Asymptotic expansions for the estimators of Lagrange multipliers and associated parameters by the maximum likelihood and weighted score methods

In this paper, inverse expansions of parameter estimators are given in terms of their true values, where the estimators are obtained by the maximum likelihood and weighted score methods with constraints placed on the parameters using Lagrange multipliers. The corresponding expansions for estimated Lagrange multipliers are also given. These expansions are derived before and after studentization. The results with studentization give one-sided confidence intervals for the parameters up to third-order accuracy. As an application of the weighted score method, a modified Jeffreys prior to remove the asymptotic biases of the Lagrange multipliers as well as the parameter estimators is obtained under canonical parametrization in the exponential family.