General matrix formulae for computing Bartlett corrections

We give general formulae using matrix notation for computing Bartlett corrections for the likelihood ratio statistics in two quite distinct situations. In the first, we consider the test of a null hypothesis, which specifies a parameter vector, in the presence of nuisance parameters. In the second, we are interested in testing a scalar parameter which is orthogonal to the remaining nuisance parameters. The formulae have advantages for numerical purposes because they require only simple operations on matrices and vectors. They are also useful in connexion with algebraic computing packages to obtain closed-form Bartlett corrections in a variety of important problems. The practical use of such formulae is illustrated.