On the Built-in Restrictions in Linear Mixed Models, with Application to Smoothing Spline Analysis of Variance

The best linear unbiased predictor (BLUP) of the random parameter in a linear mixed model satisfies a linear constraint, which has been previously termed a built-in restriction. In other literature, constraints on the random parameter itself have been introduced into the modeling framework. The present article has two goals. First, it explores the idea of imposing the built-in restrictions on the BLUP as constraints on the random parameter. Second, it investigates the built-in restrictions satisfied by certain smoothing spline analysis of variance (SSANOVA) estimators, and compares these restrictions to arguably more natural side conditions on the ANOVA decomposition.