dst {sn}

Skew-t Distribution

Description

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

Usage

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, lower.tail=TRUE, 
    log.p=FALSE, ...)
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

a positive integer representing the sample size.

log, log.p

logical; if TRUE, densities are given as log-densities and probabilities p are given as log(p)

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 5 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, depending on the other arguments.

lower.tail

logical; if TRUE (default), probabilities are P\{X\le x\}, otherwise P\{X\ge x\}

...

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

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. In all the cases, the actual computations are performed for the normalized values z=(x-xi)/omega). 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. Method 5 is targeted to tail probabilities only, and it returns NAs for non-extreme x values (those with abs(z)<=20); it is based on expressions given in Complement 4.4 of Azzalini and Capitanio (2014). Method 1 and 4 are only suitable for integer values of nu. Method 4 becomes progressively less efficient as nu increases, because the value of nu determines 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, z. 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, the 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 https://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

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)