×

REML estimation: Asymptotic behavior and related topics. (English) Zbl 0853.62022

Summary: 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. 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 the design matrix of fixed effects.
Moreover, we give a necessary and sufficient condition for asymptotic normality of Gaussian maximum likelihood estimates (MLE) in nonnormal 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 MLEs are consistent (A.N.) if and only if certain constraints on \(p\) are satisfied.

MSC:

62F12 Asymptotic properties of parametric estimators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] ANDERSON, T. W. 1973. Asy mptotically efficient estimation of covariance matrices with linear structure. Ann. Statist. 1 135 141. Z. · Zbl 0296.62022 · doi:10.1214/aos/1193342389
[2] 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. JSTOR: · Zbl 0532.62065 · doi:10.1093/biomet/71.1.85
[3] BICKEL, P. J. 1993. Estimation in semiparametric model. In Multivariate analysis: Future Z. direction C. R. Rao, ed. 55 73. North-Holland, Amsterdam. Z. · Zbl 0795.62027
[4] BROWN, K. G. 1976. Asy mptotic behavior of MINQUE-ty pe estimators of variance components. Ann. Statist. 4 746 754. Z. · Zbl 0339.62047 · doi:10.1214/aos/1176343546
[5] CHAN, N. N. and KWONG, M. K. 1985. Hermitian matrix inequalities and a conjecture. Amer. Math. Monthly 92 533 541. Z. JSTOR: · Zbl 0587.15009 · doi:10.2307/2323157
[6] CHOW, Y. S. and TEICHER, H. 1978. Probability Theory. Springer, New York. Z. · Zbl 0399.60001
[7] 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. JSTOR: · Zbl 0368.62076
[8] CRESSIE, N. 1992. REML estimation in empirical Bay es smoothing of census undercount. Survey Methodology 18 75 94. Z.
[9] CRESSIE, N. and LAHIRI, S. N. 1993. The asy mptotic distribution of REML estimators. J. Multivariate Anal. 45 217 233. Z. · Zbl 0772.62008 · doi:10.1006/jmva.1993.1034
[10] DAS, K. 1979. Asy mptotic optimality of restricted maximum likelihood estimates for the mixed model. Calcutta Statist. Assoc. Bull. 28 125 142. Z. · Zbl 0446.62075
[11] DE JONG, P. 1987. A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261 277. · Zbl 0596.60022 · doi:10.1007/BF00354037
[12] 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. · Zbl 0569.60016 · doi:10.1214/aop/1176993001
[13] 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.
[14] GREEN, P. J. 1985. Linear models for field trials, smoothing and cross-validation. Biometrika 72 527 537. Z. JSTOR: · doi:10.1093/biomet/72.3.527
[15] 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. · Zbl 0638.62022 · doi:10.1214/aos/1176350713
[16] HALL, P. and HEy DE, C. C. 1980. Martingale Limit Theory and Its Application. Academic Press, New York. Z. · Zbl 0462.60045
[17] 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.
[18] HARTLEY, H. O. and RAO, J. N. K. 1967. Maximum likelihood estimation for the mixed analysis of variance model. Biometrika 54 93 108. Z. JSTOR: · Zbl 0178.22001 · doi:10.1093/biomet/54.1-2.93
[19] HARVILLE, D. A. 1974. Bayesian inference for variance components using only error contrasts. Biometrika 61 383 385. Z. JSTOR: · Zbl 0281.62072 · doi:10.1093/biomet/61.2.383
[20] HARVILLE, D. A. 1977. Maximum likelihood approaches to variance components estimation and related problems. J. Amer. Statist. Assoc. 72 320 340. Z. JSTOR: · Zbl 0373.62040 · doi:10.2307/2286796
[21] HUBER, P. J. 1981. Robust Statistics. Wiley, New York. Z. · Zbl 0536.62025
[22] KHURI, A. I. and SAHAI, H. 1985. Variance components analysis: a selective literature survey. Internat. Statist. Rev. 53 279 300. Z. JSTOR: · Zbl 0586.62110 · doi:10.2307/1402893
[23] LAIRD, N. M. and WARE, J. M. 1982. Random effects models for longitudinal data. Biometrics 38 963 974. Z. · Zbl 0512.62107 · doi:10.2307/2529876
[24] LEHMANN, E. H. 1983. Theory of Point Estimation. Wiley, New York. Z. · Zbl 0522.62020
[25] 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. · Zbl 0473.62004 · doi:10.1214/aos/1176345516
[26] 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. · Zbl 0406.62017 · doi:10.1214/aos/1176343897
[27] NEy MAN, J. and SCOTT, E. 1948. Consistent estimates based on partially consistent observations. Econometrika 16 1 32. Z. JSTOR: · Zbl 0034.07602 · doi:10.2307/1914288
[28] PATTERSON, H. D. and THOMPSON, R. 1971. Recovery of interblock information when block sizes are unequal. Biometrika 58 545 554. Z. JSTOR: · Zbl 0228.62046 · doi:10.1093/biomet/58.3.545
[29] PFANZAGL, J. 1993. Incidental versus random nuisance parameters. Ann. Statist. 21 1663 1691. Z. · Zbl 0795.62029 · doi:10.1214/aos/1176349392
[30] RAO, C. R. and KLEFFE, J. 1988. Estimation of Variance Components and Applications. NorthHolland, Amsterdam. Z. · Zbl 0645.62073
[31] 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. · Zbl 0828.62025 · doi:10.1111/j.1467-842X.1994.tb00636.x
[32] ROBINSON, D. L. 1987. Estimation and use of variance components. The Statistician 36 3 14. Z.
[33] 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. · Zbl 0492.62056 · doi:10.1080/02331888108801586
[34] SEARLE, S. R., CASELLA, G. and MCCULLOCH, C. E. 1992. Variance Components. Wiley, New York. Z. · Zbl 0850.62007
[35] SPEED, T. P. 1986. Cumulants and partition lattices IV: a.s. convergence of generalized kstatistics. J. Austral. Math. Soc. 41 79 94. Z. · Zbl 0619.60049 · doi:10.1017/S1446788700028093
[36] 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. · Zbl 0955.62500 · doi:10.1214/ss/1177011926
[37] 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. · Zbl 0465.62069 · doi:10.1214/aos/1176345073
[38] THOMPSON, W. A., JR. 1962. The problem of negative estimates of variance components. Ann. Math. Statist. 33 273 289. · Zbl 0108.15902 · doi:10.1214/aoms/1177704731
[39] VERBy LA, A. P. 1990. A conditional derivation of residual maximum likelihood. Austral. J. Statist. 32 227 230. Z.
[40] WAHBA, G. 1990. Spline Models for Observational Data. SIAM, Philadelphia. Z. · Zbl 0813.62001
[41] WEISS, L. 1971. Asy mptotic properties of maximum likelihood estimators in some nonstandard cases. J. Amer. Statist. Assoc. 66 345 350. Z. JSTOR: · Zbl 0223.62037 · doi:10.2307/2283934
[42] 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. · Zbl 0616.62025 · doi:10.1214/aos/1176350177
[43] 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. JSTOR: · Zbl 0348.62026 · doi:10.2307/2285322
[44] CLEVELAND, OHIO 44106-7058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.