Best Nonnegative Invariant Partially Orthogonal Quadratic Estimation in Normal Regression

原文摘要

This article explores best estimation (in some sense) of the variance matrix V of the vector of disturbances in a normal linear regression model. More precisely, we consider the regression model y = Xβ + U, where U is distributed as N(0, V) and where the spectral form of V is V = Σn j1 λjRj. The R j are known and the λ j are the (possibly) distinct positive eigenvalues we wish to estimate. More generally, we wish to estimate a linear combination of the eigenvalues. This model involves, among others, the class of error components models. The estimates are required to be quadratic in y, nonnegative, and invariant with respect to β. Thus they are of the form y'Ay, where A is nonnegative definite and such that AX = 0. Given these restrictions and possibly others, we seek the minimum mean squared error estimate or the minimum variance unbiased estimate, a problem that has received much attention in the literature. Working on estimation of demands for transportation using an error components model, we noticed that prior knowledge on parameters of V was ignored; we believe that it would be a waste (translated in terms of nonoptimality) not to use it. The main goal of this article is to attempt to combine prior knowledge, or even reasonable guess, into the classical context of estimation just described. The solution of the problem is not trivial and requires somewhat complex techniques (presented in an Appendix). Fortunately, the obtained estimators are explicit and straightforward to implement: the MATLAB software is particularly well adapted for the computations. The applicability of the estimators and the statistical model go well beyond econometrics, including geology, hydrogeology, mining exploration, and cartography. An application on real data is given, to illustrate how they work.

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