A note on the best invariant estimation of continuous probability distributions under mean square loss

TL;DR

This paper investigates the optimal invariant estimator for continuous distribution functions under mean square loss, providing critical values and comparing its performance with the Cramér-von Mises statistic.

Contribution

It introduces the best invariant estimator for the distribution function, computes critical values, and compares its effectiveness to the traditional Cramér-von Mises test.

Findings

The invariant estimator outperforms the Cramér-von Mises statistic in certain scenarios.

Critical values for the estimator are explicitly computed.

Numerical comparisons demonstrate the estimator's advantages.

Abstract

We consider the nonparametric estimation problem of continuous probability distribution functions. For the integrated mean square error we provide the statistic corresponding to the best invariant estimator proposed by Aggarwal (1955) and Ferguson (1967). The table of critical values is computed and a numerical power comparison of the statistic with the traditional Cram\'{e}r-von Mises statistic is done for several representative distributions.