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

Abstract

We obtain an asymptotic expansion for the null distribution function of the gradient 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 [10]. 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. Monte Carlo simulation experiments and various examples are presented and discussed.