The Annals of Statistics

Mean squared error of empirical predictor

Kalyan Das, Jiming Jiang, and J. N. K. Rao

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The term “empirical predictor” refers to a two-stage predictor of a linear combination of fixed and random effects. In the first stage, a predictor is obtained but it involves unknown parameters; thus, in the second stage, the unknown parameters are replaced by their estimators. In this paper, we consider mean squared errors (MSE) of empirical predictors under a general setup, where ML or REML estimators are used for the second stage. We obtain second-order approximation to the MSE as well as an estimator of the MSE correct to the same order. The general results are applied to mixed linear models to obtain a second-order approximation to the MSE of the empirical best linear unbiased predictor (EBLUP) of a linear mixed effect and an estimator of the MSE of EBLUP whose bias is correct to second order. The general mixed linear model includes the mixed ANOVA model and the longitudinal model as special cases.

Article information

Ann. Statist., Volume 32, Number 2 (2004), 818-840.

First available in Project Euclid: 28 April 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62J99: None of the above, but in this section

ANOVA model EBLUP longitudinal model mixed linear model variance components


Das, Kalyan; Jiang, Jiming; Rao, J. N. K. Mean squared error of empirical predictor. Ann. Statist. 32 (2004), no. 2, 818--840. doi:10.1214/009053604000000201.

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  • Datta, G. S. and Lahiri, P. (2000). A unified measure of uncertainty of estimated best linear unbiased predictors in small area estimation problems. Statist. Sinica 10 613--627.
  • Fay, R. E. and Herriot, R. A. (1979). Estimates of income for small places: An application of James--Stein procedures to census data. J. Amer. Statist. Assoc. 74 269--277.
  • Ghosh, M. and Rao, J. N. K. (1994). Small area estimation: An appraisal (with discussion). Statist. Sci. 9 55--93.
  • Harville, D. A. and Jeske, D. R. (1992). Mean squared error of estimation or prediction under a general linear model. J. Amer. Statist. Assoc. 87 724--731.
  • Henderson, C. R. (1975). Best linear unbiased estimation and prediction under a selection model. Biometrics 31 423--447.
  • Jiang, J. (1996). REML estimation: Asymptotic behavior and related topics. Ann. Statist. 24 255--286.
  • Jiang, J. (2000). A matrix inequality and its statistical application. Linear Algebra Appl. 307 131--144.
  • Jiang, J. (2004). Dispersion matrix in balanced mixed ANOVA models. Linear Algebra Appl. 382 211--219.
  • Jiang, J., Lahiri, P. and Wan, S. (2002). A unified jackknife theory for empirical best prediction with $M$-estimation. Ann. Statist. 30 1782--1810.
  • Kackar, R. N. and Harville, D. A. (1981). Unbiasedness of two-stage estimation and prediction procedures for mixed linear models. Comm. Statist. Theory Methods 10 1249--1261.
  • Kackar, R. N. and Harville, D. A. (1984). Approximations for standard errors of estimators of fixed and random effects in mixed linear models. J. Amer. Statist. Assoc. 79 853--862.
  • Kenward, M. G. and Roger, J. H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics 53 983--997.
  • Laird, N. M. and Ware, J. M. (1982). Random effects models for longitudinal data. Biometrics 38 963--974.
  • Miller, J. J. (1977). Asymptotic properties of maximum likelihood estimates in the mixed model of the analysis of variance. Ann. Statist. 5 746--762.
  • National Research Council (2000). Small-Area Estimates of School-Age Children in Poverty. National Academy Press, Washington, DC.
  • Prasad, N. G. N. and Rao, J. N. K. (1990). The estimation of the mean squared error of small-area estimators. J. Amer. Statist. Assoc. 85 163--171.
  • Rao, C. R. and Kleffe, J. (1988). Estimation of Variance Components and Applications. North-Holland, Amsterdam.
  • Robinson, G. K. (1991). That BLUP is a good thing: The estimation of random effects (with discussion). Statist. Sci. 6 15--51.
  • Searle, S. R., Casella, G. and McCulloch, C. E. (1992). Variance Components. Wiley, New York.
  • Searle, S. R. and Henderson, H. V. (1979). Dispersion matrices for variance components models. J. Amer. Statist. Assoc. 74 465--470.