Local power of the LR, Wald, score and gradient tests in dispersion models

TL;DR

This paper derives detailed asymptotic distributions for various statistical tests in dispersion models, compares their finite-sample performance through simulations, and applies findings to real data.

Contribution

It provides the first comprehensive asymptotic expansions for likelihood ratio, Wald, score, and gradient tests in dispersion models, including their nonnull distributions.

Findings

No test is uniformly superior for regression parameter testing.

Finite-sample performance varies among tests, as shown by simulations.

Empirical application demonstrates practical relevance of the theoretical results.

Abstract

We derive asymptotic expansions up to order $n^{-1/2}$ for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of dispersion models, under a sequence of Pitman alternatives. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing the precision parameter. Based on these nonnull asymptotic expansions it is shown that there is no uniform superiority of one test with respect to the others for testing a subset of regression parameters. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.