Comparisons of asymptotic efficiency of least squares and some variants for some common statistical models.

In completely specified models, where explicit formulae are derivable for the probabilities of observed events, the method of maximum likelihood has come to be the most widely employed method of estimation due to its optimal asymptotic properties. We discuss this method along with a review of the theory in Chapter 1. There is also some introductory material on asymptotic theory included in the first part of this chapter. There is currently growing interest in the use of estimating equations, other than those based on a full likelihood specification, for estimating parameters. In many cases this approach is particularly useful because the true likelihood is unknown or intractable. For example, maximum quasi-likelihood estimation is widely applied via the GLIM computer package. It is therefore of interest to examine the efficiency of such alternative methods relative to maximum likelihood. This is the subject of the present study in which a certain class of estimating functions is applied to some common data analysis situations. In Chapter 2, we discuss the previous development of these methods and their asymptotic properties, in particular the asymptotic variances of the resulting estimators. We consider models from widely used families, of both exponential and non-exponential types, to compare the asymptotic efficiencies of the methods discussed in Chapter 2. In Chapter 3, we compare the efficiencies, relative to MLE, in the case of independently identically distributed (iid) observations. In the first part we consider the exponential family models, and in the second part the non-exponential family models. Chapters 4 and 5 are concerned with similar comparisons for the non-iid case. Chapter 4 presents results when the models are taken from the exponential family, and in Chapter 5 the results are compared for non-exponential family models. Two types of situations, (i) when some parameters are assumed known, and (ii) when all parameters are unknown, will be considered. In the first case all the methods perform well, but in the second case a singularity problem removes two of the methods from comparison. In Chapter 6 we suggest further investigations for the estimating equations and connections between the methods are developed in some cases. The concluding results of previous chapters are also examined in detail