Extended Gauss-Markov Theorem for Nonparametric Mixed-Effects Models

Su Yun Huang*, Horng-Shing Lu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


The Gauss-Markov theorem provides a golden standard for constructing the best linear unbiased estimation for linear models. The main purpose of this article is to extend the Gauss-Markov theorem to include nonparametric mixed-effects models. The extended Gauss-Markov estimation (or prediction) is shown to be equivalent to a regularization method and its minimaxity is addressed. The resulting Gauss-Markov estimation serves as an oracle to guide the exploration for effective nonlinear estimators adaptively. Various examples are discussed. Particularly, the wavelet nonparametric regression example and its connection with a Sobolev regularization is presented.

Original languageEnglish
Pages (from-to)249-266
Number of pages18
JournalJournal of Multivariate Analysis
Issue number2
StatePublished - 1 Feb 2001


  • Nonparametric mixed-effects; Gauss-Markov theorem; best linear unbiased prediction (BLUP); regularization; minimaxity; normal equations; nonparametric regression; wavelet shrinkage; deconvolution


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