Mle-equivariance, data transformations and invariant tests of fit

TL;DR

This paper introduces data transformations that preserve certain distribution classes and demonstrate that maximum likelihood estimators are invariant under these transformations, simplifying goodness-of-fit testing procedures.

Contribution

It establishes a framework linking data transformations, invariance, and MLE behavior, and applies it to improve goodness-of-fit tests, including a Monte Carlo study and real-data examples.

Findings

MLE estimators are invariant under specific data transformations.

Goodness-of-fit tests can be simplified using invariant transformations.

Empirical validation with Weibull distribution data.

Abstract

We define data transformations that leave certain classes of distributions invariant, while acting in a specific manner upon the parameters of the said distributions. It is shown that under such transformations the maximum likelihood estimators behave in exactly the same way as the parameters being estimated. As a consequence goodness--of--fit tests based on standardized data obtained through the inverse of this invariant data--transformation reduce to the case of testing a standard member of the family with fixed parameter values. While presenting our results, we also provide a selective review of the subject of equivariant estimators always in connection to invariant goodness--of--fit tests. A small Monte Carlo study is presented for the special case of testing for the Weibull distribution, along with real--data illustrations.