Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
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Goodness-of-fit tests for randomly censored Weibull distributions with estimated parameters

  • Received : 2017.07.07
  • Accepted : 2017.08.26
  • Published : 2017.09.30
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Abstract

We consider goodness-of-fit test statistics for Weibull distributions when data are randomly censored and the parameters are unknown. Koziol and Green (Biometrika, 63, 465-474, 1976) proposed the $Cram\acute{e}r$-von Mises statistic's randomly censored version for a simple hypothesis based on the Kaplan-Meier product limit of the distribution function. We apply their idea to the other statistics based on the empirical distribution function such as the Kolmogorov-Smirnov and Liao and Shimokawa (Journal of Statistical Computation and Simulation, 64, 23-48, 1999) statistics. The latter is a hybrid of the Kolmogorov-Smirnov, $Cram\acute{e}r$-von Mises, and Anderson-Darling statistics. These statistics as well as the Koziol-Green statistic are considered as test statistics for randomly censored Weibull distributions with estimated parameters. The null distributions depend on the estimation method since the test statistics are not distribution free when the parameters are estimated. Maximum likelihood estimation and the graphical plotting method with the least squares are considered for parameter estimation. A simulation study enables the Liao-Shimokawa statistic to show a relatively high power in many alternatives; however, the null distribution heavily depends on the parameter estimation. Meanwhile, the Koziol-Green statistic provides moderate power and the null distribution does not significantly change upon the parameter estimation.

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References

  1. Akritas MG (1988). Pearson-type goodness-of-fit tests: the univariate case. Journal of the American Statistical Association, 83, 222-230. https://doi.org/10.1080/01621459.1988.10478590
  2. Balakrishnan N and KateriM(2008). On the maximum likelihood estimation of parameters ofWeibull distribution based on complete and censored data. Statistics and Probability Letters, 78, 2971-2975. https://doi.org/10.1016/j.spl.2008.05.019
  3. Breslow N and Crowley J (1974). A large sample study of the life table and product limit estimates under random censorships. The Annals of Statistics, 2, 437-453. https://doi.org/10.1214/aos/1176342705
  4. Chen CH (1984). A correlation goodness-of-fit test for randomly censored data. Biometrika, 71, 315-322. https://doi.org/10.1093/biomet/71.2.315
  5. Chen YY, Hollander M, and Langberg NA (1982). Small-sample results for the Kaplan Meier estimator. Journal of the American statistical Association, 77, 141-144. https://doi.org/10.1080/01621459.1982.10477777
  6. Csorgo S and Horvath L (1981). On the Koziol-Green model for random censorship. Biometrika, 68, 391-401.
  7. D'Agostino RB (1986). Graphical analysis. In RB D'Agostino and MA Stephens (Eds), Goodnessof-fit Techniques (pp. 7-62), Marcel Dekker, New York.
  8. Efron B (1967). The two sample problem with censored data. In Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, 831-853.
  9. Freireich EJ, Gehan E, Frei E, et al. (1963). The effect of 6-mercaptopurine on the duration of steroidinduced remissions in acute leukemia: a model for evaluation of other potentially useful therapy. Blood, 21, 699-716.
  10. Hollander M and Pena EA (1992). A chi-squared goodness-of-fit test for randomly censored data. Journal of the American Statistical Association, 87, 458-463. https://doi.org/10.1080/01621459.1992.10475226
  11. Kaplan EL and Meier P (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457-481. https://doi.org/10.1080/01621459.1958.10501452
  12. Kim N (2011). Testing log normality for randomly censored data. The Korean Journal of Applied Statistics, 24, 883-891. https://doi.org/10.5351/KJAS.2011.24.5.883
  13. Kim N (2012). Testing exponentiality based on EDF statistics for randomly censored data when the scale parameter is unknown. The Korean Journal of Applied Statistics, 25, 311-319. https://doi.org/10.5351/KJAS.2012.25.2.311
  14. Kim N (2016). On the maximum likelihood estimators for parameters of a Weibull distribution under random censoring, Communications for Statistical Applications and Methods, 23, 241-250 https://doi.org/10.5351/CSAM.2016.23.3.241
  15. Kleinbaum DG and Klein M (2005). Survival Analysis : A Self-Learning Test, Springer, New York.
  16. Koziol JA (1980). Goodness-of-fit tests for randomly censored data. Biometrika, 67, 693-696. https://doi.org/10.1093/biomet/67.3.693
  17. Koziol JA and Green SB (1976). A Cramer-von Mises statistic for randomly censored data. Biometrika, 63, 465-474.
  18. Kundu D (2007). On hybrid censored Weibull distribution. Journal of Statistical Planning and Inference, 137, 2127-2142. https://doi.org/10.1016/j.jspi.2006.06.043
  19. Lee ET and Wang JW (2003). Statistical Methods for Survival Data Analysis (3rd ed), John Wiley & Sons Inc., Hoboken. Jersey.
  20. LiaoMand Shimokawa T (1999). A new goodness-of-fit test for type-I extreme value and 2-parameter Weibull distributions with estimated parameters. Journal of Statistical Computation and Simulation, 64, 23-48. https://doi.org/10.1080/00949659908811965
  21. Meier P (1975). Estimation of a distribution function from incomplete observations. In J Gani (Ed), Perspectives in Probability and Statistics (pp. 67-87), Academic Press, London.
  22. Michael JR and Schucany WR (1986). Analysis of data from censored samples, In RB D'Agostino and MA Stephens (Eds), Goodness-of-fit techniques (pp. 461-496), Marcel Dekker, New York.
  23. Nair VN (1981). Plots and tests for goodness of fit with randomly censored data. Biometrika, 68, 99-103. https://doi.org/10.1093/biomet/68.1.99
  24. Pareek B, Kundu D, and Kumar S (2009). On progressively censored competing risks data forWeibull distributions. Computational Statistics and Data Analysis, 53, 4083-4094. https://doi.org/10.1016/j.csda.2009.04.010

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