Efficiency of Estimates in Nonregular Cases

Abstract

Let {P_θ} be a family of probability measures on a measurable space (X, α) indexed by the parameter _θ where θ Θ and Θ is an open set in ℝ. We assume that P_θ is absolutely continuous with respect to some σ finite measure ʋ on ( X, α) with density function f(x,θ). Pⁿ _θ is the independent product of n identical components of Pθ with density ${{\rm{f}}_{\rm{n}}}({\rm{x,\theta }}),{\rm{x}} = ({{\rm{x}}_{\rm{1}}}, \ldots ,{{\rm{x}}_{\rm{n}}})$. One possibility to define efficiency of an estimate T_n for θ is by covering probabilities. One has to select first a class of appropriate estimates and gives then upper bounds for the probability that T_n lies within an interval (θ-t₁δ_n,θ+ t₂.δ) where t_i > 0 i = 1,2 and δ_n is a norming sequence with δ → 0 which depends on the family {Pθ}. This concept was used by Wolfowitz (1974) who defined a class of estimates in which the maximum probability estimate is optimal. Another possibility is to use the method of Bahadurand derive bounds for the class of median unbiased estimates (Pfanzagl (1970)) or more general strongly asymptotic median unbiased estimates (Michel (1978)). This method was only defined for families which are asymptotically normal. The aim of the paper is to show that this method can also be used for nonregular cases where other than normal limit structures of the loglikelihoodratios occur. In section 2 we give a general result which gives upper bounds for the covering probabilities. In section 3 we give applications to asymptotic normal distributed families and in section 4 we treat the case where the densities have discontinuities.