, Volume 27, Issue 1, pp 95–121 | Cite as

On the estimation of the characteristic function in finite populations with applications

  • M. D. Jiménez-Gamero
  • J. L. Moreno-Rebollo
  • J. A. Mayor-Gallego
Original Paper


This paper studies the estimation of the characteristic function of a finite population. Specifically, the weak convergence of the finite population empirical characteristic process is studied. Under suitable assumptions, it has the same limit as the empirical characteristic process for independent, identically distributed data from a random variable, up to a multiplicative constant depending on the sampling design. Applications of the obtained results for the two-sample problem, testing for independence and testing for symmetry are given.


Finite population Design-based inference High entropy designs Characteristic function Test for the two-sample problem Test for independence Test for symmetry 

Mathematics Subject Classification

62D05 62G10 62G09 



The authors thank the anonymous referees and the Associate Editor for their valuable time and careful comments, which improved the quality of this paper. M. D. Jiménez-Gamero acknowledges financial support from Grant MTM2014-55966-P of the Spanish Ministry of Economy and Competitiveness.

Supplementary material

11749_2016_514_MOESM1_ESM.pdf (233 kb)
Supplementary material 1 (pdf 234 KB)


  1. Alba-Fernández V, Jiménez-Gamero MD, Muñoz-García J (2008) A test for the two-sample problem based on empirical characteristic functions. Comput Stat Data Anal 52:3730–3748MathSciNetCrossRefMATHGoogle Scholar
  2. Anderson NH, Hall P, Titterington DM (1994) Two-sample tests for measuring discrepancies between two multivariate probability density functions using kernel-based density estimates. J Multivar Anal 50:41–54MathSciNetCrossRefMATHGoogle Scholar
  3. Berger YG (1998) Rate of convergence to normal distribution for the Horvith–Thompson estimator. J Stat Plan Inference 67:209–226CrossRefMATHGoogle Scholar
  4. Conti PL (2014) On the estimation of the distribution function of a finite population under high entropy sampling designs, with applications. Sankhyā 76:234–259MathSciNetCrossRefMATHGoogle Scholar
  5. Conti PL, Marella D (2015) Inference for quantiles of a finite population: asymptotic versus resampling results. Scand J Stat 42:545–561MathSciNetCrossRefMATHGoogle Scholar
  6. Csörgő S (1981) Limit behaviour of the empirical characteristic function. Ann Probab 9:130–144MathSciNetCrossRefMATHGoogle Scholar
  7. Csörgő S (1985) Testing for independence by the empirical characteristic function. J Multivar Anal 16:290–299MathSciNetCrossRefMATHGoogle Scholar
  8. Erdös P, Rényi A (1959) On the central limit theorem for samples from a finite population. Publ Math Inst Hung Acad Sci 4:49–61MathSciNetMATHGoogle Scholar
  9. Feller W (1971) An introduction to probability theory and its applications, vol 2. Wiley, New YorkMATHGoogle Scholar
  10. Feuerverger A, Mureika RA (1977) The empirical characteristic function and its applications. Ann Stat 5:88–97MathSciNetCrossRefMATHGoogle Scholar
  11. Hájek J (1964) Asymptotic theory of rejective sampling with varying probabilities from a finite population. Ann Math Stat 35:1491–1523MathSciNetCrossRefMATHGoogle Scholar
  12. Hájek J (1981) Sampling from a finite population. Marcel Dekker, New YorkMATHGoogle Scholar
  13. Henze N, Klar B, Meintanis SG (2003) Invariant tests for symmetry about an unspecified point based on the empirical characteristic function. J Multivar Anal 87:275–297MathSciNetCrossRefMATHGoogle Scholar
  14. Henze N, Klar B, Zhu LX (2005) Checking the adequacy of the multivariate semiparametric location shift model. J Multivar Anal 93:238–256MathSciNetCrossRefMATHGoogle Scholar
  15. Hlávka Z, Hušková M, Meintanis SG (2011) Tests for independence in non-parametric heteroscedastic regression models. J Multivar Anal 102:816–827MathSciNetCrossRefMATHGoogle Scholar
  16. Hušková M, Meintanis SG (2008) Tests for the multivariate k-sample problem based on the empirical characteristic function. J Nonparametr Stat 20:263–277MathSciNetCrossRefMATHGoogle Scholar
  17. Isaki C, Fuller W (1982) Survey design under the regression superpopulation model. J Am Stat Assoc 77:89–96MathSciNetCrossRefMATHGoogle Scholar
  18. Jiménez-Gamero MD, Alba-Fernández V, Muñoz-García J, Chalco-Cano Y (2009) Goodness-of-fit tests based on empirical characteristic functions. Comput Stat Data Anal 53:3957–3971MathSciNetCrossRefMATHGoogle Scholar
  19. Kankainen A, Ushakov NG (1998) A consistent modification of a test for independence based on the empirical characteristic function. J Math Sci 89:1486–1493MathSciNetCrossRefMATHGoogle Scholar
  20. Kish L (1965) Survey sampling. Wiley, New YorkMATHGoogle Scholar
  21. Marcus MB (1981) Weak convergence of the empirical characteristic function. Ann Prob 9:194–201MathSciNetCrossRefMATHGoogle Scholar
  22. Meintanis SG (2005) Permutation tests for homogeneity based on the empirical characteristic function. J Nonparametr Stat 17:583–592MathSciNetCrossRefMATHGoogle Scholar
  23. Meintanis SG, Iliopoulos G (2008) Fourier methods for testing multivariate independence. Comput Stat Data Anal 52:1884–1895MathSciNetCrossRefMATHGoogle Scholar
  24. Neuhaus G, Zhu LX (1998) Permutation tests for reflected symmetry. J Multivar Anal 67:129–153MathSciNetCrossRefMATHGoogle Scholar
  25. Särndal CE, Swenson B, Wretman J (1992) Model assisted survey sampling. Springer, New YorkGoogle Scholar
  26. Székely GJ, Rizzo M, Bakirov NK (2007) Measuring and testing dependence by correlation of distances. Ann Stat 35:2769–2794MathSciNetCrossRefMATHGoogle Scholar
  27. Tillé Y (2006) Sampling algorithms. Springer, New YorkMATHGoogle Scholar
  28. Wang J (2012) Sample distribution function based goodness-of-fit test for complex surveys. Comput Stat Data Anal 56:664–679Google Scholar
  29. Xiao Y (2017) A fast algorithm for two-dimensional Kolmogorov–Smirnov two sample tests. Comput Stat Data Anal 105:53–58MathSciNetCrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2016

Authors and Affiliations

  • M. D. Jiménez-Gamero
    • 1
  • J. L. Moreno-Rebollo
    • 1
  • J. A. Mayor-Gallego
    • 1
  1. 1.Departamento de Estadística e I.O.Universidad de SevillaSevillaSpain

Personalised recommendations