Estimation of the degrees of freedom for a Student t distribution.
Usage
studentt(link.df = "loglog")
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.
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.
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.