Maximum probability estimators and related topics / Lionel Weiss, Jacob Wolfowitz

Based on their well-known research work the authors present in this monograph an elaboration of the method and theory of maximum probability estimators. The concept of maximum probability estimation covers as a special case that of maximum likelihood estimation. The authors' maxim is that the statistical problem and the crucial question in asymptotic theory is always that of finding asymptotically efficient estimators (in terms of covering probabilities). Here, maximum likelihood estimation has certain defects: There are unreasonable restrictions on the class of competing estimators and the theory is restricted to the "regular'' case, which excludes from consideration important problems and distributions. The assumptions given in this monograph are not on the density of the observed random variables but on the behavior of the maximum probability estimator. According to the authors' experience the conditions are relatively easy to verify and it is conjectured that they hold for all reasonable statistical problems.
In Chapter 3 the maximum probability estimator is defined, its asymptotic efficiency is stated and proved (including a discussion of the assumptions), and the relation to the maximum likelihood estimator is given.
In Chapter 4 the maximum probability estimator with respect to a general loss function is defined and discussed (the loss function in Chapter 3 is the characteristic function of the complement of a bounded Borel set).
Chapter 5 deals with the asymptotic behavior of the likelihood function and introduces a new concept of asymptotic sufficiency.
Chapter 6 gives a rigorous treatment (including a set of relatively strong regularity conditions) of the classical regular case, namely the maximum likelihood estimator. Furthermore, there is a discussion of an interesting nonregular case for which the maximum likelihood estimator is efficient.
In Chapter 7 a new result in testing a hypothesis about several parameters in situations where the observations are not necessarily independent and identically distributed is proved using the basic ideas and techniques which give asymptotic optimality of the maximum probability estimator. (MathSciNet)