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\name{std}
\alias{std}
\alias{dstd}
\alias{pstd}
\alias{qstd}
\alias{rstd}
\concept{t-distribution}
\concept{Student-t distribution}
\concept{standardized Student t distribution}
\concept{standardized Student-t distribution}
\title{Standardized Student-t distribution}
\description{
Functions to compute density, distribution function, quantile function
and to generate random variates for the standardized Student-t
distribution.
}
\usage{
dstd(x, mean = 0, sd = 1, nu = 5, log = FALSE)
pstd(q, mean = 0, sd = 1, nu = 5)
qstd(p, mean = 0, sd = 1, nu = 5)
rstd(n, mean = 0, sd = 1, nu = 5)
}
\arguments{
\item{x, q}{
a numeric vector of quantiles.
}
\item{p}{
a numeric vector of probabilities.
}
\item{n}{
number of observations to simulate.
}
\item{mean}{
location parameter.
}
\item{sd}{
scale parameter.
}
\item{nu}{
shape parameter (degrees of freedom).
}
\item{log}{
logical; if \code{TRUE}, densities are given as log densities.
}
}
\details{
The standardized Student-t distribution is defined so that for a given
\code{sd} it has the same variance, \code{sd^2}, for all degrees of
freedom. For comparison, the variance of the usual Student-t
distribution is \code{nu/(nu-2)}, where \code{nu} is the degrees of
freedom. The usual Student-t distribution is obtained by setting
\code{sd = sqrt(nu/(nu - 2))}.
Argument \code{nu} must be greater than 2. Although there is a default
value for \code{nu}, it is rather arbitrary and relying on it is
strongly discouraged.
\code{dstd} computes the density,
\code{pstd} the distribution function,
\code{qstd} the quantile function,
and
\code{rstd} generates random deviates from the standardized-t
distribution with the specified parameters.
}
\value{
numeric vector
}
\references{
Fernandez C., Steel M.F.J. (2000);
\emph{On Bayesian Modelling of Fat Tails and Skewness},
Preprint, 31 pages.
Wuertz D., Chalabi Y. and Luksan L. (2006);
\emph{Parameter estimation of ARMA models with GARCH/APARCH errors: An R
and SPlus software implementation},
Preprint, 41 pages,
\url{https://github.com/GeoBosh/fGarchDoc/blob/master/WurtzEtAlGarch.pdf}
}
\author{
Diethelm Wuertz for the Rmetrics \R-port
}
\seealso{
\code{\link{stdFit}} (fit).
\code{\link{stdSlider}} (visualize),
\code{\link{absMoments}}
}
\examples{
## std -
pstd(1, sd = sqrt(5/(5-2)), nu = 5) == pt(1, df = 5) # TRUE
par(mfrow = c(2, 2))
set.seed(1953)
r = rstd(n = 1000)
plot(r, type = "l", main = "sstd", col = "steelblue")
# Plot empirical density and compare with true density:
hist(r, n = 25, probability = TRUE, border = "white", col = "steelblue")
box()
x = seq(min(r), max(r), length = 201)
lines(x, dstd(x), lwd = 2)
# Plot df and compare with true df:
plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue",
ylab = "Probability")
lines(x, pstd(x), lwd = 2)
# Compute quantiles:
round(qstd(pstd(q = seq(-1, 5, by = 1))), digits = 6)
}
\keyword{distribution}
|
