Likelihood and auxiliary statistics
Let (X, p(x;ω), Ω) be a parametric statistical model, which we shall denote by M. Here X is the sample space, Ω is the parameter space, and p(x;ω) is the model function. We presume the existence of a measure μ on X such that for each fixed value of the parameter ω the function p(x;ω) is the density with respect to μ of a probability measure Pω on X, and we term x → p(x;ω) the probability function corresponding to ω. The parameter space Ω is a subset of d-dimensional Euclidean space Rd and we denote coordinates of ω by ωr,ωs,…, the indices r,s,… thus running from 1 to d. Throughout, Ω will be either an open set or such a set with some of its boundary points added.
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