Invariant Sufficiency, Equivariance and Characterizations of the Gamma Distribution

In the first section it is shown for any sub-σ-algebra G of the σ-algebra of all scale invariant Borel subsets of IRn that an equi-variant statistic Sis G-partially sufficient iff the generated σ-alge- bra S−1(B) and G are independent and that S being invariantly sufficient and equivariant, the Pitman estimator is given by S E₁(S)/E₁(S2). For independent X₁,...,X_n the existence of an invariantly sufficient statistic ∑ c_j X_jk is characterized by X₁k,...X_nk having gamma distributions. In the second section there are established some characterizations of the gamma distribution by properties (admissibility, optimality) of the minimum variance unbiased linear estimator where X₁,...,X_n are required to be independent. Finally, the indepen-dence of X₁,...,X_n is replaced by a certain linear framework and a method is presented for carrying over the characterizations previously stated for the independent case to this linear set-up.