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The Gauss–Markov theorem for linear models is extended to nonlinear models. This is accomplished by showing that the theorem holds for a Riemannian manifold which is locally isometric to a direct product of two submanifolds. The fact that the derivative of the square of the distance function is the parallel translation along the geodesic connecting the two points implies that the linear functional on the tangent space at the center of gravity, defined as the mean in that space (exp gives the local 1:1 correspondence), is identically zero. This fact is used to define a class of parametric functions. The linear version of the theorem is applied in the tangent space at the center of gravity and the results are mapped via exp to the direct product. Then it is shown that every sample space containing a nonlinear model which admits a least squares estimate can be locally factored into a direct product (isometrically). A simple example is presented.

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.

In this note we extend the Gauss–Markov theorem to the case of singular covariances and give an explicit formula for the BLUE of an estimable parameter. The formula reduces to the usual one when the covariance is nonsingular. An easy by-product of this result reestablishes known conditions for coincidence of the BLUE and naive least squares estimate.

Given the general linear model $y = X\beta + e$ having the covariance matrix $\sigma ^2 V$ of the errors, with $\sigma ^2 > 0$, V known and nonnegative (possibly singular), we specify the complete nonempty class $\mathcal{V}$ of conditional inverses of V such that, for any estimable parametric function $\lambda '\beta $ and any $V^ * $ in $\mathcal{V}$, a best linear unbiased estimator of $\lambda '\beta $ is given by $\lambda '\hat \beta $, where $\hat \beta $ is any solution to the general normal equations $X'V^ * X\beta = X'V^ * y$. Properties of the solutions $\hat \beta $ are presented. It is further verified that if y is distributed as a multivariate normal variable then $\lambda '\hat \beta $ is the maximum likelihood estimator of $\lambda '\beta $. A procedure for testing hypotheses, using solutions to the general normal equations, is also presented.