Small area estimation of general parameters with application to poverty indicators: a hierarchical Bayes approach. (English) Zbl 1454.62489

Molina, Isabel; Nandram, Balgobin; Rao, J. N. K.

Ann. Appl. Stat. 8, No. 2, 852-885 (2014).

Summary: Poverty maps are used to aid important political decisions such as allocation of development funds by governments and international organizations. Those decisions should be based on the most accurate poverty figures. However, often reliable poverty figures are not available at fine geographical levels or for particular risk population subgroups due to the sample size limitation of current national surveys. These surveys cannot cover adequately all the desired areas or population subgroups and, therefore, models relating the different areas are needed to “borrow strength” from area to area. In particular, the Spanish Survey on Income and Living Conditions (SILC) produces national poverty estimates but cannot provide poverty estimates by Spanish provinces due to the poor precision of direct estimates, which use only the province specific data. It also raises the ethical question of whether poverty is more severe for women than for men in a given province. We develop a hierarchical Bayes (HB) approach for poverty mapping in Spanish provinces by gender that overcomes the small province sample size problem of the SILC. The proposed approach has a wide scope of application because it can be used to estimate general nonlinear parameters. We use a Bayesian version of the nested error regression model in which Markov chain Monte Carlo procedures and the convergence monitoring therein are avoided. A simulation study reveals good frequentist properties of the HB approach. The resulting poverty maps indicate that poverty, both in frequency and intensity, is localized mostly in the southern and western provinces and it is more acute for women than for men in most of the provinces.

MSC:

62P20	Applications of statistics to economics
62D05	Sampling theory, sample surveys
62F15	Bayesian inference
91B82	Statistical methods; economic indices and measures

Keywords:

hierarchical Bayes; mixed linear model; nested error linear regression model; noninformative priors; poverty mapping; small area estimation

Software:

WinBUGS

References:

[1]	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.
[2]	Bell, W. (1997). Models for county and state poverty estimates. Preprint, Statistical Research Division, U.S. Census Bureau, Washington, DC.
[3]	Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis , 2nd ed. Springer, New York. · Zbl 0572.62008
[4]	Betti, G., Cheli, B., Lemmi, A. and Verma, V. (2006). Multidimensional and longitudinal poverty: An integrated fuzzy approach. In Fuzzy Set Approach to Multidimensional Poverty Measurement (A. Lemmi and G. Betti, eds.) 111-137. Springer, New York.
[5]	Box, G. E. P. (1980). Sampling and Bayes’ inference in scientific modelling and robustness (Discussion by Professor Seymour Geisser). J. Roy. Statist. Soc. Ser. A 143 416-417. · Zbl 0471.62036
[6]	Breidenbach, J. and Astrup, R. (2012). Small area estimation of forest attributes in the Norwegian national forest inventory. Eur. J. For. Res. 131 1255-1267.
[7]	Chen, M.-H. and Shao, Q.-M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. J. Comput. Graph. Statist. 8 69-92.
[8]	Datta, G. S. (2009). Model-based approach to small area estimation. In Handbook of Statistics 29 A : Sample Surveys : Design , Methods and Applications (D. Pfeffermann and C. R. Rao, eds.) 251-288. Elsevier, Amsterdam.
[9]	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
[10]	Elbers, C., Lanjouw, J. O. and Lanjouw, P. (2003). Micro-level estimation of poverty and inequality. Econometrica 71 355-364. · Zbl 1184.91163
[11]	Ferretti, C. and Molina, I. (2012). Fast EB method for estimating complex poverty indicators in large populations. J. Indian Soc. Agricultural Statist. 66 105-120.
[12]	Foster, J., Greer, J. and Thorbecke, E. (1984). A class of decomposable poverty measures. Econometrica 52 761-766. · Zbl 0555.90029
[13]	Gelfand, A. E., Dey, D. K. and Chang, H. (1992). Model determination using predictive distributions with implementation via sampling-based methods. In Bayesian Statistics , 4 ( Peñíscola , 1991) 147-167. Oxford Univ. Press, New York.
[14]	Jiang, J. and Lahiri, P. (2006). Mixed model prediction and small area estimation. TEST 15 1-96. · Zbl 1149.62320
[15]	Marhuenda, Y., Molina, I. and Morales, D. (2013). Small area estimation with spatio-temporal Fay-Herriot models. Comput. Statist. Data Anal. 58 308-325. · Zbl 1365.62037
[16]	Mohadjer, L., Rao, J. N. K., Liu, B., Krenzke, T. and Van de Kerckhove, W. (2012). Hierarchical Bayes small area estimates of adult literacy using unmatched sampling and linking models. J. Indian Soc. Agricultural Statist. 66 55-63, 232-233.
[17]	Molina, I. and Morales, D. (2009). Small area estimation of poverty indicators. Bol. Estad. Investig. Oper. 25 218-225.
[18]	Molina, I. and Rao, J. N. K. (2010). Small area estimation of poverty indicators. Canad. J. Statist. 38 369-385. · Zbl 1235.62140
[19]	Nandram, B. and Choi, J. W. (2005). Hierarchical Bayesian nonignorable nonresponse regression models for small areas: An application to the NHNES data. Surv. Methodol. 31 73-84.
[20]	Nandram, B. and Choi, J. W. (2010). A Bayesian analysis of body mass index data from small domains under nonignorable nonresponse and selection. J. Amer. Statist. Assoc. 105 120-135. · Zbl 1397.62463
[21]	Nandram, B., Sedransk, J. and Pickle, L. W. (2000). Bayesian analysis and mapping of mortality rates for chronic obstructive pulmonary disease. J. Amer. Statist. Assoc. 95 1110-1118. · Zbl 1004.62091
[22]	Natarajan, R. and Kass, R. E. (2000). Reference Bayesian methods for generalized linear mixed models. J. Amer. Statist. Assoc. 95 227-237. · Zbl 0996.62021
[23]	Neri, L., Ballini, F. and Betti, G. (2005). Poverty and inequality in transition countries. Statistics in Transition 7 135-157.
[24]	Ntzoufras, I. (2009). Bayesian Modeling Using WinBUGS . Wiley, Hoboken, NJ. · Zbl 1218.62015
[25]	Pettit, L. I. (1990). The conditional predictive ordinate for the normal distribution. J. R. Stat. Soc. Ser. B Stat. Methodol. 52 175-184. · Zbl 0699.62031
[26]	Pfeffermann, D. (2013). New important developments in small area estimation. Statist. Sci. 28 40-68. · Zbl 1332.62038
[27]	Pfeffermann, D. and Sverchkov, M. (2007). Small-area estimation under informative probability sampling of areas and within the selected areas. J. Amer. Statist. Assoc. 102 1427-1439. · Zbl 1333.62023
[28]	Rao, J. N. K. (2003). Small Area Estimation . Wiley, Hoboken, NJ. · Zbl 1026.62003
[29]	Toto, Ma. C. S. and Nandram, B. (2010). A Bayesian predictive inference for small area means incorporating covariates and sampling weights. J. Statist. Plann. Inference 140 2963-2979. · Zbl 1203.62029
[30]	Wang, J. C., Holan, S. H., Nandram, B., Barboza, W., Toto, C. and Anderson, E. (2012). A Bayesian approach to estimating agricultural yield based on multiple repeated surveys. J. Agric. Biol. Environ. Stat. 17 84-106. · Zbl 1302.62284
[31]	You, Y. and Zhou, Q. M. (2011). Hierarchical Bayes small area estimation under a spatial model with application to health survey data. Surv. Methodol. 37 25-37.