Parameter structures in the exponential class

Abstract

Maximum likelihood estimation of parameter structures in the exponential class is considered by using a procedure for maximum likelihood estimation under constraints. This procedure does not require maximization of the likelihood function or derivation of the likelihood equations. Constraints are imposed on the expected values of the canonical statistics and not specifically on the parameters themselves. It provides a solution to maximum likelihood estimation in a restricted parameter space. Maximum likelihood estimation under constraints provides a flexible procedure for the simultaneous modelling of specific structures of means and variances - in both the univariate and multivariate case. In the univariate case, it includes linear models for means and variances, heteroskedastic linear regression, analysis of variance (the balanced and unbalanced case where the assumption of homoscedasticity is not met), gamma regression and linear mixed models. Heteroscedastic and/or autocorrelated disturbances in econometric applications are dealt with by treating correlated observations as a single multivariate observation. The same procedure is used for maximum likelihood estimation of parameter structures in the case of multivariate normal samples. The procedure provides a new statistical methodology for the analysis of specific structures in mean vectors and covariance matrices - including the case where the sample size is small relative to the dimension of the observations. Special cases include different variations of the Behrens-Fisher problem, proportional covariance matrices and proportional mean vectors. Maximum likelihood estimation under constraints as a general approach to estimation in a variety of problems concerning covariance matrices, is also considered. Estimation of Wishart matrices is a special case of the general procedure. Specific structures are illustrated with real data examples. Maximum likelihood estimation under constraints provides a convenient method to obtain maximum likelihood estimates of mean and (co)variance structures directly - also in problematic situations where the likelihood equations need to be solved numerically.