Maximum likelihood estimation when modelling in terms of constraints

Abstract

A maximum likelihood (ML) estimation procedure is developed to find the mean of the exponential family subject to the constraints g(μ) = 0, where g is a vector valued function of the mean μ, satisfying the usual regularity constraints. This result forms the basis of an iterative procedure whereby the ML estimates of the mean values of a particular model are found. The constraints on the mean vector may be linear or non-linear. The application of the procedure provides a very :flexible method for modelling data either directly in terms of certain constraints or in terms of the implied constraints of the appropriate model. The approach accommodates any choice of model assuming any predetermined distribution of the error terms, provided that the covariance matrix of the error terms can be computed.