Student t-distribution parameter/s and MLE

Question

So I always thought of the Student t-distribution as having only 1 parameter, v, the degrees of freedom (as described by wikipedia). When I searched however on how to find the MLE of v I keep coming across questions mentioning mu and sigma as parameters as well. So

1) Is the Student-t Distribution a special case of some "t-distribution"'s family, maybe where mu=0 and sigma=1.

2) Is there such a thing as solely estimating an MLE for the degree of freedom. I.e. is there a closed form solution to the MLE of the degree of freedom parameter.

Answer 1

When used as the basis for a test or confidence interval the variable for which the t-distribution is being applied has already been studentized -- so there's clearly no need for location and scale parameters in that case. This is regarded as the "standard" form akin to the standard normal.

When used for modelling data, the location-scale family based on the same distribution is used. This is no different from the distinction between the standard normal that arises in testing and CI applications (e.g. in large-sample situations) and the location-scale family of the normal that arises in modelling applications.

Usually it's clear from context which is needed -- the standard form or the location-scale-family form.

There is no closed-form ML estimate of the df parameter.

You indicate you're dealing with the standard t. For the standard t the df is readily estimated from the variance in closed form -- essentially method-of-moments (but that estimate can be quite noisy in practice, and doesn't work at all for small df).

Answer 2

You can add location and scale parameters to any symmetric distribution with support in ${\mathbb R}$, in particular, the Student-$t$. The MLEs do not exist in closed-form, but there are many packages to do so.

In particular, the R library MASS contains the command fitdistr that can be used for this purpose:

library(MASS)
set.seed(123)
data = 3 + 0.75*rt(1000,df=2)
fitdistr(data,"t")

Answer 3

Addressing your second question, if the location and scale parameters are known, then there is a nonlinear equation that defines the relationship between the scale and the degree of freedom (Dof) based on the geometric mean of the samples. See the following paper for details of the relationship. In this paper, I assumed that the shape (inverse of the DoF) is known and that the scale is being estimated but the reverse is also possible.

https://www.sciencedirect.com/science/article/pii/S0378437118311816

In a more recent paper, I showed that the location and scale can be estimated with closed-form linear equations. Once these are estimated, the result above can be used to estimate the DoF. Thus all the parameters can be estimated in closed-form, and the result is shown to achieve the maximum likelihood.

The new method works by selecting Independent Approximates (IAs). The IAs are subsamples of pairs or triplets that are approximately equal. These subsamples are distributed by the square or cube power of the original distribution respectively. The IA-pairs are guaranteed to have a mean and can be used to estimate the location. The IA-triplets are guaranteed to have a finite second moment and can be use to estimate the scale.

https://arxiv.org/abs/2012.11026