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Estimation of small area counts with the benchmarking property

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Abstract

Estimation of small area totals makes use of auxiliary variables to borrow strength from related areas through a model. Precision of final small area estimates depends on the validity of such a model. To protect against possible model failures, benchmarking procedures make the sum of small area estimates match a design consistent estimate of the total of a larger area. This is also particularly important for national institutes of statistics to ensure coherence between small area estimates and direct estimates produced at higher level planned domains. In this paper we propose a self-benchmarked estimator of small area totals which is based on a unit level logistic mixed model for a binary response. In particular, we use a plug-in approach and add a constraint to the maximum penalized quasi-likelihood (PQL) procedure considered in Saei and Chambers (Working paper M03/15, Southampton Statistical Sciences Research Institute. University of Southampton, 2003) to accommodate benchmarking. An analytic estimator for the mean squared error of the final small area estimator is also proposed following the ad-hoc procedure proposed by González-Manteiga et al. (Comput Stat Data Anal 51:2720–2733, 2007). We then compare the performance of the proposed benchmarked estimator with several competing estimators through a set of simulation studies.

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References

  1. Asparouhov, T.: Generalized multi-level modeling with sampling weights. Commun. Stat. Theory Methods 35, 439–446 (2006)

    Article  MathSciNet  Google Scholar 

  2. Bell, W., Datta, G., Ghosh, M.: Benchmarking small-area estimates. Biometrika 100, 189–202 (2013)

    Article  MathSciNet  Google Scholar 

  3. Battese, G.E., Harter, R.M., Fuller, W.A.: An error-components model for prediction of county crop areas using survey and satellite data. J. Am. Stat. Assoc. 83, 28–36 (1988)

    Article  Google Scholar 

  4. Berg, E.J., Fuller, W.A.: A small area procedure for estimating population counts. In: Proceedings of the Section on Survey Research Methods. American Statistical Association, pp. 3811–3825 . American Statistical Association, Alexandria (2009)

  5. Berg E.J., Fuller, W.A., Erciulescu, A.L.: Benchmarked small area prediction. In: Proceedings of of the Section on Survey Research Methods. American Statistical Association, pp. 4296–4309. American Statistical Association, Alexandria (2012)

  6. Boubeta, M., Lombardìa, M.J., Morales, D.: Empirical best prediction under area-level Poisson mixed models. Test 25, 548–569 (2016)

    Article  MathSciNet  Google Scholar 

  7. Breslow, N.E., Clayton, D.G.: Approximate inference to generalized linear mixed models. J. Am. Stat. Assoc. 88, 9–25 (1993)

    MATH  Google Scholar 

  8. Chandra, H., Kumar, S., Aditya, K.: Small area estimation of proportions with different levels of auxiliary data. Biometr. J. 60, 395–415 (2018)

    Article  MathSciNet  Google Scholar 

  9. D’Alò, M., Di Consiglio, L., Falorsi, S., Ranalli, M.G., Solari, F.: Use of spatial information in small area models for unemployment rate estimation at sub-provincial areas in Italy. J. Indian Soc. Agric. Stat. 66, 43–54 (2012)

    MathSciNet  Google Scholar 

  10. Datta, G.S., Ghosh, M., Steorts, R., Maples, J.: Bayesian benchmarking with applications to small area estimation. Test 20, 574–588 (2011)

    Article  MathSciNet  Google Scholar 

  11. Ghosh, M., Steorts, R.C.: Two-stage benchmarking as applied to small area estimation. Test 22, 670–687 (2013)

    Article  MathSciNet  Google Scholar 

  12. González-Manteiga, W., Lombardía, M.J., Molina, I., Morales, D., Santamaría, L.: Estimation of the mean squared error of estimators of small area linear parameters under a logistic mixed model. Comput. Stat. Data Anal. 51, 2720–2733 (2007)

    Article  Google Scholar 

  13. Grilli, L., Pratesi, M.: Weighted estimation in multilevel ordinal and binary models in the presence of informative sampling designs. Survey Methodol. 30, 93–103 (2004)

    Google Scholar 

  14. Harville, D.A.: Maximum likelihood approaches to variance component estimation and related problems. J. Am. Stat. Assoc. 72, 320–40 (1977)

    Article  MathSciNet  Google Scholar 

  15. Hobza, T., Morales, D.: Empirical best prediction under unit-level logit mixed models. J. Off. Stat. 32, 661–692 (2016)

    Article  Google Scholar 

  16. Jiang, J.: Consistent estimators in generalized linear mixed models. J. Am. Stat. Assoc. 93, 720–729 (1998)

    Article  MathSciNet  Google Scholar 

  17. Jiang, J.: Empirical best prediction for small-area inference based on generalized linear mixed models. J. Stat. Plan. Inference 111, 117–127 (2003)

    Article  MathSciNet  Google Scholar 

  18. Jiang, J.: Linear and generalized linear mixed models and their applications. Springer, New York (2007)

    MATH  Google Scholar 

  19. Jiang, J., Lahiri, P.: Empirical best prediction for small area inference with binary data. Ann. Inst. Statist. Math. 530, 217–243 (2001)

    Article  MathSciNet  Google Scholar 

  20. Jiang, J., Lahiri, P.: Mixed model prediction and small area estimation (with discussion). Test 15, 1–96 (2006)

    Article  MathSciNet  Google Scholar 

  21. Korn, E.L., Graubard, B.I.: Estimating variance components by using survey data. J. R. Stat. Soc. Ser. B 65, 175–190 (2003)

    Article  MathSciNet  Google Scholar 

  22. Lin, X., Breslow, N.E.: Bias correction in generalized linear mixed models with multiple components of dispersion. J. Am. Stat. Assoc. 91, 1007–1016 (1996)

