Maximum-Likelihood Estimation Procedures and Associated Tests of Significance

Summary The essence of many statistical problems, including most standard techniques, is to test whether or not the unknown parameters of an appropriate statistical model satisfy certain restrictions; and the outcome of such a test dictates whether it is necessary to provide estimates of these parameters which also satisfy the restrictions. In this paper we discuss and illustrate the relative merits, as practical tools for the consulting statistician, of two large-sample techniques of wide applicability to such situations: (i) unrestricted maximum-likelihood estimation with its associated Wald test, (ii) restricted maximum-likelihood estimation with its associated Lagrange-multiplier test. The discussion falls into two main sections corresponding to two methods of specifying restrictions, as constraint equations in the parameters, or as freedom equations expressing the parameters in terms of a second smaller set of parameters. The methods are modified by a simple device to apply to the case where constraints on the parameters are necessary to allow their identification.