VGAM (version 0.7-5)

studentt: 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.

Value

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

Details

The density function is $$f(y) = \frac{\Gamma((\nu+1)/2)}{\sqrt{\nu \pi} \Gamma(\nu/2)} \left(1 + \frac{y^2}{\nu} \right)^{-(\nu+1)/2}$$ 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.

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)