  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.
