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

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