The Annals of Statistics

A unified jackknife theory for empirical best prediction with M-estimation

Jiming Jiang, P. Lahiri, and Shu-Mei Wan

Full-text: Open access

Abstract

The paper presents a unified jackknife theory for a fairly general class of mixed models which includes some of the widely used mixed linear models and generalized linear mixed models as special cases. The paper develops jackknife theory for the important, but so far neglected, prediction problem for the general mixed model. For estimation of fixed parameters, a jackknife method is considered for a general class of M-estimators which includes the maximum likelihood, residual maximum likelihood and ANOVA estimators for mixed linear models and the recently developed method of simulated moments estimators for generalized linear mixed models. For both the prediction and estimation problems, a jackknife method is used to obtain estimators of the mean squared errors (MSE). Asymptotic unbiasedness of the MSE estimators is shown to hold essentially under certain moment conditions. Simulation studies undertaken support our theoretical results.

Article information

Source
Ann. Statist., Volume 30, Number 6 (2002), 1782-1810.

Dates
First available in Project Euclid: 23 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aos/1043351257

Digital Object Identifier
doi:10.1214/aos/1043351257

Mathematical Reviews number (MathSciNet)
MR1969450

Zentralblatt MATH identifier
1020.62025

Subjects
Primary: 62G09: Resampling methods 62D05: Sampling theory, sample surveys

Keywords
Empirical best predictors mean squared errors $M$-estimators mixed linear models mixed logistic models small-area estimation uniform consistency variance components

Citation

Jiang, Jiming; Lahiri, P.; Wan, Shu-Mei. A unified jackknife theory for empirical best prediction with M -estimation. Ann. Statist. 30 (2002), no. 6, 1782--1810. doi:10.1214/aos/1043351257. https://projecteuclid.org/euclid.aos/1043351257


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References

  • ARVESEN, J. N. (1969). Jackknifing U-statistics. Ann. Math. Statist. 40 2076-2100.
  • BATTESE, G. E., HARTER, R. M. and FULLER, W. A. (1988). An error-components model for prediction of county crop areas using survey and satellite data. J. Amer. Statist. Assoc. 83 28-36.
  • BRESLOW, N. E. and CLAy TON, D. G. (1993). Approximate inference in generalized linear mixed models. J. Amer. Statist. Assoc. 88 9-25.
  • BUTAR, F. B. (1997). Empirical Bay es methods in survey sampling. Ph.D. dissertation, Dept. Mathematics and Statistics, Univ. Nebraska-Lincoln.
  • CHATTOPADHy AY, M., LAHIRI, P., LARSEN, M. and REIMNITZ, J. (1999). Composite estimation of drug prevalences for sub-state areas. Survey Methodology 25 81-86.
  • 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.
  • EFRON, B. and MORRIS, C. (1973). Stein's estimation rule and its competitors: An empirical Bay es approach. J. Amer. Statist. Assoc. 68 117-130.
  • EFRON, B. and MORRIS, C. (1975). Data analysis using Stein's estimator and its generalizations. J. Amer. Statist. Assoc. 70 311-319.
  • EFRON, B. and TIBSHIRANI, R. J. (1993). An Introduction to the Bootstrap. Chapman and Hall, London.
  • ERICSON, W. A. (1969). A note on the posterior mean of a population mean. J. Roy. Statist. Soc. Ser. B 31 332-334.
  • 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 LAHIRI, P. (1987). Robust empirical Bay es estimation of means from stratified samples. J. Amer. Statist. Assoc. 82 1153-1162.
  • GHOSH, M. and MEEDEN, G. (1986). Empirical Bay es estimation in finite population sampling. J. Amer. Statist. Assoc. 81 1058-1062.
  • HUBER, P. J. (1981). Robust Statistics. Wiley, New York.
  • JIANG, J. (1996). REML estimation: Asy mptotic behavior and related topics. Ann. Statist. 24 255-286.
  • JIANG, J. (1998). Consistent estimators in generalized linear mixed models. J. Amer. Statist. Assoc. 93 720-729.
  • JIANG, J. (1999). Jackknifing MSE of empirical best predictor: A theoretical sy nthesis. Technical report, Dept. Statist., Case Western Reserve Univ.
  • JIANG, J. and LAHIRI, P. (2001). Empirical best prediction for small area inference with binary data. Ann. Inst. Statist. Math. 53 217-243.
  • LAHIRI, P. (1995). A jackknife measure of uncertainty of linear empirical Bay es estimators. Unpublished manuscript.
  • LAHIRI, P. and RAO, J. N. K. (1995). Robust estimation of mean squared error of small area estimators. J. Amer. Statist. Assoc. 90 758-766.
  • LAIRD, N. M. and LOUIS, T. A. (1987). Empirical Bay es confidence intervals based on bootstrap samples (with discussion). J. Amer. Statist. Assoc. 82 739-757.
  • LEE, Y. and NELDER, J. A. (1996). Hierarchical generalized linear models (with discussion). J. Roy. Statist. Soc. Ser. B 58 619-678.
  • LEHMANN, E. L. (1983). Theory of Point Estimation. Wiley, New York.
  • MALEC, D., SEDRANSK, J., MORIARITY, C. L. and LE CLERE, F. B. (1997). Small area inference for binary variables in the National Health Interview Survey. J. Amer. Statist. Assoc. 92 815-826.
  • MCFADDEN, D. (1989). A method of simulated moments for estimation of discrete response models without numerical integration. Econometrica 57 995-1026.
  • MORRIS, C. N. (1983). Parametric empirical Bay es inference: Theory and applications. J. Amer. Statist. Assoc. 78 47-59.
  • PRASAD, N. G. N. and RAO, J. N. K. (1988). Robust tests and confidence intervals for error variance in a regression model and for functions of variance components in an unbalanced random one-way model. Comm. Statist. Theory Methods 17 1111-1133.
  • PRASAD, N. G. N. and RAO, J. N. K. (1990). The estimation of mean squared errors of small area estimators. J. Amer. Statist. Assoc. 85 163-171.
  • RAO, J. N. K. and PRASAD, N. G. N. (1986). Discussion of "Jackknife, bootstrap and other resampling methods in regression analysis," by C. F. J. Wu. Ann. Statist. 14 1320-1322.
  • SEARLE, S. R., CASELLA, G. and MCCULLOCH, C. E. (1992). Variance Components. Wiley, New York.
  • SHAO, J. and TU, D. (1995). The Jackknife and Bootstrap. Springer, New York.
  • DAVIS, CALIFORNIA 95616 E-MAIL: jiang@wald.ucdavis.edu P. LAHIRI JOINT PROGRAM IN SURVEY METHODOLOGY UNIVERSITY OF MARy LAND 1218 LEFRAK HALL
  • COLLEGE PARK, MARy LAND 20742 E-MAIL: plahiri@survey.umd.edu S.-M. WAN DEPARTMENT OF FINANCE LUNGHWA UNIVERSITY OF SCIENCE AND TECHNOLOGY 300, WAN-SO ROAD, SEC. 1, KWEI-SAN SHANG TOU-YUANG COUNTY
  • TAIWAN, REPUBLIC OF CHINA 333