Covariance Estimators and Adjusted Pseudo Maximum Likelihood Method

In this paper, we introduce the concept of covariance estimators. These estimators are obtained by solving the empirical counterpart of some noncorrelation conditions characterizing the interest parameters. The statistical properties of the covariance estimators are studied in a general framework and a special attention is given to a large class of decomposable models. In a second step, we introduce the concept of adjusted pseudo maximum likelihood estimators. These estimators are the solutions of centered pseudo maximum likelihood equations, Le. equations in which the bias due to the non zero expected score is corrected by introducing terms involving its empirical mean. Then, we give the simplified form of the asymptotic covariance matrix of the covariance estimators of the adjusted PML approach for decomposable models. In particular we analyse in details models with translation and scale parameters. As a by-product it is proved that misspecified pseudo-maximum likelihood methods lead to consistent estimators of the coefficients of the explanatory variables as soon as some auxiliary translation and scale parameters have been introduced in the model.