A New Goodness of Fit Test for Complete or Type II Right Censored Samples

Year 2024, Volume: 28 Issue: 2, 240 - 250, 23.08.2024

Anıl Koyuncu , Mehmet Karahasan

https://doi.org/10.19113/sdufenbed.1479284

Abstract

This study proposes a new goodness-of-fit test based on the empirical distribution function for complete or type II right-censored random samples, which are drawn from either the exponential or log-normal distributions. Some simulation studies were conducted to compare the newly proposed test with some of the well-known goodness-of-fit tests, such as Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling, in terms of power over various sample sizes and censoring rates. The simulation results show that the newly proposed goodness of fit test generally seems to perform well compared to the other goodness of fit tests considered. In addition, the newly proposed test and the other goodness of fit tests are illustrated by applying them to some real data sets obtained from the relevant literature.

Keywords

Goodness of fit tests, Empirical distribution, Type II right censoring, Scale family, Location-scale family

Tam ya da II. Tür Sağdan Durdurulmuş Örneklemler için Yeni Bir Uyum İyiliği Testi

Abstract

Bu çalışmada, üstel veya log-normal dağılımlardan alınan tam veya II. tür sağdan durdurulmuş rastgele örneklemler için deneysel dağılım fonksiyonuna dayalı yeni bir uyum iyiliği testi önerilmektedir. Yeni önerilen uyum iyiliği testi Kolmogorov-Smirnov, Cramer-von Mises ve Anderson-Darling gibi iyi bilinen bazı uyum iyiliği testleri ile çeşitli örneklem büyüklükleri ve durdurma oranları üzerinde güç açısından karşılaştırmak için bazı simülasyon çalışmaları yapılmıştır. Simülasyon sonuçları, yeni önerilen uyum iyiliği testinin, dikkate alınan diğer uyum iyiliği testlerine kıyasla genel olarak iyi performans gösterdiğini ortaya koymaktadır. Ayrıca, yeni önerilen uyum iyiliği testi ve diğer uyum iyiliği testleri, ilgili literatürden elde edilen bazı gerçek veri setlerine uygulanarak gösterilmiştir.

Keywords

Uyum iyiliği testleri, Deneysel dağılım, II. Tür Sağdan Durdurma, Ölçek Ailesi, Konum-Ölçek ailesi

Ethical Statement

In this study, we undertake that all the rules required to be followed within the scope of the "Higher Education Institutions Scientific Research and Publication Ethics Directive" are complied with, and that none of the actions stated under the heading "Actions Against Scientific Research and Publication Ethics" are not carried out.

