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

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.