Equalities between the BLUEs and BLUPs under the partitioned linear fixed model and the corresponding mixed model

Abstract

In this article we consider the partitioned fixed linear model F : y = X1β1 + X2β2 + ε" and the corresponding mixed model M : y =X1β1+X2u+ ε, where ε is a random error vector and u is a random effect vector. In 2006, Isotalo, M¨ols, and Puntanen found conditions under which an arbitrary representation of the best linear unbiased estimator (BLUE) of an estimable parametric function of β1 in the fixed model F remains BLUE in the mixed model M . In this paper we extend the results concerning further equalities arising from models F and M.

Discussion

“…The use of specialized sample survey software necessary for analysis that properly incorporates the survey structure can often be delayed until initial models (which for linear models may then for simplicity of implementation include stepwise fitting of regressors) have been checked and the most important predictor variables and their interactions have been found, at least provisionally. The use of contextual variables as fixed effects that reduce variation in random effects may have technical ramifications (Haslett et al [17] ), but in the context of SAE modelling, where it is the predictions rather than the parameter estimates that are important, these issues are minor. Similarly, because they are usually swamped by aggregation or averaging, outliers in unit-level data (where they apply at the individual or household level) are much less of an issue than outliers in area-level models.…”