# 척도모수가 미지인 임의중도절단자료의 EDF 통계량을 이용한 지수 검정

• 김남현 (홍익대학교 기초과학과)
• Accepted : 2012.03.09
• Published : 2012.04.30
• 94 27

#### 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.

#### Keywords

Goodness of fit;random censorship;Cram$\acute{e}$r-von Mises statistic;Kolmogorov-Smirnov statistic;Kaplan-Meier product limit estimate

#### Acknowledgement

Supported by : 한국연구재단

#### 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.

#### Cited by

1. Goodness-of-fit tests for randomly censored Weibull distributions with estimated parameters vol.24, pp.5, 2017, https://doi.org/10.5351/CSAM.2017.24.5.519