The Annals of Statistics

REML estimation: asymptotic behavior and related topics

Jiming Jiang

Full-text: Open access

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

Permanent link to this document
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

Keywords
Mixed models restricted maximum likelihood MLE ANOVA consistency asymptotic normality

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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References

  • ANDERSON, T. W. 1973. Asy mptotically efficient estimation of covariance matrices with linear structure. Ann. Statist. 1 135 141. Z.
  • AZZALINI, A. 1984. Estimation and hy pothesis testing for collection of autoregressive time series. Biometrika 71 85 90. Z. BARNDORFF-NIELSEN, O. 1983. On a formula for the distribution of the maximum likelihood estimator. Biometrika 70 343 365. Z.
  • BICKEL, P. J. 1993. Estimation in semiparametric model. In Multivariate analysis: Future Z. direction C. R. Rao, ed. 55 73. North-Holland, Amsterdam. Z.
  • BROWN, K. G. 1976. Asy mptotic behavior of MINQUE-ty pe estimators of variance components. Ann. Statist. 4 746 754. Z.
  • CHAN, N. N. and KWONG, M. K. 1985. Hermitian matrix inequalities and a conjecture. Amer. Math. Monthly 92 533 541. Z.
  • CHOW, Y. S. and TEICHER, H. 1978. Probability Theory. Springer, New York. Z.
  • COOPER, D. M. and THOMPSON, R. 1977. A note on the estimation of the parameters of autoregressive moving average process. Biometrika 64 625 628. Z.
  • CRESSIE, N. 1992. REML estimation in empirical Bay es smoothing of census undercount. Survey Methodology 18 75 94. Z.
  • CRESSIE, N. and LAHIRI, S. N. 1993. The asy mptotic distribution of REML estimators. J. Multivariate Anal. 45 217 233. Z.
  • DAS, K. 1979. Asy mptotic optimality of restricted maximum likelihood estimates for the mixed model. Calcutta Statist. Assoc. Bull. 28 125 142. Z.
  • DE JONG, P. 1987. A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261 277.
  • FOX, R. and TAQQU, M. S. 1985. Noncentral limit theorems for quadratic forms in random variables having long-range dependence. Ann. Probab. 13 428 446. Z. Z.
  • GLEESON, A. C. and CULLIS, B. R. 1987. Residual maximum likelihood REML estimation of a neighbour model for field experiments. Biometrics 43 277 287. Z.
  • GREEN, P. J. 1985. Linear models for field trials, smoothing and cross-validation. Biometrika 72 527 537. Z.
  • GUTTORP, P. and LOCKHART, R. A. 1988. On the asy mptotic distribution of quadratic forms in uniform order statistics. Ann. Statist. 16 433 449. Z.
  • HALL, P. and HEy DE, C. C. 1980. Martingale Limit Theory and Its Application. Academic Press, New York. Z.
  • HAMMERSTROM, T. 1978. On the asy mptotic optimality of tests and estimates in the presence of increasing numbers of nuisance parameter. Ph.D. dissertation, Univ. California, Berkeley. Z.
  • HARTLEY, H. O. and RAO, J. N. K. 1967. Maximum likelihood estimation for the mixed analysis of variance model. Biometrika 54 93 108. Z.
  • HARVILLE, D. A. 1974. Bayesian inference for variance components using only error contrasts. Biometrika 61 383 385. Z.
  • HARVILLE, D. A. 1977. Maximum likelihood approaches to variance components estimation and related problems. J. Amer. Statist. Assoc. 72 320 340. Z.
  • HUBER, P. J. 1981. Robust Statistics. Wiley, New York. Z.
  • KHURI, A. I. and SAHAI, H. 1985. Variance components analysis: a selective literature survey. Internat. Statist. Rev. 53 279 300. Z.
  • LAIRD, N. M. and WARE, J. M. 1982. Random effects models for longitudinal data. Biometrics 38 963 974. Z.
  • LEHMANN, E. H. 1983. Theory of Point Estimation. Wiley, New York. Z.
  • MAKELAINEN, T., SCHMIDT, K. and STy AN, G. P. H. 1981. On the existence and uniqueness of the ¨ ¨ maximum likelihood estimates of a vector-valued parameter in fixed-size samples. Ann. Statist. 9 758 767. Z.
  • MILLER, J. J. 1977. Asy mptotic properties of maximum likelihood estimates in the mixed model of the analysis of variance. Ann. Statist. 5 746 762. Z.
  • NEy MAN, J. and SCOTT, E. 1948. Consistent estimates based on partially consistent observations. Econometrika 16 1 32. Z.
  • PATTERSON, H. D. and THOMPSON, R. 1971. Recovery of interblock information when block sizes are unequal. Biometrika 58 545 554. Z.
  • PFANZAGL, J. 1993. Incidental versus random nuisance parameters. Ann. Statist. 21 1663 1691. Z.
  • RAO, C. R. and KLEFFE, J. 1988. Estimation of Variance Components and Applications. NorthHolland, Amsterdam. Z.
  • RICHARDSON, A. M. and WELSH, A. H. 1994. Asy mptotic properties of restricted maximum Z. likelihood REML estimates for hierarchical mixed linear models. Austral. J. Statist. 36 31 43. Z.
  • ROBINSON, D. L. 1987. Estimation and use of variance components. The Statistician 36 3 14. Z.
  • SCHMIDT, W. H. and THRUM, R. 1981. Contributions to asy mptotic theory in regression models with linear covariance structure. Math. Operationsforsch. Statist. Ser. Statist. 12 243 269. Z.
  • SEARLE, S. R., CASELLA, G. and MCCULLOCH, C. E. 1992. Variance Components. Wiley, New York. Z.
  • SPEED, T. P. 1986. Cumulants and partition lattices IV: a.s. convergence of generalized kstatistics. J. Austral. Math. Soc. 41 79 94. Z.
  • SPEED, T. P. 1991. Comment on ``That BLUP is a good thing The estimation of random effects'' by G. K. Robinson. Statist. Sci. 6 42 44. Z.
  • SZATROWSKI, T. H. and MILLER, J. J. 1980. Explicit maximum likelihood estimates from balanced data in the mixed model of the analysis of variance. Ann. Statist. 8 811 819. Z.
  • THOMPSON, W. A., JR. 1962. The problem of negative estimates of variance components. Ann. Math. Statist. 33 273 289.
  • VERBy LA, A. P. 1990. A conditional derivation of residual maximum likelihood. Austral. J. Statist. 32 227 230. Z.
  • WAHBA, G. 1990. Spline Models for Observational Data. SIAM, Philadelphia. Z.
  • WEISS, L. 1971. Asy mptotic properties of maximum likelihood estimators in some nonstandard cases. J. Amer. Statist. Assoc. 66 345 350. Z.
  • WESTFALL, P. H. 1986. Asy mptotic normality of the ANOVA estimates of components of variance in the nonnormal, unbalanced hierarchal mixed model. Ann. Statist. 14 1572 1582. Z.
  • ZELLNER, A. 1976. Bayesian and non-Bayesian analysis of the regression model with multivariate Student-t error terms. J. Amer. Statist. Assoc. 71 400 405.
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