, Volume 78, Issue 1, pp 59–83 | Cite as

Data transformations and goodness-of-fit tests for type-II right censored samples

  • Christian Goldmann
  • Bernhard Klar
  • Simos G. Meintanis


We suggest several goodness-of-fit (GOF) methods which are appropriate with Type-II right censored data. Our strategy is to transform the original observations from a censored sample into an approximately i.i.d. sample of normal variates and then perform a standard GOF test for normality on the transformed observations. A simulation study with several well known parametric distributions under testing reveals the sampling properties of the methods. We also provide theoretical analysis of the proposed method.


Empirical characteristic function Empirical distribution function Goodness-of-fit test Censored data 


  1. Arcones MA, Wang Y (2006) Some new tests for normality based on U-processes. Stat Probab Lett 76:69–82CrossRefMathSciNetMATHGoogle Scholar
  2. Bargal AI, Thomas DR (1983) Smooth goodness of fit tests for the Weibull distribution with singly censored data. Commun Stat Theory Methods 12:1431–1447CrossRefMATHGoogle Scholar
  3. Baringhaus L, Danschke R, Henze N (1989) Recent and classical tests for normality. A comparative study. Commun Stat Simul Comput 18:363–379CrossRefMathSciNetMATHGoogle Scholar
  4. Barr DR, Davidson Teddy (1973) A Kolmogorov–Smirnov test for censored samples. Technometrics 15:739–757CrossRefMathSciNetMATHGoogle Scholar
  5. Brain CW, Shapiro SS (1983) A regression test for exponentiality. Technometrics 25:69–76CrossRefMATHGoogle Scholar
  6. Castro-Kuriss C (2011) On a goodness-of-fit test for censored data from a location-scale distribution with applications. Chil J Stat 2:115–136MathSciNetGoogle Scholar
  7. Castro-Kuriss C, Kelmansky DM, Leiva V, Martínez EJ (2010) On a goodness-of-fit test for normality with unknown parameters and type-II censored data. J Appl Stat 37:1193–1211CrossRefMathSciNetGoogle Scholar
  8. Chen, G (1991) Empirical processes based on regression residuals: Theory and Application. Ph.D. Thesis-Simon Fraser UniversityGoogle Scholar
  9. Chen G, Balakrishnan N (1995) A general purpose approximate goodness-of-fit test. J Qual Technol 27:154–161Google Scholar
  10. D’Agostino R, Massaro JM (1992) Goodness-of-fit tests. In: Balakrishnan N (ed) Handbook of the logistic distribution. Marcel Dekker Inc, New York, pp 291–371Google Scholar
  11. D’Agostino R, Stephens M (1986) Goodness-of-fit techniques. Marcel Dekker Inc, New YorkMATHGoogle Scholar
  12. Dufour R, Maag UR (1978) Distribution results for modified Kolmogorov–Smirnov statistics for truncated or censored samples. Technometrics 20:29–32MATHGoogle Scholar
  13. Durbin J (1973) Weak convergence of the sample distribution function when parameters are estimated. Ann Stat 1:279–290CrossRefMathSciNetMATHGoogle Scholar
  14. Epps TW (2005) Tests for location-scale families based on the empirical characteristic function. Metrika 62:99–114CrossRefMathSciNetMATHGoogle Scholar
  15. Epps TW, Pulley LB (1983) A test for normality based on the empirical characteristic function procedures. Biometrika 70:723–726CrossRefMathSciNetMATHGoogle Scholar
  16. Fischer T, Kamps U (2011) On the existence of transformations preserving the structure of order statistics in lower dimensions. J Stat Plan Inference 141:536–548CrossRefMathSciNetMATHGoogle Scholar
  17. Glen AG, Foote BL (2009) An inference methodology for life tests with complete samples or type-II right censoring. IEEE Trans Reliab 58:597–603CrossRefGoogle Scholar
  18. Grané A (2012) Exact goodness-of-fit tests for censored data. Ann Inst Stat Math 64:1187–1203CrossRefMATHGoogle Scholar
  19. Gupta AK (1952) Estimation of the mean and standard deviation of a normal population from a censored sample. Biometrika 39:266–273CrossRefGoogle Scholar
