Families of densities with non-constant carriers which have finite dimensional sufficient statistics

Abstract

Let X₁,X₂,....,X_n be independently and identically distributed random variables with common density function f_θ(x). Let the carrier X_θ = {x: f_θ(x) > 0} of the density f_θ (x) be an open interval (a(θ), b(θ)). If a(θ), b(θ) are continuous and monotonic in opposite sense and if there is a continuous real-valued statistic which is sufficient for the given family of densities for a sample of size n ≥ 2, then we prove f_θ(x) has the following form f_θ(x) = g(θ).h(x). In our second theorem we have the same conclusion where a(θ), b(θ) are constinuous and monotonic in the same sense and there is a continuous statistic taking values in R², which is sufficient for a sample of size n ≥ 3. In the last section we investigate cases when a(θ), b(θ) are monotonic and the sufficient statistic is k-dimensional, k ≥ 2.