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

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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

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.

beta

Skewness parameter beta, default is 1.

nu

Shape parameter nu, default is 1.

param

Specifying the parameters as a vector of the form
c(mu,delta,beta,nu).

log,log.p

Logical; if log = TRUE, probabilities are given as log(p).

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 lower.tail = TRUE, the cumulative density is taken from the lower tail.

tolerance

Specified level of tolerance when checking if parameter beta is equal to 0.

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 pskewhyp 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 qskewhyp for cubic spline interpolation of the distribution function.

uniTol

Value of tol in calls to uniroot. See uniroot.

...

Passes additional arguments to integrate in pskewhyp and qskewhyp, and to uniroot in qskewhyp.

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) = (2^((1-nu)/2) delta^nu abs(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 != 0, and

f(x)=(gamma((nu+1)/2)/(sqrt(pi) delta gamma(nu/2)))(1+((x-mu)^2)/(delta^2))^(-(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

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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")
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)
hist(data1, freq = FALSE, add = TRUE)
title("Density and Histogram\n when nu is small")
logHist(data1, main = "Log-Density and Log-Histogram\n when nu is small")
curve(dskewhyp(x, param = param1, log = TRUE),
      range(data1)[1], range(data1)[2],
      n = 500, add = TRUE, col = 2)