An optimal property of the Gauss-Markoff estimator

AbstractConsider the linear model Y = Xβ + ε, where Y is the response variable of order (n×1), X is an (n×p) matrix of known constants, β is a (p×1) unknown parameter, ε is an (n×1) error variable with E(ε) = 0, and E(εε′) = σ2Λ, where Λ is a known (n×n) positive definite matrix and σ2 is a positive scalar, possibly unknown. Suppose that θ is a linear function of the components of β. It is shown that when ε is assumed to have a distribution belonging to the class of elliptical distributions (i.e., distributions having constant density or equiprobable surfaces on homothetic ellipsoids as in the case of the multivariate normal distributions), the probability of the Gauss-Markoff estimator of θ falling inside any fixed ellipsoid centered at θ is greater than or equal to the probability that any linear unbiased estimator of θ falls inside the same ellipsoid