AST: Asymmetric Student-t Distribution

In dan9401/st: Asymmetric Student t-Distributions

Asymmetric Student-t Distribution

Description

Probablity density function(PDF), Cumulative distribution function(CDF), Quantile function and Random generation of the AST distribution

Usage

dast(x, mu = 0, s = 1, alpha = 0.5, nu1 = Inf, nu2 = Inf, pars = NULL)

past(q, mu = 0, s = 1, alpha = 0.5, nu1 = Inf, nu2 = Inf, pars = NULL)

qast(p, mu = 0, s = 1, alpha = 0.5, nu1 = Inf, nu2 = Inf, pars = NULL)

rast(n, mu = 0, s = 1, alpha = 0.5, nu1 = Inf, nu2 = Inf, pars = NULL)

Arguments

x, q	vector of quantiles
mu	location parameter
s	scale parameter, s > 0
alpha	skewness parameter, 0 < alpha < 1
nu1	degrees of freedom / tail parameter for the left tail, nu1 > 0
nu2	degrees of freedom / tail parameter for the right tail, nu2 > 0
pars	a vector that contains mu, s, alpha, nu1, nu2, if pars is specified, mu, s, alpha, nu1, nu2 should not be specified
p	vector of probablilities
n	number of observations for random generation

Details

The 'asymmetric' in AST distribution, not only suggests skewness in the distribution, but also the asymmetry in the two tail powers of the distribution.

• Location parameter mu is the mode, but not necessarily the mean of the distribution.

• Scale parameter s is not necessarily the standard deviation.

• The distribution skews to the right when the skewness parameter alpha < 0.5, skews to the left when alpha > 0.5.

• The location paramter mu always locates at the \alpha-th percentile of the distribution. The two degrees of freedom / tail parameters each controls one tail of the distribution, separated at the location paramter mu. The left tail parameter nu1 only affects the left half(0th to \alpha-th percentile) of the distribution, while the right tail paramter nu2 only affects the right half(\alpha-th to 100-th percentile) of the distribution.

Value

dast gives the density, past gives the distribution function, qast gives the quantile function, and rast generates random samples for AST distribution.

References

Zhu, D., & Galbraith, J. W. (2010). A generalized asymmetric Student-t distribution with application to financial econometrics. Journal of Econometrics, 157(2), 297-305. \Sexpr[results=rd]{tools:::Rd_expr_doi("https://doi.org/10.1016/j.jeconom.2010.01.013")}

Examples

d <- dast(0.12, 0.12, 0.6, 0.6, 3, 5)
p <- past(0.12, 0.12, 0.6, 0.6, 3, 5)
q <- qast(0.4, 0.12, 0.6, 0.6, 3, 5)
data <- rast(1000, 0.12, 0.6, 0.6, 3, 5)
hist(data, breaks = 50, probability = TRUE)

pars <- c(0.12, 0.6, 0.6, 3, 5)
x <- seq(-3, 3, 0.01)
y <- dast(x, pars = pars)
lines(x, y, col = 4)

dan9401/st documentation built on Jan. 6, 2025, 8:28 p.m.