The Annals of Statistics

Functional central limit theorems for single-stage sampling designs

Hélène Boistard, Hendrik P. Lopuhaä, and Anne Ruiz-Gazen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For a joint model-based and design-based inference, we establish functional central limit theorems for the Horvitz–Thompson empirical process and the Hájek empirical process centered by their finite population mean as well as by their super-population mean in a survey sampling framework. The results apply to single-stage unequal probability sampling designs and essentially only require conditions on higher order correlations. We apply our main results to a Hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation.

Article information

Source
Ann. Statist. Volume 45, Number 4 (2017), 1728-1758.

Dates
Received: September 2015
Revised: August 2016
First available in Project Euclid: 28 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1498636872

Digital Object Identifier
doi:10.1214/16-AOS1507

Zentralblatt MATH identifier
06773289

Subjects
Primary: 62D05: Sampling theory, sample surveys

Keywords
Design and model-based inference Hájek Process Horvitz–Thompson process rejective sampling Poisson sampling high entropy designs poverty rate

Citation

Boistard, Hélène; Lopuhaä, Hendrik P.; Ruiz-Gazen, Anne. Functional central limit theorems for single-stage sampling designs. Ann. Statist. 45 (2017), no. 4, 1728--1758. doi:10.1214/16-AOS1507. https://projecteuclid.org/euclid.aos/1498636872


