Matrix formulae for computing improved score tests

We give simple matrix formulae for computing Bartlett-type corrections for the score statistic for testing a general composite null hypothesis. These formulae, involving only simple operations on matrices and vectors, have computational advantages over the tensor notation of Harris' (1985) asymptotic expansion for the null distribution of the score statistic. Two situations are discussed. First, the test of a general composite null hypothesis is considered. Second, the test of a scalar parameter is studied under the assumption of global orthogonality of the parameter of interest and the vector of remaining parameters. The formulae can be implemented in algebraic computer systems in order to obtain closed-form expressions for the corrections in a wide range of problems. Two illustrative examples are presented.