A Gauss–Markov Theorem for Infinite-Dimensional Regression Models with Possibly Singular Covariance
Thomas D. Morley · SIAM Journal on Applied Mathematics · 1979
Consider the linear model \[ {\bf z} = Bx + {\bf v} \] where x is a vector, B a linear operator, and v is a random variable with mean 0 and covariance $V^2 = \operatorname{cov} ({\bf v},{\bf v})$. A linear functional $(g, \cdot )$ is called a best linear unbiased estimator for c if (i) $B * g = c$, and (ii) $\| {V^2 g} \|^2 $ is as small as possible, subject to (i). The classical Gauss–Markov theorem gives a formula for the best linear unbiased estimator in finite dimensions, under the condition that $V^2 $ is invertible. We extend the Gauss–Markov theorem to Hilbert space without the condition that the covariance is invertible.