Skew-t Distribution

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Description

Density function, distribution function, quantiles and random number generation for the skew-t (ST) distribution

Usage

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dst(x, xi=0, omega=1, alpha=0, nu=Inf, dp=NULL, log=FALSE) 
pst(x, xi=0, omega=1, alpha=0, nu=Inf, dp=NULL, method=0, ...)
qst(p, xi=0, omega=1, alpha=0, nu=Inf, tol=1e-08, dp=NULL, method=0, ...)
rst(n=1, xi=0, omega=1, alpha=0, nu=Inf, dp=NULL)

Arguments

x

vector of quantiles. Missing values (NAs) are allowed.

p

vector of probabililities.

xi

vector of location parameters.

omega

vector of scale parameters; must be positive.

alpha

vector of slant parameters. With pst and qst, it must be of length 1.

nu

a single positive value representing the degrees of freedom; it can be non-integer. Default value is nu=Inf which corresponds to the skew-normal distribution.

dp

a vector of length 4, whose elements represent location, scale (positive), slant and degrees of freedom, respectively. If dp is specified, the individual parameters cannot be set.

n

sample size

log

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

tol

a scalar value which regulates the accuracy of the result of qsn, measured on the probability scale.

method

an integer value between 0 and 4 which selects the computing method; see ‘Details’ below for the meaning of these values. If method=0 (default value), an automatic choice is made among the four actual computing methods, which depends on the other arguments.

...

additional parameters passed to integrate or pmst

Value

Density (dst), probability (pst), quantiles (qst) and random sample (rst) from the skew-t distribution with given xi, omega, alpha and nu parameters.

Details

Typical usages are

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dst(x, xi=0, omega=1, alpha=0, nu=Inf, log=FALSE)
dst(x, dp=, log=FALSE)
pst(x, xi=0, omega=1, alpha=0, nu=Inf, method=0, ...)
pst(x, dp=, log=FALSE)
qst(p, xi=0, omega=1, alpha=0, nu=Inf, tol=1e-8,  method=0, ...)
qst(x, dp=, log=FALSE)
rst(n=1, xi=0, omega=1, alpha=0, nu=Inf)
rst(x, dp=, log=FALSE)

Background

The family of skew-t distributions is an extension of the Student's t family, via the introduction of a alpha parameter which regulates skewness; when alpha=0, the skew-t distribution reduces to the usual Student's t distribution. When nu=Inf, it reduces to the skew-normal distribution. When nu=1, it reduces to a form of skew-Cauchy distribution. See Chapter 4 of Azzalini & Capitanio (2014) for additional information. A multivariate version of the distribution exists; see dmst.

Details

For evaluation of pst, and so indirectly of qst, four different methods are employed. Method 1 consists in using pmst with dimension d=1. Method 2 applies integrate to the density function dst. Method 3 again uses integrate too but with a different integrand, as given in Section 4.2 of Azzalini & Capitanio (2003), full version of the paper. Method 4 consists in the recursive procedure of Jamalizadeh, Khosravi and Balakrishnan (2009), which is recalled in Complement 4.3 on Azzalini & Capitanio (2014); the recursion over nu starts from the explicit expression for nu=1 given by psc. Of these, Method 1 and 4 are only suitable for integer values of nu. Method 4 becomes progressively less efficient as nu increases, because its value corresponds to the number of nested calls, but the decay of efficiency is slower for larger values of length(x). If the default argument value method=0 is retained, an automatic choice among the above four methods is made, which depends on the values of nu, alpha, length(x). The numerical accuracy of methods 1, 2 and 3 can be regulated via the ... argument, while method 4 is conceptually exact, up to machine precision.

If qst is called with nu>1e4, computation is transferred to qsn.

References

Azzalini, A. and Capitanio, A. (2003). Distributions generated by perturbation of symmetry with emphasis on a multivariate skew-t distribution. J.Roy. Statist. Soc. B 65, 367–389. Full version of the paper at http://arXiv.org/abs/0911.2342.

Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-normal and Related Families. Cambridge University Press, IMS Monographs series.

Jamalizadeh, A., Khosravi, M., and Balakrishnan, N. (2009). Recurrence relations for distributions of a skew-$t$ and a linear combination of order statistics from a bivariate-$t$. Comp. Statist. Data An. 53, 847–852.

See Also

dmst, dsn, dsc

Examples

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pdf <- dst(seq(-4, 4, by=0.1), alpha=3, nu=5)
rnd <- rst(100, 5, 2, -5, 8)
q <- qst(c(0.25, 0.50, 0.75), alpha=3, nu=5)
pst(q, alpha=3, nu=5)  # must give back c(0.25, 0.50, 0.75)
#
p1 <- pst(x=seq(-3,3, by=1), dp=c(0,1,pi, 3.5))
p2 <- pst(x=seq(-3,3, by=1), dp=c(0,1,pi, 3.5), method=2, rel.tol=1e-9)

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