1988 | OriginalPaper | Chapter
The Basu Theorems
Author : J. K. Ghosh
Published in: Statistical Information and Likelihood
Publisher: Springer New York
Included in: Professional Book Archive
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The theorems are related to the notions of sufficiency, ancillarity and conditional independence. Let X denote the sample and θ the parameter that completely specifies the sampling distribution Pθ of X. An event E is ancillary if Pθ (E) is θ-free, i.e., Pθ(E) = Pθ’(E) for all θ’ ε θ, the parameter space. A statistic Y = Y(X) is ancillary if every Y-event (i.e., a measurable set defined in terms of Y) is ancillary (see ANCILLARY STATISTICS). A statistic T is sufficient if, for every event E, there exists a θ-free version of the conditional probability function Pθ (E |T) (see SUFFICIENCY). The event E is (conditionally) independent of T if, for each θ ε θ, the conditional probability function Pθ(E |T) is Pθ -essentially equal to the constant Pθ (E). The statistic Y is independent of T if every Y-event is inde-pendent of T. (Independence is a symmetric relationship between two statistics.)