A formula to improve score test statistics

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Abstract
Bartlett corrections are widely used to improve the chi–squared approximation to the log–likelihood ratio statistic. Bartlett-type corrections are a recent extension of Bartlett corrections to statistics other than the log–likelihood ratio. For any test statistic having a chi-squared distribution asymptotically, Cordeiro and Ferrari (1991) derived a polynomial transformation in the original test statistic whose distribution function to O(n -1)is chi–squared, but it does not preserve monotonicity, where n is the sample size. We present a formula for the Bartlett-type correction which is equivalent to order n -1 to this polynomial and is a monotone transformation of the original statistic. Our formula has some advantages for obtaining improved score test statistics. The formula is investigated by means of simulation and is compared with both the polynomial transfondation and the original score statistic.