The Annals of Statistics

Wald consistency and the method of sieves in REML estimation

Jiming Jiang

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We prove that for all unconfounded balanced mixed models of the analysis of variance, estimates of variance components parameters that maximize the (restricted) Gaussian likelihood are consistent and asymptotically normal--and this is true whether normality is assumed or not. For a general (nonnormal) mixed model, we show estimates of the variance components parameters that maximize the (restricted) Gaussian likelihood over a sequence of approximating parameter spaces (i.e., a sieve) constitute a consistent sequence of roots of the REML equations and the sequence is also asymptotically normal. The results do not require the rank p of the design matrix of fixed effects to be bounded. An example shows that, in some unbalanced cases, estimates that maximize the Gaussian likelihood over the full parameter space can be inconsistent, given the condition that ensures consistency of the sieve estimates.

Article information

Ann. Statist., Volume 25, Number 4 (1997), 1781-1803.

First available in Project Euclid: 9 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators

Mixed models restricted maximum likelihood Wald consistency the method of sieves


Jiang, Jiming. Wald consistency and the method of sieves in REML estimation. Ann. Statist. 25 (1997), no. 4, 1781--1803. doi:10.1214/aos/1031594742.

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