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Nonparametric estimation of mean-squared prediction error in nested-error regression models. (English) Zbl 1246.62106
Summary: Nested-error regression models are widely used for analyzing clustered data. For example, they are often applied to two-stage sample surveys, and in biology and econometrics. Prediction is usually the main goal of such analyses, and the mean-squared prediction error is the main way in which prediction performance is measured. We suggest a new approach to estimating the mean-squared prediction error. We introduce a matched-moment, double-bootstrap algorithm, enabling the notorious underestimation of the naive mean-squared error estimator to be substantially reduced. Our approach does not require specific assumptions about the distributions of the errors. Additionally, it is simple and easy to apply. This is achieved through using Monte Carlo simulations to implicitly develop formulae which, in a more conventional approach, would be derived laboriously by mathematical arguments.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62G09 Nonparametric statistical resampling methods 62G05 Nonparametric estimation 65C05 Monte Carlo methods
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##### References:
  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.  Bell, W. (2001). Discussion of “Jackknife in the Fay–Herriot model with an application,” by Jiang, Lahiri, Wan and Wu. In Proc. Seminar on Funding Opportunity in Survey Research 98–104. Council of Professional Associations on Federal Statistics, Washington.  Booth, J. G. and Hobert, J. P. (1998). Standard errors of prediction in generalized linear mixed models. J. Amer. Statist. Assoc. 93 262–272. JSTOR: · Zbl 1068.62516  Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186. JSTOR: · Zbl 0673.62033  Chen, S. and Lahiri, P. (2003). A comparison of different MSPE estimators of EBLUP for the Fay–Herriot model. Proc. Survey Research Methods Section 905–911. Amer. Statist. Assoc., Alexandria, VA.  Das, K., Jiang, J. and Rao, J. N. K. (2004). Mean squared error of empirical predictor. Ann. Statist. 32 818–840. · Zbl 1092.62063  Datta, G. S. and Ghosh, M. (1991). Bayesian prediction in linear models: Applications to small area estimation. Ann. Statist. 19 1748–1770. · Zbl 0738.62030  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. · Zbl 1054.62566  Delaigle, A. and Gijbels, I. (2004). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249–267. · Zbl 1429.62125  Domínguez, M. A. and Lobato, I. N. (2003). Testing the martingale difference hypothesis. Econometric Rev. 22 351–377. · Zbl 1030.62066  El-Amraoui, A. and Goffinet, B. (1991). Estimation of the density of $$G$$ given observations of $$Y=G+E$$. Biometrical J. 33 347–355. · Zbl 0734.62041  Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272. · Zbl 0729.62033  Fan, J. (1992). Deconvolution with supersmooth distributions. Canad. J. Statist. 20 155–169. JSTOR: · Zbl 0754.62020  Fan, Y. and Li, Q. (2002). A consistent model specification test based on the kernel sum of squares of residuals. Econometric Rev. 21 337–352. · Zbl 1033.62038  Flachaire, E. (2002). Bootstrapping heteroskedasticity consistent covariance matrix estimator. Comput. Statist. 17 501–506. · Zbl 1037.62023  González Manteiga, W., Martínez Miranda, M. D. and Pérez González, A. (2004). The choice of smoothing parameter in nonparametric regression through wild bootstrap. Comput. Statist. Data Anal. 47 487–515. · Zbl 1429.62139  Hall, P. and Maiti, T. (2005). Nonparametric estimation of mean-squared prediction error in nested-error regression models. Available at http://arxiv.org/abs/math/0509493. · Zbl 1246.62106  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. JSTOR: · Zbl 0763.62039  Jiang, J., Lahiri, P. and Wan, S.-M. (2002). A unified jackknife theory for empirical best prediction with $$M$$-estimation. Ann. Statist. 30 1782–1810. · Zbl 1020.62025  Kacker, R. 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. JSTOR: · Zbl 0557.62066  Kauermann, G. and Opsomer, J. D. (2003). Local likelihood estimation in generalized additive models. Scand. J. Statist. 30 317–337. · Zbl 1053.62084  Lahiri, P. (2003). On the impact of bootstrap in survey sampling and small-area estimation. Statist. Sci. 18 199–210. · Zbl 1331.62076  Lahiri, P. (2003). A review of empirical best linear unbiased prediction for the Fay–Herriot small-area model. Philippine Statistician 52 1–15.  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. JSTOR: · Zbl 0826.62008  Li, Q., Hsiao, C. and Zinn, J. (2003). Consistent specification tests for semiparametric/nonparametric models based on series estimation methods. J. Econometrics 112 295–325. · Zbl 1027.62027  Li, T. and Vuong, Q. (1998). Nonparametric estimation of the measurement error model using multiple indicators. J. Multivariate Anal. 65 139–165. · Zbl 1127.62323  Prasad, N. G. N. and Rao, J. N. K. (1990). The estimation of mean squared error of small-area estimators. J. Amer. Statist. Assoc. 85 163–171. JSTOR: · Zbl 0719.62064  Prášková, Z. (2003). Wild bootstrap in RCA(1) model. Kybernetika ( Prague ) 39 1–12. · Zbl 1248.62155  Rao, J. N. K. (2003). Small Area Estimation . Wiley, Hoboken, NJ. · Zbl 1026.62003  Rao, J. N. K. and Choudhry, G. H. (1995). Small area estimation: Overview and empirical study. In Business Survey Methods (B. G. Cox, D. A. Binder, B. N. Chinnappa, A. Christianson, M. J. Colledge and P. S. Kott, eds.) 527–542. Wiley, New York. · Zbl 0851.90023  Stukel, D. M. and Rao, J. N. K. (1997). Estimation of regression models with nested error structure and unequal error variances under two and three stage cluster sampling. Statist. Probab. Lett. 35 401–407. · Zbl 0955.62071  Wang, J. and Fuller, W. A. (2003). The mean squared error of small area predictors constructed with estimated area variances. J. Amer. Statist. Assoc. 98 716–723. · Zbl 1046.62071
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