| SkewHyperbolicDistribution {SkewHyperbolic} | R Documentation |
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
x, q |
Vector of quantiles. |
p |
Vector of probabilities. |
n |
Number of random variates to be generated. |
mu |
Location parameter |
delta |
Scale parameter |
beta |
Skewness parameter |
nu |
Shape parameter |
param |
Specifying the parameters as a vector of the form |
log, log.p |
Logical; if |
method |
Character. If |
lower.tail |
Logical. If |
tolerance |
Specified level of tolerance when checking if
parameter |
subdivisions |
The maximum number of subdivisions used to integrate the density and determine the accuracy of the distribution function calculation. |
intTol |
Value of |
valueOnly |
Logical. If |
nInterpol |
Number of points used in |
uniTol |
Value of |
... |
Passes additional arguments to |
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)