R: Skew-t Distribution

dst {sn} | R Documentation

## 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, ...)
    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 (`NA`s) 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` | 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, depending 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

    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 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, 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

    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)

[Package sn version 1.5-3 Index]