Gradient statistic: higher-order asymptotics and Bartlett-type correction

TL;DR

This paper develops higher-order asymptotic expansions for the gradient statistic's null distribution, proposing a Bartlett-type correction to improve its chi-square approximation in hypothesis testing.

Contribution

It introduces a novel Bartlett-type correction for the gradient statistic using a Bayesian asymptotic expansion, enhancing accuracy in finite samples.

Findings

The corrected statistic follows a chi-square distribution with error of order o(n^{-1})

Monte Carlo simulations confirm improved approximation accuracy

Examples demonstrate practical applicability of the correction method

Abstract

We obtain an asymptotic expansion for the null distribution function of thegradient statistic for testing composite null hypotheses in the presence of nuisance parameters. The expansion is derived using a Bayesian route based on the shrinkage argument described in Ghosh and Mukerjee (1991). Using this expansion, we propose a Bartlett-type corrected gradient statistic with chi-square distribution up to an error of order o(n^{-1}) under the null hypothesis. Further, we also use the expansion to modify the percentage points of the large sample reference chi-square distribution. A small Monte Carlo experiment and various examples are presented and discussed.