Most statisticians have had occasion to rely on asymptotic or large sample arguments. This author is no exception. However, it happened that for a number of years, he also had to teach the subject. It appeared then that the theory was a bewildering maze without any logical structure. Out of pedagogical necessity an attempt was made to organize at least part of the subject. It turned out that one could indeed organize a part, and perhaps a sizable part, around very few essential ideas and elements.This book is the outcome of such an organizational effort. The ideas and techniques used reflect first and foremost the influence of Abraham Wald's writings. Another very direct influence was that of Jerzy Neyman, who asked a variety of questions but who also promoted my academic career. Some other easily discernible influences are those of Jaroslav Hajek and Charles Stein. Not so visible, but indispensable, were the teachings of Etienne Halphen, who attempted (without success) to convert us to the Bayesian creed long before it became fashionable, but who also taught us a great deal about the interplay between theory and practice.My first notes on "asymptotics" were written around 1955-56 with the help of Thomas S. Ferguson. They were "classical" in character, and it quickly became evident that we could not provide a comprehensive and logical account of the theory without some additional thinking. This took a number of years. As a result, the work went through many revisions, subjecting many secretaries to thankless work. Among the people who deserve high praise for turning my indecipherable manuscripts into elegant typescript, I must particularly thank Mrs. Julia Rubalcava, Miss Gail Coe, Mrs. Patricia Hardy, and Madame Micheline Marano. I also wish to extend belated thanks to Mrs. Ginette Henkin who, at one time, typed for me a treatise in French on Statistical Decision Theory. The work was never published; a few traces of it are left in Chapter 2 of this work. CHAPTER 3 Likelihood Ratios and Conical Measures

Maximnm likelihood estimates are reported to be best under all circumstances. Yet there are numerous simple examples where they plainly misbehave. One gives some eranmples for problems that had not been invented for the purpose of annoying ms,aximunm likelihood fans. Another example, imitated from B'hadu'r, has been specially created with just such a purpose in mind. Next, we present a list of principles leading to the construction of good estimates. The main principle says that one should not believe in principles but study each problem for its own sake.One of the most widely used methods of statistical estimation is that of maximum likelihood. Opinions on who was the first to propose the method differ. However Fisher is usually credited with the invention of the name 'maximum likelihood', with a major effort intended to spread its use and with the derivation of the optimality properties of the resulting estimates.Qualms about the general validity of the optimality properties have been expressed occasionally. However as late as 1970 L.J. Savage could imply in his 'Fisher lecture' that the difficulties arising in some examples would have rightly been considered 'mathematical caviling' by R.A. Fisher.Of course nobody has been able to prove that maximum likelihood estimates are 'best' under all circumstances. The lack of any such proof is not sufficient by itself to invalidate Fisher's claims. It might simply mean that we have not yet translated into mathematics the basic principles which underlied Fisher's intuition.The present author has, unwittingly, contributed to the confusion by writing two papers which have been interpreted by some as attempts to substantiate Fisher's claims.To clarify the situation we present a few known facts which should be kept in mind as one proceeds along through the various proofs of consistency, asymptotic normality or asymptotic optimality of maximum likelihood estimates.The examples given here deal mostly with the case of independent identically distributed observations. They are intended to show that maximum likelihood does possess disquieting features which rule out the possibility of existence of undiscovered underlying principles which could be used to justify it. One of the very gross forms of misbehavior can be stated as follows.Maximum likelihood estimates computed with all the information available may turn out to be inconsistent. Throwing away a substantial part of the information may render them consistent.The examples show that, in spite of all its presumed virtues, the maximum likelihood procedure cannot be universally recommended. This does not mean that we advocate L. LE CAM some other principle instead, although we give a few guidelines in ? 6.

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