The Basu Theorems

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 P_θ of X. An event E is ancillary if P_θ (E) is θ-free, i.e., P_θ(E) = P_θ’(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 P_θ (E |T) (see SUFFICIENCY). The event E is (conditionally) independent of T if, for each θ ε θ, the conditional probability function P_θ(E |T) is P_θ -essentially equal to the constant P_θ (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.)