Computing the asymptotic power of a Euclidean-distance test for goodness-of-fit

TL;DR

This paper analyzes the asymptotic power of a Euclidean-distance goodness-of-fit test, providing a numerical method to evaluate its effectiveness against large-sample alternatives and illustrating its performance with examples.

Contribution

It introduces an efficient numerical approach to assess the asymptotic power of the Euclidean-distance goodness-of-fit test for large samples, specifically under certain alternative distributions.

Findings

Provides a numerical method for cdf evaluation of the test statistic

Plots asymptotic power curves for various examples

Demonstrates the test's effectiveness in large-sample regimes

Abstract

A natural (yet unconventional) test for goodness-of-fit measures the discrepancy between the model and empirical distributions via their Euclidean distance (or, equivalently, via its square). The present paper characterizes the statistical power of such a test against a family of alternative distributions, in the limit that the number of observations is large, with every alternative departing from the model in the same direction. Specifically, the paper provides an efficient numerical method for evaluating the cumulative distribution function (cdf) of the square of the Euclidean distance between the model and empirical distributions under the alternatives, in the limit that the number of observations is large. The paper illustrates the scheme by plotting the asymptotic power (as a function of the significance level) for several examples.