Correction: Linearity of Unbiased Linear Model Estimators

Abstract

The author presented a proof that a regression estimator unbiased for all distributions in a sufficiently broad family F0 must be linear. The family was taken to consist of the convolutions of all two-point distributions with a scale-family of smooth densities tending to a point mass at zero. The basic calculations were based solely on the discrete distributions, but the convolutions were introduced so that the family would be a subset of some standard, smooth nonparametric families. For example, adaptive estimation requires enough smoothness so that the Cramér-Rao bound provides optimal asymptotics. Some further comments appear in the supplemental material. The proof required that convergence of the smooth densities to zero implied that the expectations under the smooth convolutions converged to the expectation under the two-point distribution. This requires that the estimate be continuous, and there was a major error in the proof that unbiasedness implies continuity. It appears that continuity cannot be proved using unbiasedness over F0 , but it can be proved using unbiasedness over families of discrete distributions (see supplemental material). Thus, either the estimator must be assumed to be continuous, or a result using only discrete distributions is required. In trying to correct the error, a much simpler proof of the linearity result was found. This proof takes F0 to consist only of discrete distributions. The details are also presented in the supplemental material, but the basic idea is relatively simple: Consider a simplex with center at zero. Each point in the simplex (say −y ) is a convex combination of the vertices of the simplex; that is, −y is the expectation for a (discrete) distribution putting probability pi on vertex zi . Thus, the distribution putting probability 2 on y and 1 2 pi on zi has mean zero. Therefore, by unbiasedness, T(y) is a convex combination of the {T(zi)} ; that is, T(y) is the matrix whose columns are T(zi) times a vector, p , of probabilities. By basic properties of a simplex, p is an affine function of −y ; and so T(y) is an affine function of y . By unbiasedness under a point mass at zero, T(0) = 0 ; and so the affine constant is zero and T must be linear.