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Published online by Cambridge University Press: 04 February 2011
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.
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