Density function, distribution function, quantiles and random number generation for the skew-t (ST) distribution
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vector of quantiles. Missing values (
vector of probabililities.
vector of location parameters.
vector of scale parameters; must be positive.
vector of slant parameters. With
a single positive value representing the degrees of freedom;
it can be non-integer. Default value is
a vector of length 4, whose elements represent location, scale
(positive), slant and degrees of freedom, respectively. If
a positive integer representing the sample size.
logical; if TRUE, densities are given as log-densities
a scalar value which regulates the accuracy of the result of
an integer value between
additional parameters passed to
dst), probability (
pst), quantiles (
and random sample (
rst) from the skew-t distribution with given
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)
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.
nu=Inf, it reduces to the skew-normal distribution.
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
For evaluation of
pst, and so indirectly of
qst, four different methods are employed.
Method 1 consists in using
pmst with dimension
Method 2 applies
integrate to the density function
Method 3 again uses
integrate too but with a different integrand,
as given in Section 4.2 of Azzalini & Capitanio (2003), full version of
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
Of these, Method 1 and 4 are only suitable for integer values of
Method 4 becomes progressively less efficient as
because its value corresponds to the number of nested calls, but the
decay of efficiency is slower for larger values of
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.
qst is called with
nu>1e4, computation is transferred to
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.
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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)