VGAM (version 0.7-3)

studentt: Student t Distribution

Description

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 $$f(y)=\frac{\mathrm{\Gamma }((\nu +1)/2)}{\sqrt{\nu \pi }\mathrm{\Gamma }(\nu /2)}{(1+\frac{y^2}{\nu })}^{-(\nu +1)/2}$$ 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.

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(1)))
fit = vglm(y ~ 1, studentt)
coef(fit, matrix=TRUE)
Coef(fit)