## The Annals of Statistics

### REML estimation: asymptotic behavior and related topics

Jiming Jiang

#### Abstract

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

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

Dates
First available in Project Euclid: 26 September 2002

https://projecteuclid.org/euclid.aos/1033066209

Digital Object Identifier
doi:10.1214/aos/1033066209

Mathematical Reviews number (MathSciNet)
MR1389890

Zentralblatt MATH identifier
0853.62022

Subjects
Primary: 62F12: Asymptotic properties of estimators

#### Citation

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

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