An optimal estimating equation based on the first three cumulants

For improving on quasilikelihood estimation two types of quadratic estimating equations have been proposed, one based on the Edgeworth expansion, the other on the generalisation of the quasi-score. The first requires that the skewness of observations has a small departure from the exponential family; the second requires the knowledge of both skewness and kurtosis. We introduce an optimal quadratic estimating equation applicable when the skewness is not small and the kurtosis is unknown. Apart from optimality, the manner in which skewness is incorporated ensures that its misspecification does not affect √n-consistency, and that the estimator enjoys an invariance property akin to that of the bias-corrected maximum likelihood estimate. Simulations indicate a solid improvement in accuracy.