st.cumulants {sn} | R Documentation |
Cumulants of the skew-t distribution and inverse matching
st.cumulants(location = 0, scale = 1, shape = 0, df = Inf, n = 4) st.cumulants(dp=, n = 4) st.cumulants.inversion(cum, abstol = 1e-08)
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) |
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.
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
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. Note however, that for
stability reasons, this is not adopted as the standard method
for producing initial values of MLE search.
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.
sn.cumulants
,dst
,
sample.centralmoments
, optim
,
nlminb
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