Regression of polynomial statistics on the sample mean and natural exponential families

Abstract

A polynomial Q = Q(X1, …, Xn) of degree m in independent identically distributed random variables with distribution function F is an unbiased estimator of a functional q(α1(F), …, αm(F)), where q(u1, …, um) is a polynomial in u1, …, um and αj(F) is the jth moment of F (assuming the necessary moment of F exists). It is shown that the relation E(Q | X1 + … + Xn) = 0 holds if and only if q(α1(θ), …, αm(θ)) ≡ 0, where αj(θ) is the jth moment of the natural exponential family generated by F. This result, based on the fact that X1 + … + Xn is a complete sufficient statistic for a parameter θ in a sample from a natural exponential family of distributions Fθ(x) = ∫−∞xeθu−k(θ)dF(u), explains why the distributions appearing as solutions of regression problems are the same as solutions of problems for natural exponential families though, at the first glance, the latter seem unrelated to the former.