A general expression for second-order covariance matrices-an application to dispersion models

We present a general expression that allows the calculation of both the n(-2) asymptotic covariance matrices of the maximum likelihood estimator (MLE) and the first-order bias corrected MLE, where n is the sample size. The formula is presented in a matrix notation which has numerical advantages since it requires only simple operations on matrices and vectors. The usefulness of the formula is to construct better Wald statistics. We apply our findings to dispersion models and develop simulation studies which show that modification in the Wald statistic effectively removes size distortions of the type I error probability with no power loss. For illustrative purposes, a real data application is considered to support our theoretical results.