studentt {VGAM}R Documentation

Student t Distribution

Description

Estimation of the degrees of freedom for a Student t distribution.

Usage

studentt(link.df = "loglog", earg=list(), idf=NULL, nsimEIM=100)

Arguments

link.df Parameter link function for the degrees of freedom nu. See Links for more choices. The default ensures the parameter is greater than unity.
earg List. Extra argument for the link. See earg in Links for general information.
idf Optional initial value. If given, its value must be greater than 1. The default is to compute an initial value internally.
nsimEIM See CommonVGAMffArguments.

Details

The density function is

f(y) = (gamma((nu+1)/2) / (sqrt(nu*pi) gamma(nu/2))) * (1 + y^2 / nu)^{-(nu+1)/2}

for all real y. Then E(Y)=0 if nu>1 (returned as the fitted values), and Var(Y)= nu/(nu-2) for nu > 2. When nu=1 then the Student t-distribution corresponds to the standard Cauchy distribution. The degrees of freedom is treated as a parameter to be estimated, and as real and not integer.

Simulation is used to estimate the EIM. Consequently the results will be reproducible only if a function such as set.seed is used. Increasing the value of nsimEIM will give more accurate results. In general convergence will be slow, especially when there are covariates.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Note

Practical experience has shown reasonably good initial values are required. If convergence failure occurs try using idf. Local solutions are also possible.

A standard normal distribution corresponds to a t distribution with infinite degrees of freedom. Consequently, if the data is close to normal, there may be convergence problems.

Author(s)

T. W. Yee

References

Evans, M., Hastings, N. and Peacock, B. (2000) Statistical Distributions, New York: Wiley-Interscience, Third edition.

Student (1908) The probable error of a mean. Biometrika, 6, 1–25.

See Also

normal1, loglog, TDist.

Examples

n = 500
y = rt(n, df=exp(exp(0.5)))
fit = vglm(y ~ 1, studentt, trace=TRUE)
coef(fit, matrix=TRUE)
Coef(fit) 

[Package VGAM version 0.7-7 Index]