, Volume 76, Issue 2, pp 161–177 | Cite as

On the goodness-of-fit procedure for normality based on the empirical characteristic function for ranked set sampling data



The behaviour of the goodness-of-fit procedure for normality based on weighted integrals of the empirical characteristic function, discussed in the case of i.i.d. data, for instance, in Epps and Pulley (Biometrika 70:723–726, 1983), is considered here in the context of ranked set sampling (RSS) data. In the RSS context, we obtain the limiting distribution of the empirical characteristic process and perform a power study, against a broad set of alternatives, that enables an evaluation of the gain in power that occurs when a simple random sample is replaced by RSS data. The adaptation of the results obtained in the Gaussian RSS setting to the case of other important location-scale families is also discussed.


Ranked set sampling Goodness-of-fit Empirical characteristic function 


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  1. Azzalini A (1986) Further results on a class of distributions which includes the normal ones. Statistica 46: 199–208MathSciNetMATHGoogle Scholar
  2. Barabesi L, El-Shaarawi AH (2001) The efficiency of ranked set sampling for parameter estimation. Stat Probab Lett 53(2): 189–199MATHCrossRefGoogle Scholar
  3. Baringhaus L, Henze N (1988) A consistent test for multivariate normality based on the empirical characteristic function. Metrika 35(1): 339–348MathSciNetMATHCrossRefGoogle Scholar
  4. Chen Z, Bai Z, Sinha BK (2004) Ranked set sampling. Theory and applications. Lecture notes in statistics, 176. Springer, New YorkGoogle Scholar
  5. DasGupta A (2008) Asymptotic theory of statistics and probability. Springer, New YorkMATHGoogle Scholar
  6. Dell TR, Clutter JL (1972) Ranked set sampling theory with order statistics background. Biometrika 28: 545–555MATHGoogle Scholar
  7. Epps TW, Pulley LB (1983) A test for normality based on the empirical characteristic function. Biometrika 70: 723–726MathSciNetMATHCrossRefGoogle Scholar
  8. Epps TW, Singleton KJ (1986) An omnibus test for the two sample problem using the empirical characteristic function. J Stat Comput Simul 26(3–4): 177–203MathSciNetMATHCrossRefGoogle Scholar
  9. Feuerverger A, Mureika RA (1977) The empirical characteristic function and its applications. Ann Stat 5(1): 88–97MathSciNetMATHCrossRefGoogle Scholar
  10. Henze N, Wagner T (1997) A new approach to the BHEP test for multivariate normality. J Multivar Anal 62(1): 1–23MathSciNetMATHCrossRefGoogle Scholar
  11. Ihaka R, Gentleman R (1996) R: a language for data analysis and graphics. J Comput Graph Stat 5(3): 299–314Google Scholar
  12. Kaur A, Patil GP, Sinha AK, Taillie C (1995) Ranked set sampling: an annotated bibliography. Environ Ecol Stat 2: 25–54CrossRefGoogle Scholar
  13. Koutrouvelis IA (1980) A goodness-of-fit test of simple hypothesis based on the empirical characteristic function. Biometrika 67: 238–240MathSciNetMATHCrossRefGoogle Scholar
  14. Koutrouvelis IA, Kellermeier J (1981) A goodness-of-fit test based on the empirical characteristic function when parameters must be estimated. J R Stat Soc Ser B 43: 173–176MathSciNetMATHGoogle Scholar
  15. Li T, Balakrishnan N (2008) Some simple nonparametric methods to test for perfect ranking in ranked set sampling. J Stat Plan Inference 138: 1325–1338MathSciNetMATHCrossRefGoogle Scholar
  16. Mahdizadeh M, Arghami NR (2009) Efficiency of ranked set sampling in entropy estimation and goodness-of-fit testing for the inverse Gaussian law. J Stat Comput Simul. doi: 10.1080/00949650902773551
  17. Manzotti A, Quiroz AJ (2001) Spherical harmonics in quadratic forms for testing multivariate normality. Test 10(1): 87–104MathSciNetMATHCrossRefGoogle Scholar
  18. McIntyre GA (1952) A method for unbiased selective sampling, using ranked sets. Aust J Agric Res 3: 385–390CrossRefGoogle Scholar
  19. Ni Chuiv N, Sinha B (1998) On some aspects of ranked set sampling in parametric estimation. In: Balakrishnan N, Rao CR (eds) Handbook of statistics, vol 17. Elsevier, Amsterdam, pp 337–377Google Scholar
  20. Patil GP (2002) Ranked set sampling. In: El-Shaarawi AH, Piegorsch WW (eds) Encyclopedia of environmetrics, vol 3. Wiley, Chichester, pp 1684–1690Google Scholar
  21. Stokes SL (1980) Estimation of variance using judgement ordered ranked set samples. Biometrics 36: 35–42MathSciNetMATHCrossRefGoogle Scholar
  22. Stokes SL (1995) Parametric ranked set sampling. Ann Inst Stat Math 47: 465–482MathSciNetMATHGoogle Scholar
  23. Stokes SL, Sager TW (1988) Characterization of a ranked-set-sample with application to estimating distribution functions. J Am Stat Assoc 83(402): 374–381MathSciNetMATHCrossRefGoogle Scholar
  24. Takahasi K, Wakimoto K (1968) On unbiased estimates of the population mean based on the sample stratified by means of ordering. Ann Inst Stat Math 20: 1–31MathSciNetMATHCrossRefGoogle Scholar
  25. Tenreiro C (2009) On the choice of the smoothing parameter for the BHEP goodness-of-fit test. Comput Stat Data Anal 53(4): 1038–1053MathSciNetMATHCrossRefGoogle Scholar
  26. Ushakov NG (1999) Selected topics in characteristic functions. VSP BV, UtrechtMATHCrossRefGoogle Scholar
  27. van der Vaart AW (1998) Asymptotic statistics. Cambridge University Press, CambridgeMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • N. Balakrishnan
    • 1
    • 2
    • 3
  • M. R. Brito
    • 4
    • 6
  • A. J. Quiroz
    • 5
    • 6
  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  2. 2.King Saud UniversityRiyadhSaudi Arabia
  3. 3.National Central UniversityZhongliTaiwan
  4. 4.Departamento de Matemáticas Puras y AplicadasUniversidad Simón BolívarCaracasVenezuela
  5. 5.Departamento de Cómputo Científico y EstadísticaUniversidad Simón BolívarCaracasVenezuela
  6. 6.Departamento de MatemáticasUniversidad de Los AndesBogotáColombia

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