Summary

In this chapter, we apply some of the basic results obtained in Chapter 2 to the problem of the asymptotic efficiency of a sequence of estimates of the unknown parameter θ ∈ Θ. The asymptotic efficiency adopted in this chapter is the one proposed by Wolfowitz [1] (see Definition 1.1). The main results obtained herein are in the nature of establishing a certain upper bound for the limiting probability of concentration of various estimates under consideration. All these results (Theorems 4.1, 5.1 and 5.2), except for one (Theorem 4.2), are established for the case that Θ is an open subset of R. In Sections 6 and 7, we consider the asymptotic efficiency from the classical point of view and show that standard results for both the one-dimensional and multidimensional case (see Theorems 6.1 and 7.1) are obtained either as special cases of, or are closely related to, the main results mentioned above.

W-efficiency – preliminaries

The classical approach of proving asymptotic efficiency (a.eff.) of estimates has been geared towards showing that an MLE is a. eff. More specifically, under suitable regularity conditions, an MLE, properly normalized, is asymptotically normal with mean zero and variance the inverse of Fisher's information number. One then considers the class of all estimates which, properly normalized, are asymptotically normal with mean zero, and calls the one with the smallest variance (of the limiting normal distribution) an a.eff. estimate, if such an estimate exists.

Summary

The main purpose of this chapter is to present the concept of contiguity (see Definition 2.1) introduced by LeCam [4] and study some alternative characterizations of it (see Theorem 6.1). In the process of doing so, some auxiliary concepts such as weak convergence, relative compactness and tightness of a sequence of probability measures are needed. These concepts are introduced in this chapter, as we go along, and also some of their relationships are stated and/or proved. For the omitted proofs, the reader is always referred to appropriate sources. The various characterizations of continguity provide alternative methods one may employ in establishing the presence (or absence) of contiguity in a given case. Some concrete examples are used for illustrative purposes.

Contiguity is a concept of ‘nearness’ of sequences of probability measures. It would then be appropriate to relate it to other more familiar concepts of the same nature such as ‘nearness’ of two sequences of probability measures expressed by the norm (L1-norm) associated with convergence in variation. By means of examples, it is shown, as one would expect, that ‘nearness’ of two sequences of probability measures expressed by contiguity is weaker than that expressed by the L1-norm.

Some attention is also focused to possible relationships between contiguity on the one hand, and mutual absolute continuity and tightness on the other. In connection with this, it is shown, by means of examples, that mutual absolute continuity of the (corresponding) measures in two sequences of probability measures need not imply contiguity of the sequences.

Summary

In this chapter, some fundamental results regarding the asymptotic expansion, in the probability sense, and also the asymptotic distribution of certain likelihood functions are derived. These results constitute the backbone of the remaining chapters in this monograph and their derivation rests heavily on the material discussed in Chapter 1.

The underlying probability model involved in our discussions is that of a Markov process satisfying certain reasonable regularity conditions. This model includes, of course, as a special but important case the model consisting of independent identically distribution (i.i.d.) random variables (r.v.s) which is assumed more often in statistical literature.

We now proceed to present a brief outline of what is done in this chapter, since the various derivations are rather involved and the reader might lose sight of the essence of the results. In Section 2, we gather together the various assumptions which are used in the present chapter and which also are basic for what is discussed in the subsequent chapters. The new element here is the assumption of differentiability in quadratic mean of the square root of the probability density function. It replaces the assumption usually made in statistical literature about the existence of two or three pointwise derivatives of the logarithm of the density. As is shown by LeCam [6], the classical Cramér type assumptions imply the one made here. The underlying conditions are then verified in a number of examples which are used throughout this monograph for illustrative purposes.

This monograph represents a modest attempt on my part to introduce the concept of contiguity, elaborate on the mathematical theory behind it, and also indicate some of its statistical applications. It lays no claim in containing an exhaustive discussion of results pertaining to contiguity. In fact, there are already new results available which, however, could not have been included in this book. It is simply the result of an attempt to make the concept of contiguity and some of its statistical applications more familiar to several kinds of research workers. These include Theoretical Statisticians, Probabilists, Mathematicians whose primary interest lies in measure theory or approximation theory, and perhaps to practitioners of Statistics as well. It is my belief that contiguity deserves more attention than it has received. I hope that this monograph will be a step in that direction, anticipating the appearance of a more comprehensive treatise on the subject.

The concept of contiguity was introduced by Professor Lucien LeCam as a criterion of nearness of sequences of probability measures. In addition to its purely mathematical interest, contiguity is a powerful and very useful tool in Statistics, when one is concerned with asymptotic theory, leading to elegant derivations of the asymptotic properties of tests and estimates under less restrictive assumptions than usual.

Introduction to Part 2

Harmonic analysis might be said to comprise the study of functions and function spaces defined over a topological group G, special reference being paid to the functional operators of translation arising from the group structure of G; see the discussion in Edwards [3], Chapter 2. This description is correct as far as it goes, but it is unlikely to convey much except to those who are already acquainted with the subject (and who therefore have no great need of a description anyway). The only effective way to discover what harmonic analysis is about, is to dip into it, taking stock of just enough (but not too much) detail. Our treatment attempts to present in such a style something from which the taster may choose.

As seems entirely natural, the display offered refers to one of the technically simpler cases: that in which the underlying group G is compact and Hausdorff. (The case in which G is also Abelian is even simpler, and some readers may wish to concentrate on this situation, which still offers many challenging problems.)

Even with the restrictions mentioned in 2. 0.1, there remain a number of approaches to abstract harmonic analysis. Their relative merits depend in part on how much is assumed about the underlying group G (always assumed locally compact and Hausdorff), though there are no sharp dividing lines. For compact groups, the various approaches are much on a par with each other and the choice is largely a matter of taste. We mention a few of the possible approaches.

This set of notes is the result of fusing two sets of skeletal notes, one headed ‘The Riesz representation theorem’ and the other ‘Harmonic analysis on compact groups’, the aim being to end up with a reasonably self-contained introduction to portions of analysis on compact spaces and, more especially, on compact groups.

The term ‘introduction’ requires emphasis. These notes are not (and cannot be) expected to do much more than convey a general picture, even though a few aspects are treated in some detail. In particular, a good many proofs easily accessible in standard texts have been omitted; and many of the proofs included are presented in a somewhat condensed form and may require further attention from readers who decide to study in more detail the areas under discussion. These features arise from a deliberate attempt to avoid too much detail; they are also to some extent inevitable consequences of an attempt to survey rapidly a fairly large body of material.

The substructure of Part 2 has (I am told) been found useful as a lead-in by research students whose subsequent interest has been in specialised topics in harmonic analysis. Part 1 has, I think, filled a similar role in relation to abstract integration theory. If the readers have been attracted by the topics presented, they have pressed on to study some of the more detailed items listed in the bibliography. (In respect of Part 2, there is little doubt that the second volume of Hewitt and Ross [1] is the main follow-up to these notes.)