Superefficiency

Abstract

We review the history and several proofs of the famous result of Le Cam that a sequence of estimators can be superefficient on at most a Lebesgue null set. 27.1 Introduction The method of maximum likelihood as a general method of estimation in statistics was introduced and developed by Fisher (1912, 1922, 1925, 1934). It gained popularity as it appeared that the method automatically produces efficient estimators if the number of observations is large. The concept of asymptotic efficiency was invented by Fisher as early as 1922 roughly in the form as we use it for regular models today: a sequence of statistics is efficient if it tends to a normal distribution with the least possible standard deviation. In the 1930s and 1940s there were many steps in the direction of a rigorous foundation of Fisher's remarkable insights. These consisted both of proofs of the asymptotic normality of maximum likelihood estimators and of obtaining lower bounds for the variance of estimators. Chapters 32 and 33 of Cramér (1946) give a summary of the state of affairs in the mid 1940s, even though some work carried out in the early war years, notably Wald's, had been unavailable to him. Chapter 32 gives a rigorous proof of what we now know as the Cramér-Rao inequality and next goes on to define the asymptotic efficiency of an estimator as the quotient of the inverse Fisher information and the asymptotic variance. Next Chapter 33 gives a rigorous proof of asymptotic normality of the maximum likelihood estimator, based on work by Dugué (1937). Cramér defines an estimator sequence to be asymptotically efficient if its asymptotic efficiency (the quotient mentioned previously) equals one. Thus combination of the results of the two chapters leads to the correct conclusion that the method of maximum likelihood produces asymptotically efficient estimators, under some regularity conditions on the underlying densities. Apparently the conceptual hole in the definition was not fully recognized until 1951, even though the difficulty must have been clear to several authors who had worked on establishing efficiency within restricted classes of estimators.