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