fGarch (version 290.76)

sstd: Skew Student-t Distribution and Parameter Estimation

Description

Functions to compute density, distribution function, quantile function and to generate random variates for the standardized skew Student-t distribution. In addition maximum likelihood estimators are available to fit the parameters of the distribution. The functions are: ll{ [dpqr]std Symmetric Student-t Distribution, [dpqr]sstd Skew Student-t Distribution, stdFit MLE parameter fit for a Sudent-t distribution, sstdFit MLE parameter fit for a skew Sudent-t distribution, sstdSlider Displays interactively skew GED distribution. }

Usage

dstd(x, mean = 0, sd = 1, nu = 5)
pstd(q, mean = 0, sd = 1, nu = 5)
qstd(p, mean = 0, sd = 1, nu = 5)
rstd(n, mean = 0, sd = 1, nu = 5)
dsstd(x, mean = 0, sd = 1, nu = 5, xi = 1.5)
psstd(q, mean = 0, sd = 1, nu = 5, xi = 1.5)
qsstd(p, mean = 0, sd = 1, nu = 5, xi = 1.5)
rsstd(n, mean = 0, sd = 1, nu = 5, xi = 1.5)
stdFit(x, ...)
sstdFit(x, ...)
sstdSlider(type = c("dist", "rand"))

Arguments

mean, sd, nu, xi

location parameter mean, scale parameter sd, shape parameter nu, skewness parameter xi.

n

the number of observations.

p

a numeric vector of probabilities.

type

a character string denoting which interactive plot should be displayed. Either a distribution plot type="dist", the default value, or a random variates plot, type="rand".

x, q

a numeric vector of quantiles.

...

parameters parsed to the optimization function nlm.

Value

• d* returns the density, p* returns the distribution function, q* returns the quantile function, and r* generates random deviates, all values are numeric vectors. [s]stdFit returns a list with the following components:
• par
  The best set of parameters found.
• objective
  The value of objective corresponding to par.
• convergence
  An integer code. 0 indicates successful convergence.
• message
  A character string giving any additional information returned by the optimizer, or NULL. For details, see PORT documentation.
• iterations
  Number of iterations performed.
• evaluations
  Number of objective function and gradient function evaluations.

Details

Parameter Estimation: The function nlminb is used to minimize the "negative" maximum log-likelihood function. nlminb carries out a minimization using a Newton-type algorithm.

References

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

Examples

## sstd -
   par(mfrow = c(2, 2))
   set.seed(1953)
   r = rsstd(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, dsstd(x), lwd = 2)
   
   # Plot df and compare with true df:
   plot(sort(r), (1:1000/1000), main = "Probability", col = "steelblue",
     ylab = "Probability")
   lines(x, psstd(x), lwd = 2)
   
   # Compute quantiles:
   round(qsstd(psstd(q = seq(-1, 5, by = 1))), digits = 6)
       
## sstdFit -
   sstdFit(r, print.level = 2)