  20. Henze N (1990) An approximation to the limit distribution of the Epps–Pulley test statistic for normality. Metrika 37:7–18CrossRefMathSciNetMATHGoogle Scholar
  21. Henze N, Wagner T (1997) A new approach to the BHEP tests for multivariate normality. J Multivar Anal 62:1–23CrossRefMathSciNetMATHGoogle Scholar
  22. Klar B, Meintanis SG (2012) Specification tests for the response distribution in generalized linear models. Comput Stat 27:251–267CrossRefMathSciNetGoogle Scholar
  23. Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1. Wiley, New YorkMATHGoogle Scholar
  24. Lin C-T, Huang Y-L, Balakrishnan N (2008) A new method for goodness-of-fit testing based on type-II censored samples. IEEE Trans Reliab 57:633–642CrossRefGoogle Scholar
  25. Loynes RM (1980) The empirical distribution function of residuals from generalised regression. Ann Stat 8:285–298CrossRefMathSciNetMATHGoogle Scholar
  26. Meintanis SG (2009) Goodness-of-fit testing by transforming to normality: comparison between classical and characteristic function-based methods. J Stat Comput Simul 79:205–212CrossRefMathSciNetMATHGoogle Scholar
  27. Michael JR, Schucany WR (1979) A new approach to testing goodness of fit for censored samples. Technometrics 21:435–441Google Scholar
  28. Mihalko DP, Moore DS (1980) Chi-square tests of fit for type-II censored data. Ann Stat 8:625–644CrossRefMathSciNetMATHGoogle Scholar
  29. O’Reilly FJ, Stephens MA (1988) Transforming censored samples for testing fit. Technometrics 30:79–86CrossRefMathSciNetMATHGoogle Scholar
  30. Pettitt AN (1976) Cramér-von Mises statistics for testing normality with censored samples. Biometrika 63:475–481MathSciNetMATHGoogle Scholar
  31. Pettitt AN (1977) Tests for the exponential distribution with censored data using the Cramér-von Mises statistics. Biometrika 64:629–632MathSciNetMATHGoogle Scholar
  32. Peña EA (1995) Residuals from type II censored samples. In: Balakrishnan N (ed) Recent advances in life-testing and reliability. CRC Press, London, pp 523–543Google Scholar
  33. R Core Team (2012) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, AustriaGoogle Scholar
  34. Rao JS, Sethuraman J (1975) Weak convergence of empirical distribution functions of random variables subject to perturbations and scale factors. Ann Stat 3:299–313CrossRefMathSciNetMATHGoogle Scholar
  35. Tenreiro C (2009) On the choice of the smoothing parameter for the BHEP goodness-of-fit test. Comput Stat Data Anal 53:1038–1053CrossRefMathSciNetMATHGoogle Scholar
  36. Thode HC (2002) Testing for normality. Marcel Dekker Inc, New YorkCrossRefMATHGoogle Scholar
  37. Tiku ML (1967) Estimating the mean and standard deviation from a censored normal sample. Biometrika 54:155–165CrossRefMathSciNetGoogle Scholar
  38. Tiku ML, Tan WY, Balakrishnan N (1986) Robust inference. Marcel Dekker Inc, New YorkMATHGoogle Scholar
  39. van der Vaart AW, Wellner JA (2002) Weak convergence and empirical processes. Springer, New YorkGoogle Scholar
  40. Wilk MB, Gnanadesikan R, Huyett MJ (1962) Estimation of parameters of the gamma distribution using order statistics. Biometrika 49:525–545CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Christian Goldmann
    • 1
  • Bernhard Klar
    • 2
  • Simos G. Meintanis
    • 3
    • 4
  1. 1.Mathematical Methods in Dynamics and DurabilityFraunhofer Institute for Industrial Mathematics ITWMKaiserslauternGermany
  2. 2.Department of MathematicsKarlsruhe Institute of Technology (KIT)KarlsruheGermany
  3. 3.Department of EconomicsNational and Kapodistrian University of AthensAthensGreece
  4. 4.Unit for Business Mathematics and InformaticsNorth-West UniversityPotchefstroomSouth Africa

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