ABSTRACT

In the Fay–Herriot model, we consider estimators of the linking variance obtained using different types of resampling schemes. The usefulness of this approach is that even when the estimator from the original data falls below zero or any other specified threshold, several of the resamples can potentially yield values above the threshold. We establish asymptotic consistency of the resampling-based estimator of the linking variance for a wide variety of resampling schemes and show the efficacy of using the proposed approach in numeric examples.

References

1. Barbe, P., & Bertail, P (1995). The weighted bootstrap, volume 98 of Lecture Notes in Statistics. New York, NY: Springer-Verlag. [Google Scholar]
2. Chatterjee, S. (1998). Another look at the jackknife: Further examples of generalized bootstrap. Statistics and Probability Letters, 40(4), 307–319. [Crossref], [Web of Science ®], [Google Scholar]
3. Chatterjee, S. (2018). On modifications to linking variance estimators in the Fay-Herriot model that induce robustness. Statistics And Applications, 16(1), 289–303. [Web of Science ®], [Google Scholar]
4. Chatterjee, S., & Bose, A. (2000). Variance estimation in high dimensional regression models. Statistica Sinica, 10(2), 497–516. [Web of Science ®], [Google Scholar]
5. Chatterjee, S., & Bose, A. (2002). Dimension asymptotics for generalised bootstrap in linear regression. Annals of the Institute of Statistical Mathematics, 54(2), 367–381. [Crossref], [Web of Science ®], [Google Scholar]
6. Chatterjee, S., & Bose, A. (2005). Generalized bootstrap for estimating equations. The Annals of Statistics, 33(1), 414–436. [Crossref], [Web of Science ®], [Google Scholar]
7. Chatterjee, S., Lahiri, P., & Li, H. (2008). Parametric bootstrap approximation to the distribution of EBLUP and related prediction intervals in linear mixed models. The Annals of Statistics, 36(3), 1221–1245. [Crossref], [Web of Science ®], [Google Scholar]
8. Das, K., Jiang, J., & Rao, J. N. K. (2004). Mean squared error of empirical predictor. The Annals of Statistics, 32(2), 818–840. [Crossref], [Web of Science ®], [Google Scholar]
9. Datta, G. S., Hall, P., & Mandal, A. (2011). Model selection by testing for the presence of small-area effects, and application to area-level data. Journal of the American Statistical Association, 106(493), 362–374. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
10. Datta, G. S., Rao, J. N. K., & Smith, D. D. (2005). On measuring the variability of small area estimators under a basic area level model. Biometrika, 92(1), 183–196. [Crossref], [Web of Science ®], [Google Scholar]
11. Efron, B. (1979). Bootstrap methods: Another look at the jackknife. The Annals of Statistics, 7(1), 1–26. [Crossref], [Web of Science ®], [Google Scholar]
12. Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. Boca Raton, FL: Chapman & Hall/CRC press. [Crossref], [Google Scholar]
13. Fay III, R. E., & Herriot, R. A. (1979). Estimates of income for small places: an application of James-Stein procedures to census data. Journal of the American Statistical Association, 74(366), 269–277. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
14. Hirose, M. Y., & Lahiri, P (2017). A new model variance estimator for an area level small area model to solve multiple problems simultaneously. arXiv preprint arXiv:1701.04176. [Google Scholar]
15. Jiang, J., & Lahiri, P. (2006). Mixed model prediction and small area estimation (with discussion). Test, 15(1), 1–96. [Crossref], [Web of Science ®], [Google Scholar]
16. Jiang, J., Lahiri, P., & Wan, S.-M. (2002). A unified jackknife theory for empirical best prediction with M-estimation. The Annals of Statistics, 30(6), 1782–1810. [Crossref], [Web of Science ®], [Google Scholar]
17. Kubokawa, T., & Nagashima, B. (2012). Parametric bootstrap methods for bias correction in linear mixed models. Journal of Multivariate Analysis, 106, 1–16. [Crossref], [Web of Science ®], [Google Scholar]
18. Li, H., & Lahiri, P. (2010). An adjusted maximum likelihood method for solving small area estimation problems. Journal of Multivariate Analysis, 101(4), 882–892. [Crossref], [Web of Science ®], [Google Scholar]
19. Molina, I., Rao, J. N. K., & Datta, G. S. (2015). Small area estimation under a Fay–Herriot model with preliminary testing for the presence of random area effects. Survey Methodology, 41(1), 1–19. [Web of Science ®], [Google Scholar]
20. Pfeffermann, D. (2013). New important developments in small area estimation. Statistical Science, 28(1), 40–68. [Crossref], [Web of Science ®], [Google Scholar]
21. Pfeffermann, D., & Glickman, H. (2004). Mean square error approximation in small area estimation by use of parametric and nonparametric bootstrap. ASA Section on Survey Research Methods Proceedings, Alexandria, VA, pp. 4167–4178. [Google Scholar]
22. Prasad, N. G. N., & Rao, J. N. K. (1990). The estimation of the mean squared error of small-area estimators. Journal of the American Statistical Association, 85(409), 163–171. [Taylor & Francis Online], [Web of Science ®], [Google Scholar]
23. Præstgaard, J., & Wellner, J. A. (1993). Exchangeably weighted bootstraps of the general empirical process. The Annals of Probability, 21(4), 2053–2086. [Crossref], [Web of Science ®], [Google Scholar]
24. Rao, J. N. K. (2015). Inferential issues in model-based small area estimation: Some new developments. Statistics in Transition, 16(4), 491–510. [Crossref], [Google Scholar]
25. Rao, J. N. K., & Molina, I. (2015). Small area estimation. New York, NY: Wiley. [Crossref], [Google Scholar]
26. Rubin-Bleuer, S., & You, Y. (2016). Comparison of some positive variance estimators for the Fay–Herriot small area model. Survey Methodology, 42(1), 63–85. [Web of Science ®], [Google Scholar]
27. Shao, J., & Tu, D. (1995). The jackknife and bootstrap. New York, NY: Springer-Verlag. [Crossref], [Google Scholar]
28. Yoshimori, M., & Lahiri, P. (2014a). A new adjusted maximum likelihood method for the Fay–Herriot small area model. Journal of Multivariate Analysis, 124, 281–294. [Crossref], [Web of Science ®], [Google Scholar]
29. Yoshimori, M., & Lahiri, P. (2014b). A second-order efficient empirical Bayes confidence interval. The Annals of Statistics, 42(4), 1233–1261. [Crossref], [Web of Science ®], [Google Scholar]