Statistics Independent of a Complete Sufficient Statistic

Abstract

If {P_θ}, θ ε Ω, be a family of probability measures on an abstract sample space S and T be a sufficient statistic for θ then for a statistic T₁ to be stochastically independent of T it is necessary that the probability distribution of T₁ be independent of θ. The condition is also sufficient if T be a boundedly complete sufficient statistic. Certain well-known results of distribution theory follow immediately from the above considerations. For instance, if x₁, x₂,..., x_n, are independent N(μ, σ)’s then the sample mean x and the sample variance s² are mutually independent and are jointly independent of any statistic f (real or vector valued) that is independent of change of scale and origin. It is also deduced that if x₁, x₂,..., x_n are independent random variables such that their joint distribution involves an unknown location parameter θ then there can exist a linear boundedly complete sufficient statistic for θ only if the x’s are all normal.Similar characterizations for the Gamma distribution also are indicated.