Comparison of the several two-parameter exponential distributed group means in the presence of outliers

Abstract

The two-parameter exponential distribution is often used to model the lifetime of a product. The comparison of the mean lifetimes of several products is a main concern in reliability applications. In this study, the performance of the methods to compare the mean lifetimes of several products based on generalized p-value, parametric bootstrap, and fiducial approach are compared in the presence of outliers. The results of Monte-Carlo simulations clearly indicate that there is no uniformly powerful test. The parametric bootstrap test is superior to the others except in the case of the lower number of groups and the presence of outliers. An illustrative example of testing the equality lifetimes of a component is given to perform the proposed tests. The considered tests are implemented in an R package doex.

Keywords

• ANOVA
• parametric bootstrap penalized power
• exponential distribution
• lifetime data

References

1. Tsui, K., Weerahandi, S., Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters, Journal of the American Statistical Association, 84 (1989), 602–607. https://doi.org/10.2307/2289949
2. Cavus, M., Yazici, B., Sezer, A., Modified tests for comparison of group means under heteroskedasticity and non-normality caused by outlier(s), Hacettepe Journal of Mathematics and Statistics, 46 (2017), 602–607. https://doi.org/10.15672/HJMS.2017.417
3. Cavus, M., Yazici, B., Sezer, A., A revised generalized F-test for testing the equality of group means under non-normality caused by skewness, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70 (2021), 1036–1054. https://doi.org/10.31801/cfsuasmas.800452
4. Cavus, M., Yazici, B., Sezer, A., Analyzing regional export data by the modified generalized F-test, International Journal of Economic and Administrative Studies, (2018), 541–552. https://doi.org/10.18092/ulikidince.348070
5. Tian, L., Wu, J., Inferences on the common mean of several log-normal populations: the generalized variable approach, Biometrical Journal, 49 (2007), 944–951. https://doi.org/10.1002/bimj.200710391
6. Tian, L., Testing equality of inverse gaussian means under heterogeneity based on generalized test variable, Computational Statistics and Data Analysis, 51 (2006), 1156–1162. https://doi.org/10.1016/j.csda.2005.11.012
7. Ma, C., Tian, L., A parametric bootstrap approach for testing equality of inverse gaussian means under heterogeneity, Communications in Statistics-Simulation and Computation, 38 (2009), 1156–1162. https://doi.org/10.1080/03610910902833470
8. Niu, C., Guo, X., Xu, W., Zhu, L., Comparison of several Birbaum-Saunders distributions, Journal of Statistical Computation and Simulation, 84 (2014), 2721–2733. https://doi.org/10.1080/00949655.2014.881814

9. Ghosh, M., Razmpour, A., Estimation of the common location parameter of several exponentials, Sankhya: The Indian Journal of Statistics, 46 (1984), 383–394. https://www.jstor.org/stable/25050498
10. Chen, H., A new range statistic for comparisons of several exponential location parameters, Biometrika, 69 (1982), 257–260. https://doi.org/10.2307/2335881
11. Singh, N., The likelihood ratio test for the equality of location parameters of exponential populations based on Type II censored samples, Technometrics, 25 (1983), 193–195. https://doi.org/10.1080/00401706.1983.10487852
12. Kambo, N., Awad, A., Testing equality of location parameters of k exponential distributions, Communications in Statistics - Theory and Methods, 14 (1985), 567–583.
13. Hsieh, H., An exact test for comparing location parameters of k exponential distributions with unequal scaled based on Type II censored data, Technometrics, 28 (1986), 157–164. https://doi.org/10.2307/1270452
14. Vaughan, D., Tiku, M., Testing the equality of location parameters of exponential populations from censored samples, Communication in Statistics-Theory and Methods, 22 (1993), 2567–2581. https://doi.org/10.1080/03610928308831169
15. Tiku, M. Vaughan, D., Testing equality of location parameters of two exponential distributions from censored samples, Communication in Statistics-Theory and Methods, 20 (1991), 929–944. https://doi.org/10.1080/03610929108830540
16. Ananda, M., Weerahandi, S., Testing the difference of two exponential means using generalized p-values, Communications in Statistics - Simulation and Computation, 25 (1996), 521–532. https://doi.org/10.1080/03610919608813327
17. Wu, S., One stage multiple comparisons with the control for exponential mean lifetimes based on doubly censored samples under heteroscedasticity, Communications in Statistics - Simulation and Computation, 50 (2021), 1473-1483. https://doi.org/10.1080/03610918.2019.1584302
18. Malekzadeh, A., Jafari, A., Inference on the equality means of several twoparameter exponential distributions under progressively Type II censoring, Communications in Statistics - Simulation and Computation, 49 (2020), 3196-3211. https://doi.org/10.1080/03610918.2018.1538452
19. Rahman, M., Pearson, L. M., Estimation in two-parameter exponential distributions, Journal of Statistical Computation and Simulation, 70 (2001), 371–386. https://doi.org/10.1080/00949650108812128
20. Viveros, R., Balakrishnan, N., Interval estimation of parameters of life from progressively censored data, Technometrics, 36 (1994), 84–91. https://doi.org/10.2307/1269201
21. Cochran, W. G., Problem arising in the analysis of a series of similar experiments, Journal of the Royal Statistical Society, 4 (1937), 102–118. https://www.jstor.org/stable/2984123
22. Weerahandi, S., ANOVA under unequal error variances, Biometrics, 51 (1995), 102–118. https://doi.org/10.2307/2532947
23. Krishnamoorthy, K., Lu, F., Mathew, T., A parametric bootstrap approach for ANOVA with unequal variances: fixed and random models, Computational Statistics and Data Analysis, 51 (2007), 5731–5742. https://doi.org/10.1016/j.csda.2006.09.039
24. Li, X., Wang, J., Liang, H., Comparison of several means: A fiducial based approach, Computational Statistics and Data Analysis, 55 (2011), 1993–2002. https://doi.org/10.1016/j.csda.2010.12.009
25. Cavus, M., Yazici, B., Sezer, A., Penalized power approach to compare the power of the tests when Type I error probabilities are different, Communications in Statistics - Simulation and Computation, 50 (2021), 1912-1926. https://doi.org/10.1080/03610918.2019.1588310
26. Cavus, M., Yazici, B., Testing the equality of normal distributed and independent groups’ means under unequal variances by doex package, The R Journal, 12 (2021), 134-154. https://doi.org/10.32614/RJ-2021-008
27. Cavus, M., Yazici, B., doex: The one-way heteroscedastic ANOVA tests, R package, v.1.2.