The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
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Testing Exponentiality Based on EDF Statistics for Randomly Censored Data when the Scale Parameter is Unknown

척도모수가 미지인 임의중도절단자료의 EDF 통계량을 이용한 지수 검정

  • Received : 2012.01.02
  • Accepted : 2012.03.09
  • Published : 2012.04.30
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Abstract

The simplest and the most important distribution in survival analysis is exponential distribution. Koziol and Green (1976) derived Cram$\acute{e}$r-von Mises statistic's randomly censored version based on the Kaplan-Meier product limit estimate of the distribution function; however, it could not be practical for a real data set since the statistic is for testing a simple goodness of fit hypothesis. We generalized it to the composite hypothesis for exponentiality with an unknown scale parameter. We also considered the classical Kolmogorov-Smirnov statistic and generalized it by the exact same way. The two statistics are compared through a simulation study. As a result, we can see that the generalized Koziol-Green statistic has better power in most of the alternative distributions considered.

수명시간 분석에서 가장 간단하고 또한 자주 이용되는 분포는 지수분포이다. Koziol과 Green (1976)은 Cram$\acute{e}$r-von Mises 통계량을 Kaplan-Meier의 product limit 경험분포함수를 이용하여 임의중도절단자료에 대해서 일반화하였다. 그러나 이 통계량은 모수의 값이 주어진 단순귀무가설을 가정하고 있으므로 실제 자료에 적용하기에는 어려운 점이 있다. 본 논문에서는 척도모수가 미지인 지수분포의 적합도 검정에 모수를 추정하여 Koziol-Green 통계량을 적용하였다. 그리고 같은 방법으로, 전통적인 Kolmogorov-Smirnov 검정통계량을 일반화하고 두 가지 통계량의 검정력을 모의실험을 통하여 비교하였다. 그 결과 전반적으로 일반화된 Koziol-Green 통계량이 Kolmogorov-Smirnov 통계량보다 지수분포의 검정에 있어서는 좀 더 좋은 검정력을 보여주었다.

Keywords

References

  1. Akritas, M. G. (1988). Pearson type goodness of fit test: The univariate case, Journal of the American Statistical Association, 83, 222-230. https://doi.org/10.1080/01621459.1988.10478590
  2. 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
  3. Chen, C. (1984). A correlation goodness-of-fit test for randomly censored data, Biometrika, 71, 315-322. https://doi.org/10.1093/biomet/71.2.315
  4. Chen, Y. Y., Hollander, M. and Langberg, N. A. (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
  5. Csorgo, S. and Faraway, J. J. (1996). The exact and asymptotic distributions of Cramer-von Mises Statistics, Journal of the Royal Statistical Society, Series B, 58, 221-234.
  6. Csorgo, S. and Horvath, L. (1981). On the Koziol-Green Model for random censorship, Biometrika, 68, 391-401.
  7. Efron, B. (1967). The two sample problem with censored data, In Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, 4, 831-853.
  8. Filliben, J. J. (1975). The probability plot correlation coefficient test for normality, Technometrics, 17, 111-117. https://doi.org/10.1080/00401706.1975.10489279
  9. Hollander, M. and Pena, E. A. (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
  10. Kaplan, E. L. 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
  11. 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
  12. Koziol, J. A. (1978). A two sample Cramer-von Mises test for randomly censored data, Biometrical Journal, 20, 603-608. https://doi.org/10.1002/bimj.4710200608
  13. Koziol, J. A. (1980). Goodness-of-fit tests for randomly censored data,n Biometrika, 67, 693-696. https://doi.org/10.1093/biomet/67.3.693
  14. Koziol, J. A. and Green, S. B. (1976). A Cramer-von Mises statistic for randomly censored data, Biometrika, 63, 465-474.
  15. Lee, E.T. and Wang, J. W. (2003). Statistical Methods for Survival Data Analysis, John Wiley & Sons, Inc. New Jersey.
  16. Meier, P. (1975). Estimation of a distribution function from incomplete observations, In Perspectives in Probability and Statistics, Ed. J. Gani, 67-87. London, Academic Press.
  17. Michael, J. R. and Schucany, W. R. (1986). Analysis of data from censored samples, In Goodness of Fit Techniques, (Edited by D'Agostino, R. B. and Stephens, M. A.), Chapter 11, Marcel Dekker, New York.
  18. Nair, V. N. (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
  19. Stephens, M. A. (1974). EDF statistics for goodness of fit and some comparisons, Journal of the American Statistical Association, 69, 730-737. https://doi.org/10.1080/01621459.1974.10480196
  20. Stephens, M. A. (1986). Tests based on EDF statistics, In Goodness-of-Fit Techniques (Edited by D'Agostino, R. B. and Stephens, M. A.), Chapter 5, Marcel Dekker, New York.

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