The Annals of Statistics

REML estimation: asymptotic behavior and related topics

Jiming Jiang

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The restricted maximum likelihood (REML) estimates of dispersion parameters (variance components) in a general (non-normal) mixed model are defined as solutions of the REML equations. In this paper, we show the REML estimates are consistent if the model is asymptotically identifiable and infinitely informative under the (location) invariant class, and are asymptotically normal (A.N.) if in addition the model is asymptotically nondegenerate. The result does not require normality or boundedness of the rank p of design matrix of fixed effects. Moreover, we give a necessary and sufficient condition for asymptotic normality of Gaussian maximum likelihood estimates (MLE) in non-normal cases. As an application, we show for all unconfounded balanced mixed models of the analysis of variance the REML (ANOVA) estimates are consistent; and are also A.N. provided the models are nondegenerate; the MLE are consistent (A.N.) if and only if certain constraints on p are satisfied.

Article information

Ann. Statist., Volume 24, Number 1 (1996), 255-286.

First available in Project Euclid: 26 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators

Mixed models restricted maximum likelihood MLE ANOVA consistency asymptotic normality


Jiang, Jiming. REML estimation: asymptotic behavior and related topics. Ann. Statist. 24 (1996), no. 1, 255--286. doi:10.1214/aos/1033066209.

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