likelihoodfunctions.t: Functions for Maximum Likelihood Estimation of Student's T...

In christopherggreen/cggmisc: Christopher G Green's Miscellaneous Functions

Description

These functions are useful for fitting a location-scale model based on Student's T distribution via maximum likelihood.

Usage

likelihood.t(x, obs)
score.t(x, obs)
info.exp.t(x)
asy.var.t(x)

Arguments

x	vector (of length 3) of parameters for the T distribution. The first entry is the degrees of freedom parameter nu; the second entry is the location parameter mu; and the third entry is the scale parameter sigma.
obs	a vector of observations. obs should not include missing or non-finite values.

Details

The distributional model in use here assumes that the random variable X follows a location-scale model based on the Student's T distribution; that is,

(X - mu)/(sigma) ~ T_{nu},

where mu and sigma are location and scale parameters, respectively, and nu is the degrees of freedom parameter of the T distribution.

Value

likelihood.t returns the value of the likelihood function for the T distribution evaluated at the given parameter set x and observations obs.

score.t returns a vector of length 3 containing the values of the partial derivatives of the likelihood function with respect to each of the parameters.

info.exp.t returns the 3-by-3 expected information matrix evaluated at the given parameter set x.

asy.var.t returns the 3-by-3 asymptotic variance matrix evaluated at the given parameter set x.

Author(s)

Christopher G. Green [e-mail]

References

The likelihood and score functions used were calculated as part of one of the author's qualifying exams. See Chapter 3 of the paper below.

Green, C. G. (2005) Heavy-Tailed Distributions in Finance: An Empirical Study. Qualifying Exam, Department of Statitics, University of Washington. Available from http://students.washington.edu/cggreen/uwstat/papers/computing_prelim_2005_green.pdf

More general calculations for the information matrix of general multivariate elliptic distributions can be found in the work of Ann F. S. Mitchell.

Mitchell, Ann F. S. (1989) The Information Matrix, Skewness Tensor and alpha-Connections for the General Multivariate Elliptic Distribution. Annals of the Institute for Statistics and Math, 41 (2), pp 289-304.

See Also

fit.mle.t

There are several other packages that offer functions to fit the Student's t and related distributions using other approaches.

• The MASS package contains the fitdistr function, which can fit many univariate distributions via maximum likelihood.

• The PearsonDS package fits general Pearson distributions, of which the Student's t distribution is a special case;

• The GeneralizedHyperbolic package fits the generalized hyperbolic distribution, of which the Student's t distribution is a special case.

• The SkewHyperbolic package fits the skew hyperbolic distribution, of which the skewed t distribution is a special case.

• The fGarch package fits the Student's t distribution via the stdFit function.

• The function tFit in the fBasics package fits the Student's t distribution.

Examples

a <- -5 + 2*rt(1000, df=4)
# should be close to (0,0,0) indicating
# fit.mle.t found an extremum
score.t(unlist(fit.mle.t(a)),a)

christopherggreen/cggmisc documentation built on May 13, 2019, 7:04 p.m.