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

x, q

a numeric vector of quantiles.

p

a numeric vector of probabilities.

n

number of observations to simulate.

mean

location parameter.

sd

scale parameter.

nu

shape parameter (degrees of freedom).

log

logical; if TRUE, densities are given as log densities.

## Details

The standardized Student-t distribution is defined so that for a given sd it has the same variance, sd^2, for all degrees of freedom. For comparison, the variance of the usual Student-t distribution is nu/(nu-2), where nu is the degrees of freedom. The usual Student-t distribution is obtained by setting sd = sqrt(nu/(nu - 2)).

Argument nu must be greater than 2. Although there is a default value for nu, it is rather arbitrary and relying on it is strongly discouraged.

dstd computes the density, pstd the distribution function, qstd the quantile function, and rstd generates random deviates from the standardized-t distribution with the specified parameters.

numeric vector

## References

Fernandez C., Steel M.F.J. (2000); On Bayesian Modelling of Fat Tails and Skewness, Preprint, 31 pages.

Wuertz D., Chalabi Y. and Luksan L. (2006); Parameter estimation of ARMA models with GARCH/APARCH errors: An R and SPlus software implementation, Preprint, 41 pages, https://github.com/GeoBosh/fGarchDoc/blob/master/WurtzEtAlGarch.pdf

## Author

Diethelm Wuertz for the Rmetrics R-port

stdFit (fit). stdSlider (visualize),

absMoments

## Examples

## std -

pstd(1, sd = sqrt(5/(5-2)), nu = 5) == pt(1, df = 5) # TRUE
#> [1] 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)
#> [1] -1  0  1  2  3  4  5