Solved – Student t-distribution parameter/s and MLE

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.

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).

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