Export citation

References

  • [1] Barrett, G. F. and Donald, S. G. (2009). Statistical inference with generalized Gini indices of inequality, poverty, and welfare.J. Bus. Econom. Statist.271–17.
  • [2] Berger, Y. G. (1998). Rate of convergence for asymptotic variance of the Horvitz–Thompson estimator.J. Statist. Plann. Inference74149–168.
  • [3] Berger, Y. G. (1998). Rate of convergence to normal distribution for the Horvitz–Thompson estimator.J. Statist. Plann. Inference67209–226.
  • [4] Berger, Y. G. (2011). Asymptotic consistency under large entropy sampling designs with unequal probabilities.Pakistan J. Statist.27407–426.
  • [5] Berger, Y. G. and Skinner, C. J. (2003). Variance estimation for a low income proportion.J. R. Stat. Soc. Ser. C. Appl. Stat.52457–468.
  • [6] Berger, Y. G. and Skinner, C. J. (2005). A jackknife variance estimator for unequal probability sampling.J. R. Stat. Soc. Ser. B. Stat. Methodol.6779–89.
  • [7] Bertail, P., Chautru, E. and Clémençon, S. (2016). Empirical processes in survey sampling.Scand. J. Stat..DOI:10.1111/sjos.12243.
  • [8] Bhattacharya, D. (2007). Inference on inequality from household survey data.J. Econometrics137674–707.
  • [9] Bhattacharya, D. and Mazumder, B. (2011). A nonparametric analysis of black–white differences in intergenerational income mobility in the United States.Quant. Econ.2335–379.
  • [10] Bickel, P. J. and Freedman, D. A. (1984). Asymptotic normality and the bootstrap in stratified sampling.Ann. Statist.12470–482.
  • [11] Billingsley, P. (1999).Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [12] Binder, D. and Roberts, G. (2009). Design- and model-based inference for model parameters. InHandbook of Statistics29B:Sample Surveys:Design,Methods and Applications(D. Pfefferman and C. R. Rao, eds.) 33–54. Elsevier, Amsterdam.
  • [13] Boistard, H., Lopuhaä, H. P. and Ruiz-Gazen, A. (2012). Approximation of rejective sampling inclusion probabilities and application to high order correlations.Electron. J. Stat.61967–1983.
  • [14] Boistard, H., Lopuhaä, H. P. and Ruiz-Gazen, A. (2017). Supplement to “Functional central limit theorems for single-stage samplings designs.”DOI:10.1214/16-AOS1507SUPP.
  • [15] Breidt, F. J. and Opsomer, J. D. (2000). Local polynomial regresssion estimators in survey sampling.Ann. Statist.281026–1053.
  • [16] Breslow, N. E. and Wellner, J. A. (2007). Weighted likelihood for semiparametric models and two-phase stratified samples, with application to Cox regression.Scand. J. Stat.3486–102.
  • [17] Cardot, H., Chaouch, M., Goga, C. and Labruère, C. (2010). Properties of design-based functional principal components analysis.J. Statist. Plann. Inference14075–91.
  • [18] Chauvet, G. (2015). Coupling methods for multistage sampling.Ann. Statist.432484–2506.
  • [19] Conti, P. L. (2014). On the estimation of the distribution function of a finite population under high entropy sampling designs, with applications.Sankhya B76234–259.
  • [20] Conti, P. L., Marella, D. and Mecatti, F. (2015). Recovering sampling distributions of statistics of finite populations via resampling: A predictive approach. Preprint.
  • [21] Davidson, R. (2009). Reliable inference for the Gini index.J. Econometrics15030–40.
  • [22] Dell, F. and d’Haultfœuille, X. (2008). Measuring the evolution of complex indicators: Theory and application to the poverty rate in France.Ann. Écon. Stat.90259–290.
  • [23] Deville, J.-C. and Särndal, C.-E. (1992). Calibration estimators in survey sampling.J. Amer. Statist. Assoc.87376–382.
  • [24] Dorfman, A. H. (2009). Chapter 36: Inference on distribution functions and quantiles. InHandbook of Statistics29B:Sample Surveys:Design,Methods and Applications(D. Pfefferman and C. R. Rao, eds.) 371–395. Elsevier, Amsterdam.
  • [25] Dupačová, J. (1979). A note on rejective sampling. InContributions to Statistics71–78. Reidel, Dordrecht.
  • [26] Escobar, E. L. and Berger, Y. G. (2013). A jackknife variance estimator for self-weighted two-stage samples.Statist. Sinica23595–613.
  • [27] Francisco, C. A. and Fuller, W. A. (1991). Quantile estimation with a complex survey design.Ann. Statist.19454–469.
  • [28] Fuller, W. A. (2009).Sampling Statistics. Wiley, New York.
  • [29] Graf, E. and Tillé, Y. (2014). Variance estimation using linearization for poverty and social exclusion indicators.Surv. Methodol.4061–79.
  • [30] Hájek, J. (1959). Optimum strategy and other problems in probability sampling.Čas. Pěst. Mat.84387–423.
  • [31] Hájek, J. (1964). Asymptotic theory of rejective sampling with varying probabilities from a finite population.Ann. Math. Stat.351491–1523.
  • [32] Isaki, C. T. and Fuller, W. A. (1982). Survey design under the regression superpopulation model.J. Amer. Statist. Assoc.7789–96.
  • [33] Korn, E. L. and Graubard, B. I. (1998). Variance estimation for superpopulation parameters.Statist. Sinica81131–1151.
  • [34] Krewski, D. and Rao, J. N. K. (1981). Inference from stratified samples: Properties of the linearization, jackknife and balanced repeated replication methods.Ann. Statist.91010–1019.
  • [35] Lin, D. Y. (2000). On fitting Cox’s proportional hazards models to survey data.Biometrika8737–47.
  • [36] Oguz-Alper, M. and Berger, Y. G. (2015). Variance estimation of change of poverty based upon the Turkish EU-SILC survey.J. Off. Stat.31155–175.
  • [37] Pfeffermann, D. and Sverchkov, M. (2009). Inference under informative sampling. InHandbook of Statistics29B:Sample Surveys:Inference and Analysis(D. Pfefferman and C. R. Rao, eds.) 455–487. Elsevier, Amsterdam.
  • [38] Præstgaard, J. and Wellner, J. A. (1993). Exchangeably weighted bootstraps of the general empirical process.Ann. Probab.212053–2086.
  • [39] Prásková, Z. and Sen, P. K. (2009). Asymptotic in finite population sampling. InHandbook of Statistics29B:Sample Surveys:Inference and Analysis(D. Pfefferman and C. R. Rao, eds.) 489–522. Elsevier, Amsterdam.
  • [40] Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1994). Estimation of regression coefficients when some regressors are not always observed.J. Amer. Statist. Assoc.89846–866.
  • [41] Rubin-Bleuer, S. and Schiopu Kratina, I. (2005). On the two-phase framework for joint model and design-based inference.Ann. Statist.332789–2810.
  • [42] Saegusa, T. and Wellner, J. A. (2013). Weighted likelihood estimation under two-phase sampling.Ann. Statist.41269–295.
  • [43] Silverman, B. W. (1986).Density Estimation for Statistics and Data Analysis. Chapman & Hall, London.
  • [44] Thompson, M. E. (1997).Theory of Sample Surveys. Monographs on Statistics and Applied Probability74. Chapman & Hall, London.
  • [45] van der Vaart, A. W. (1998).Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics3. Cambridge Univ. Press, Cambridge.
  • [46] van der Vaart, A. W. and Wellner, J. A. (1996).Weak Convergence and Empirical Processes:With Applications to Statistics. Springer, New York.
  • [47] Víšek, J. Á. (1979). Asymptotic distribution of simple estimate for rejective, Sampford and successive sampling. InContributions to Statistics263–275. Reidel, Dordrecht.
  • [48] Wang, J. C. (2012). Sample distribution function based goodness-of-fit test for complex surveys.Comput. Statist. Data Anal.56664–679.

Supplemental materials

  • Supplement to “Functional central limit theorems for single-stage samplings designs.”. Appendix A: Proofs for results in the main text. This supplement contains detailed proofs of lemmas, propositions and corollaries for results in the main text that are not present in Section 9. Appendix B: Additional technicalities. This supplement contains detailed proofs of some remarks and additional lemmas. Appendix C: Fixed size sampling designs with deterministic inclusion probabilities. This supplement contains results for fixed size sampling designs with deterministic inclusion probabilities, obtained under alternative conditions for (C2)–(C4).