    Article  MathSciNet  Google Scholar 

  23. Marino M.F., Ranalli M.G., Salvati N., Alfò M.: Semi-parametric empirical best prediction for small area estimation of unemployment indicators. Ann. Appl. Stat. arXiv:1704.02220v2 (in press)

  24. McCulloch, C.E., Searle, S.R.: Generalized, Linear, and Mixed Model. Wiley, New York (2001)

    MATH  Google Scholar 

  25. Militino, A.F., Ugarte, M.D., Goicoa, T.: Deriving small area estimates from information technology business surveys. J. R. Stat. Soc. Ser. A 178, 1051–1067 (2015)

    Article  MathSciNet  Google Scholar 

  26. Molina, I., Saei, A., Lombardia, M.J.: Small area estimates of labour force participation under a multinomial logit mixed model. J. R. Stat. Soc. Ser. A 170, 975–1000 (2007)

    Article  MathSciNet  Google Scholar 

  27. Pfeffermann, D., Barnard, C.H.: Some new estimators for small-area means with application to the assessment of farmland values. J. Bus. Econ. Stat. 90, 73–84 (1991)

    Google Scholar 

  28. Pfeffermann, D., Skinner, C.J., Holmes, D.J., Goldstein, H., Rasbash, J.: Weighting for unequal selection probabilities in multi-level models. J. R. Stat. Soc. Ser. B 60, 23–56 (1998)

    Article  MathSciNet  Google Scholar 

  29. Prasad, N.G.N., Rao, J.N.K.: The estimation of the mean squared error of small area estimation. J. Am. Stat. Assoc. 85, 163–171 (1990)

    Article  Google Scholar 

  30. Rabe-Hesketh, S., Skrondal, A.: Multilevel modelling of complex survey data. J. R. Stat. Soc. Ser. A 169, 805–827 (2006)

    Article  MathSciNet  Google Scholar 

  31. Rao, J.N., Molina, I.: Small area estimation. Wiley, Hoboken (2015)

    Book  Google Scholar 

  32. Rao, J.N.K., Verret, F., Hidiroglou, M.A.: A weighted composite likelihood approach to inference for two-level models from survey data. Survey Methodol. 39, 263–282 (2013)

    Google Scholar 

  33. Saei A., Chambers R.: Small area estimation under linear and generalized linear model with time and area effects. In: Working Paper M03/15, Southampton Statistical Sciences Research Institute. University of Southampton (2003)

  34. Schall, R.: Estimation in generalized linear models with random effects. Biometrika 78, 719–727 (1991)

    Article  Google Scholar 

  35. Ugarte, M.D., Militino, A.F., Goicoa, T.: Benchmarked estimates in small areas using linear mixed models with restrictions. Test 18, 342–364 (2009)

    Article  MathSciNet  Google Scholar 

  36. Wang, J., Fuller, W.A., Qu, Y.: Small area estimation under a restriction. Survey Methodol. 34, 29–36 (2008)

    Google Scholar 

  37. Yi, G., Rao, J.N.K., Li, H.: A weighted composite likelihood approach for analysis of survey data under two-level models. Stat. Sin. 26, 569–587 (2016)

    MathSciNet  MATH  Google Scholar 

  38. You, Y., Rao, J.N.K.: A pseudo-empirical best linear unbiased prediction approach to small area estimation using survey weights. Can. J. Stat. 30, 431–439 (2002)

    Article  MathSciNet  Google Scholar 

  39. You, Y., Rao, J.N.K., Dick, P.: Benchmarking hierarchical Bayes small area estimators in the Canadian census undercoverage estimation. Stat. Transit. 6, 631–640 (2004)

    Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Scienze Politiche, Universitá degli Studi di Perugia, Perugia, Italy

    M. Giovanna Ranalli & Giorgio E. Montanari

  2. Direzione Centrale per le Statistiche Congiunturali, ISTAT, Rome, Italy

    Cecilia Vicarelli

  3. Direzione Centrale per la Contabilita’ Nazionale, Rome, Italy

    Cecilia Vicarelli

Authors
  1. M. Giovanna Ranalli
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  2. Giorgio E. Montanari
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  3. Cecilia Vicarelli
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Correspondence to M. Giovanna Ranalli.

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The work of Ranalli has been partially developed under the support of the project PRIN-SURWEY (Grant 2012F42NS8, Italy).

Appendix: Tables of results from simulation study

Appendix: Tables of results from simulation study

The following tables report for each estimator and survey variable

  • the median (MeA), the 3rd quartile (Q3A), and the maximum value (MaxA) of the assolute relative bias (ARB) across all areas;

  • the median (Me), the 3rd quartile (Q3), and the maximum value (Max) of the relative root mean squared error (RRMSE) across all areas (see Tables 3, 4, 5, 6, 7, and 8)

Table 3 Estimators’ performance for survey variables \(k=1, 2\)
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Table 4 Estimators’ performance for survey variables \(k=3, 4\)
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Table 5 Estimators’ performance for survey variables \(k=5, 6\)
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Table 6 Estimators’ performance for survey variables \(k=7, 8\)
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Table 7 Estimators’ overall performance (average across the eight survey variables of the percent ratio between the values of the median, third quartile, and maximum of ARB and of RRMSE of an estimator and the respective minimum value among all estimators for each survey variable)
Full size table
Table 8 MeA ARB and Me RRMSE for survey variables \(k=1,2,3,4\) over the 10 smallest areas
Full size table

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Ranalli, M.G., Montanari, G.E. & Vicarelli, C. Estimation of small area counts with the benchmarking property. METRON 76, 349–378 (2018). https://doi.org/10.1007/s40300-018-0146-2

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