Best prediction in linear models with mixed integer/real unknowns: theory and application

Abstract

In this contribution, we extend the existing theory of minimum mean squared error prediction (best prediction). This extention is motivated by the desire to be able to deal with models in which the parameter vectors have real-valued and/or integer-valued entries. New classes of predictors are introduced, based on the principle of equivariance. Equivariant prediction is developed for the real-parameter case, the integer-parameter case, and for the mixed integer/real case. The best predictors within these classes are identified, and they are shown to have a better performance than best linear (unbiased) prediction. This holds true for the mean squared error performance, as well as for the error variance performance. We show that, in the context of linear model prediction, best predictors and best estimators come in pairs. We take advantage of this property by also identifying the corresponding best estimators. All of the best equivariant estimators are shown to have a better precision than the best linear unbiased estimator. Although no restrictions are placed on the probability distributions of the random vectors, the Gaussian case is derived separately. The best predictors are also compared with least-squares predictors, in particular with the integer-based least-squares predictor introduced in Teunissen (J Geodesy, in press, 2006).

On a stronger-than-best property for best prediction

“…After all, such predictors, when they exist, will have a smaller mean squared prediction error and a smaller error variance than the BLUP. In Teunissen (2006), it has been shown that such predictors of y 0 indeed exist. They are found in the class of equivariant predictors and in the class of integer equivariant predictors.…”

“…This is not needed per se. In Teunissen (2006), it has been shown that one can determine meaningful best predictors in classes of predictors that are larger than the class of linear unbiased predictors. Such predictors are however nonlinear.…”

“…Does the result of Teunissen (2006) make the BLUP obsolete? The answer must be yes if one prefers predictors with smaller mean squared prediction errors and smaller error variances.…”

“…It can be shown, if no restrictions are put on the class of predictors , that the best predictor is given by the conditional mean,Ĝ(y) = E(y 0 |y) (e.g. Bar-Shalom and Li 1993;Teunissen 2006). This predictor is unbiased and, in the Gaussian case, it is given, withȳ 0 = E(y 0 ) andȳ = E(y), aŝ…”

“…In Teunissen (2006), it has been shown, however, that for the same linear model on which the BLUP is based, meaningful predictors can be devised that have smaller mean squared prediction errors than the BLUP. Hence, if one really values the property of obtaining the smallest possible MSE, these predictors are to be preferred over the BLUP.…”