The estimation of residual variance in regression analysis

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Abstract
In this paper we discuss the problem of estimating the residual variance σ2 in the linear regression model . We assume that the components of the random vector ∈ are stochastically independent but we do not suppose that the k-xtosis of the distribution, β – 3, is equal to zero. I t is investigated when a quadratic estimator of σ2 is best quadratic estimator or best quadratic unbiased estimator. Especially the question is of interest under which conditions a multiple of the projection-matrix leads to such estimator. In both cases the Hsu-condition necessary and sufficient. Imposing positive-definiteness on the estimator does not weaken this eondition. The obtained estimators are show-n to be positive-semi-definite if ∣β – 3∣ < 2.For the practical applicability the question is what is to do if β is unknown. For this purpose admissible quadratic estimators of σ2 are determined. The proofs of the theorems are based on a general projection theorem for quasi-inner prodncts which is given in section 2.