, plot = FALSE, hessian = TRUE)
print(djfit)
summary(djfit)