Classical asymptotic properties of a certain estimator related to the maximum likelihood estimator

1. A general method for constructing paint estimatars In a statistical point estimation problem, the goal is to use information obtained from a sample of observations to estimate an unknown parameter (or a function of an unknown parameter) of the probability distribution which governs the variability of sampling. Ideally, we would like to construct an estimator which has the property that with probability one a correct estimate is made of the true value 0 of the unknown parameter. This goal is, of course, met only in trivial cases. More realistically, we hope to find an estimator having highest possible probability of being "close" to 0. Suppose that we have independent and identically distributed (i.i.d.) random observations 2(1, X~,.-., X~ having common distribution Po. Here we assume that each X~ is defined on a measure space (.~, ~), where 2~" is any topological space and ~ is a sigma-field of measurable sets. We assume that P0 is a member of a class {P0, 0~0} indexed by a point 0 in a subset 0 of k-dimensional Euclidean space. Further, we assume that the class {Po, 0 ~ e} is dominated by a c-finite measure defined on ~, so that each Po has a density (Radon-Nikodym derivative) f(x[O)=dPJd~ with respect to ,a. The sample (X~, X2,'", X~) is then defined on the Cartesian product space (S C~), ~c,,)) with respect to the product probability measure P~"~ which has density f[ f(xi ]0) with i=1 respect to the product measure ~(").