st.cumulants {sn}R Documentation

Cumulants of the skew-t distribution

Description

Cumulants of the skew-t distribution and inverse matching

Usage

st.cumulants(location = 0, scale = 1, shape = 0, df = Inf, n = 4)
st.cumulants(dp=, n = 4)
st.cumulants.inversion(cum, abstol = 1e-08)

Arguments

location location parameter (vector)
scale scale parameter (vector)
shape shape parameter (vector)
df degrees of freedom (scalar); default is df=Inf which corresponds to the skew-normal distribution.
dp a vector of four elements, whose elements are (location, scale, shape, df) respectively. If dp is specified, then the individual parameters must not be.
n a scalar integer of the maximal order or cumulants required; it must be from 1 to 4 and smaller than df
cum a vector of 4 elements which are taken to represent the first 4 cumulants of a skew-t distribution
abstol a scalar which regulates the accuracy of the cumulants matching (default value 1e-08)

Details

Expressions of the moments and other details on the skew-t distribution are given in the reference below. These formulae are used by st.cumulants to compute the cumulants.

st.cumulants.inversion searches the set of shape and df parameters of the skew-t family, attempting to match the third and fourth cumulants with those of the supplied vector cum. This search is done numerically twice, once using optim and a second time using nlminb, to the accuracy abstol; the best matching solution is retained. If the required accuracy of the matching is not achieved by any of the two methods, a warning message is issued. After this step, the other two parameters (location and scale) are computed via simple algebra.

Value

st.cumulants computes the cumulants up to order n of the skew-t distribution with the selected parameters. The returned object is a vector of length n if the parameters are all scalar, otherwise a matrix with n columns.
st.cumulants.inversion returns a vector with the dp parameters of the matching skew-t distribution

Note

The joint use st.cumulants.inversion and sample.centralmoments allows to fit a skew-t distribution by the methods of moments; see the example below

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.

See Also

sn.cumulants,dst, sample.centralmoments, optim, nlminb

Examples

st.cumulants(shape=c(0,3,9), df=5)
cum <- st.cumulants(dp=c(10, 2, -8, 5.2))
st.cumulants.inversion(cum)
#
data(ais, package='sn')
mom <- sample.centralmoments(ais[,"bmi"])
st.cumulants.inversion(cum=c(mom[1:3],mom[4]-3*mom[2]^2))
# parameters of the fitted ST distribution

[Package sn version 0.4-0 Index]