sn (version 0.4-3)

st.cumulants: 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 (hence the second term must be positive)

abstol

a scalar which regulates the accuracy of the cumulants matching (default value 1e-08)

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

synopsis

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

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 ST distribution fitted by method of moments