, Volume 25, Issue 4, pp 692–709 | Cite as

Simultaneous confidence bands for the distribution function of a finite population and of its superpopulation

  • Jiangyan Wang
  • Suojin Wang
  • Lijian Yang
Original Paper


Simultaneous confidence bands (SCBs) are proposed for the distribution function of a finite population and of the latent superpopulation via the empirical distribution function (nonsmooth) and kernel distribution estimator (smooth) based on a simple random sample (SRS), either with or without finite population correction. It is shown that both nonsmooth and smooth SCBs achieve asymptotically the nominal confidence level under standard assumptions. In particular, the uncorrected nonsmooth SCB for superpopulation is exactly the same as the Kolmogorov–Smirnov SCB based on an independent and identically distributed sample as long as the SRS size is infinitesimal relative to the finite population size. Extensive simulation studies confirm the asymptotic properties. As an illustration, the proposed SCBs are constructed for the population distribution of the well-known baseball data (Lohr, Sampling: design and analysis, 2nd edn. Brooks/Cole, Boston, 2009).


Bandwidth Brownian bridge Kernel Kolmogorov distribution Sample survey 

Mathematics Subject Classification

62D05 62G05 62G15 62G20 



We thank four reviewers, an Associate Editor and the Editor-in-Chief Ana F. Militino for their helpful comments and suggestions that have led to a much improved version of this paper. This research was supported in part by Jiangsu Key-Discipline Program (Statistics) ZY107002, ZY107992, National Natural Science Foundation of China award 11371272, Research Fund for the Doctoral Program of Higher Education of China award 20133201110002, Soochow University Excellent Doctoral Dissertation Project 233200113 and Jiangsu Graduate Students’ Innovative Research Project KYZZ_0331.

Supplementary material

11749_2016_491_MOESM1_ESM.pdf (109 kb)
Supplementary material 1 (pdf 108 KB)


  1. Billingsley P (1999) Convergence of probability measures, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  2. Cai L, Yang L (2015) A smooth simultaneous confidence band for conditional variance function. TEST 24:632–655MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cao G, Wang L, Li Y, Yang L (2016) Oracle-efficient confidence envelopes for covariance functions in dense functional data. Stat Sin 26:359–383MathSciNetzbMATHGoogle Scholar
  4. Cao G, Yang L, Todem D (2012) Simultaneous inference for the mean function based on dense functional data. J Nonpar Stat 24:359–377MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cardot H, Degras D, Josserand E (2013) Confidence bands for Horvitz–Thompson estimators using sampled noisy functional data. Bernoulli 19:2067–2097MathSciNetCrossRefzbMATHGoogle Scholar
  6. Cardot H, Josserand E (2011) Horvitz-Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling. Biometrika 98:107–118MathSciNetCrossRefzbMATHGoogle Scholar
  7. Chambers RL, Dunstan R (1986) Estimation distribution functions from survey data. Biometrika 73:597–604MathSciNetCrossRefzbMATHGoogle Scholar
  8. Chen J, Wu C (2002) Estimation of distribution function and quantiles using the model-calibrated pseudo empirical likelihood method. Stat Sin 12:1223–1239MathSciNetzbMATHGoogle Scholar
  9. Cheng M, Peng L (2002) Regression modeling for nonparametric estimation of distribution and quantile functions. Stat Sin 12:1043–1060MathSciNetzbMATHGoogle Scholar
  10. Degras D (2011) Simultaneous confidence bands for nonparametric regression with functional data. Stat Sin 21:1735–1765MathSciNetzbMATHGoogle Scholar
  11. Falk M (1985) Asymptotic normality of the kernel quantile estimator. Ann Stat 13:428–433MathSciNetCrossRefzbMATHGoogle Scholar
  12. Francisco C, Fuller W (1991) Quantile estimation with a complex survey design. Ann Stat 19:454–469MathSciNetCrossRefzbMATHGoogle Scholar
  13. Frey J (2009) Confidence bands for the CDF when sampling from a finite population. Comput Stat Data Anal 53:4126–4132MathSciNetCrossRefzbMATHGoogle Scholar
  14. Gu L, Wang L, Härdle W, Yang L (2014) A simultaneous confidence corridor for varying coefficient regression with sparse functional data. TEST 23:806–843MathSciNetCrossRefzbMATHGoogle Scholar
  15. Gu L, Yang L (2015) Oracally efficient estimation for single-index link function with simultaneous confidence band. Electron J Stat 9:1540–1561MathSciNetCrossRefzbMATHGoogle Scholar
  16. Liu R, Yang L (2008) Kernel estimation of multivariate cumulative distribution function. J Nonpar Stat 20:661–677MathSciNetCrossRefzbMATHGoogle Scholar
  17. Lohr s (2009) Sampling: design and analysis, 2nd edn. Brooks/Cole, BostonzbMATHGoogle Scholar
  18. Ma S, Yang L, Carroll R (2012) A simultaneous confidence band for sparse longitudinal regression. Stat Sin 22:95–122MathSciNetzbMATHGoogle Scholar
  19. O’Neill T, Stern S (2012) Finite population corrections for the Kolmogorov-Smirnov tests. J Nonpar Stat 24:497–504MathSciNetCrossRefzbMATHGoogle Scholar
  20. Reiss R (1981) Nonparametric estimation of smooth distribution funtions. Scand J Stat 8:116–119MathSciNetzbMATHGoogle Scholar
  21. Rosén B (1964) Limit theorems for sampling from finite populations. Arkiv för Matematik 5:383–424MathSciNetCrossRefzbMATHGoogle Scholar
  22. Song Q, Liu R, Shao Q, Yang L (2014) A simultaneous confidence band for dense longitudinal regression. Commun Stat Theory Methods 43:5195–5210MathSciNetCrossRefzbMATHGoogle Scholar
  23. Wang J, Cheng F, Yang L (2013) Smooth simultaneous confidence bands for cumulative distribution functions. J Nonpar Stat 25:395–407MathSciNetCrossRefzbMATHGoogle Scholar
  24. Wang J, Liu R, Cheng F, Yang L (2014) Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band. Ann Stat 42:654–668MathSciNetCrossRefzbMATHGoogle Scholar
  25. Wang J, Yang L (2009) Polynomial spline confidence bands for regression curves. Stat Sin 19:325–342MathSciNetzbMATHGoogle Scholar
  26. Wang S, Dorfman A (1996) A new estimator for the finite population distribution function. Biometrika 83:639–652MathSciNetCrossRefzbMATHGoogle Scholar
  27. Xue L, Wang J (2010) Distribution function estimation by constrained polynomial spline regression. J Nonpar Stat 22:443–457MathSciNetCrossRefzbMATHGoogle Scholar
  28. Yamato H (1973) Uniform convergence of an estimator of a distribution function. Bull Math Stat 15:69–78MathSciNetzbMATHGoogle Scholar
  29. Zheng S, Yang L, Härdle W (2014) A smooth simultaneous confidence corridor for the mean of sparse functional data. J Am Stat Assoc 109:661–673MathSciNetCrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2016

Authors and Affiliations

  1. 1.Center for Advanced Statistics and Econometrics ResearchSoochow UniversitySuzhouChina
  2. 2.Department of StatisticsTexas A&M UniversityCollege StationUSA
  3. 3.Center for Statistical Science and Department of Industrial EngineeringTsinghua UniversityBeijingChina

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