Estimation of heteroscedastic variances on a linear models

Abstract

Let Y=Xβ + e be a Gauss-Markoff linear model such that E(e)=0 and D(e), the dispersion matrix of the error vector, is a diagonal matrix Δ whose ith diagonal element is σ_i², the variance of the ith observation y_i. Some of the s_i² may be equal. The problem is to estimate all the different variances. In this article, a new method known as MINQUE (Minimum Norm Quadratic Unbiased Estimation) is introduced for the estimation of the heteroscedastic variances. This method satisfies some intuitive properties: (i) if S₁ is the MINQUE of Σp_iσ_i² and S₂ that of Σq_iσ²_i, then S₁ + S₂ is the MINQUE of Σ(p_i + q_i)σ²_i, (ii) it is invariant under orthogonal transformation, etc. Some sufficient conditions for the estimation of all linear functions of the σ²_i are given. The use of estimated variances in problems of inference on the β parameters is briefly indicated.