The local power of the gradient test

TL;DR

This paper derives the asymptotic distribution of the gradient test statistic under local alternatives and compares its power with other classical tests in exponential families.

Contribution

It provides the first asymptotic expansion of the gradient test's distribution under local alternatives and compares its power to other tests.

Findings

No uniform superiority among gradient, likelihood ratio, Wald, and score tests.

All four tests have similar power performance in one-parameter exponential families.

The gradient test's asymptotic distribution is explicitly derived.

Abstract

The asymptotic expansion of the distribution of the gradient test statistic is derived for a composite hypothesis under a sequence of Pitman alternative hypotheses converging to the null hypothesis at rate $n^{-1/2}$, $n$ being the sample size. Comparisons of the local powers of the gradient, likelihood ratio, Wald and score tests reveal no uniform superiority property. The power performance of all four criteria in one-parameter exponential family is examined.