References

• [1] Kolmogorov, A. N. 1933. Sulla Determinazione Emprica di una Legge di Distribuzione. [On the Empirical Determination of a Law of Distribution]. Giornale Dell’lstituto Italiano Degli Attuari, 4, 83-91.
• [2] Smirnov, N. 1948. Table for Estimating the Goodness Fit of Empirical Distributions. Annals of Mathematical Statistics, 19, 279-281.
• [3] Cramér, H. 1928. On the Composition of Elementary Errors. Scandinavian Actuarial Journal, 1928(1), 13-74.
• [4] Von Mises, R.E. 1928. Wahrscheinlichkeit, Statistik und Wahrheit [Probability, Statistics and Truth]. SPRINGER, Wien, 192s.
• [5] Anderson, T. W., Darling, D. A. 1954. A Test of Goodness of Fit. Journal of the American Statistical Association, 49(268), 765-769.
• [6] Kuiper, N. H. 1960. Tests Concerning Random Points on a Circle. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, A (63), 38–47.
• [7] Watson, G. S. 1961. Goodness-of-Fit Tests on a Circle. Biometrika, 48(1/2), 109-114.
• [8] D'Agostino, R. B., Stephens, M.A. 1986. Goodness-of-Fit-Techniques. MARCEL DEKKER Inc. New York and Basel, 560s.
• [9] Stephens, M. A. 1974, EDF Statistics for Goodness of Fit and Some Comparisons. Journal of the American Statistical Association, 69(347), 730-737.
• [10] Stephens, M.A. 1974. Components of Goodness-of-fit Statistics. In Annales de I’HP Probabilities et statistiques, 10(1), 37-54.
• [11] Stephens, M.A. 1977. Goodness-of-Fit for the Extreme-Value Distribution. Biometrika, 64(3), 583-588.
• [12] Green, J. R., Hegazy, Y.A.S 1976. Powerful Modified-EDF Goodness of Fit Tests. Journal of the American Statistical Association, 71(353), 204-209.
• [13] Michael, J.R., Schucany, W.R. 1979. A New Approach to Testing Goodness of Fit for Censored Samples. Technometrics, 21, 435–44.
• [14] Petitt, A.N., Stephens, M.A. 1976. Modified Cramer-von Mises Statistics for Censored Data. Biometrika, 63(2), 291-298.
• [15] Chen, G., Balakrishnan, N. 1995. A General Purpose Approximate Goodness-of-Fit Test. Journal of Quality Technology, 27(2), 154-161.
• [16] Aho, M., Bain, L.J., Englehardt, M. 1983. Goodness-of-fit Tests for the Weibull Distribution with Unknown Parameters and Censored Sampling. Journal of Statistical Computation and Simulation, 18(1), 59-68.
• [17] Aho, M., Bain, L.J., and Englehardt, M. 1985. Goodness-of-fit tests for the Weibull Distribution with Unknown Parameters and Censored Sampling. Journal of Statistical Computation and Simulation, 21(3-4), 213-225.
• [18] Bain, L.J., Englehardt, M. 1983. A Review of Model Selection Procedures Relevant to the Weibull distribution. Communications in Statistics-Theory and Methods, 12(5), 589-609.
• [19] Pakyari, R., Balakrishnan, N. 2012. A General Purpose Approximate Goodness-of-Fit Tests for Progressively Type II Censored Data. IEEE Transactions on Reliability, 61(1), 238-244.
• [20] Castro-Kuriss, C., Kelmansky, D. M., Leiva, V., and Martinez, E.J. 2010. On a Goodness-of-Fit Tests for Normality with Unknown Parameters and Type-II Censored Data. Journal of Applied Statistics, 37(7), 1193-1211.
• [21] Zhao, J., Xu, X., Ding, X. 2010. New Goodness of Fit Tests Based on Stochastic EDF. Communication in Statistics-Theory and Methods, 39(6), 1075-1094.
• [22] Laio, F. 2004. Cramer-von Mises and Anderson-Darling Goodness of Fit Tests for Extreme Value Distributions with Unknown Parameters. Water Resources Research, 40(9).
• [23] Krit, M., Gaudoin, O., Remy, E. 2021. Goodness-of-Fit Tests for the Weibull and Extreme Value Distributions:A Review and Comparative Study. Communications in Statistics-Simulation and Computation, 50(7), 1888-1911.
• [24] Fischer, T. 2010. Goodness-of-fit tests for type-II right censored data: structure preserving transformations and power studies. Aachen University, Doctoral Dissertation, 122p, Dusseldorf.
• [25] Goldmann, C., Klar, B., Meintanis, S. G. 2015. Data Transformations and Goodness-of-fit Tests for Type-II Right Censored Samples. Metrika, 78(1), 59-83.
• [26] Noughabi, H.A., Balakrishnan, N. 2014. Goodness of Fit Using a New Estimate of Kullback-Leibler Information Based on Type II Censored Data. IEEE Transactions on Reliability, 64(2), 627-635.
• [27] Proschan, F. 1963. Theoretical Explanation of Observed Decreasing Failure Rate. Technometrics, 5(3), 375-383.
• [28] Kamakura, T., Yanagimoto, T., Olkin, I. 1989. Estimating and Testing Weibull Means Based on the Method of Moments. Technical Report 266, Stanford University, Dept. of Statistics.
• [29] Harter, H. L., Dubey, S.D. 1967. Theory and Tables for Tests of Hypotheses Concerning the Mean and the Variance of a Weibull population. Vol. 67, No. 59 Aerospace Research Laboratories, Office of Aerospace Research, United States Air Force.
• [30] Elsherpieny, A.E., Ibrahim, N.S., Radwan, U.N. 2013. Discriminating Between Weibull and Log-Logistic Distributions. International Journal of Innonative Research in Science, Engineering and Technology, 2(8), 3358-3371.
• [31] Gupta, R.D., Kundu, D. 2003. Discriminating Between Weibull and Generalized Exponential Distributions. Computational Statistics & data analysis, 43(2), 179-196.
• [32] Mohd Saat, N.Z., Jemain A.A.,and Al-Mashoor, S.H. 2008. A Comparison of Weibull and Gamma Distributions in Application of Sleep Spnea. Asian Journal of Mathematics and Statistics, 1(3), 132-138.
• [33] Bromideh, A.A., Valizadeh, R. 2014. Discriminating Between Gamma and Log-Normal Distributions by Ratio of Minimized Kullback-Leibler Divergence. Pakistan Journal