Confidence Intervals for Ratios of the Coefficients of Variation of the Delta-Birnbaum-Saunders Distributions

Abstract

The Delta-Birnbaum-Saunders distribution is a combination of positive values that follow the Birnbaum-Saunders distribution and zeros that follow the binomial distribution, making it a relatively new distribution. The coefficient of variation is calculated as the ratio of the standard deviation to the mean. It is important for comparing the dispersion of datasets. Therefore, this paper aims to generate confidence intervals for ratios of coefficients of variation under the Delta-Birnbaum-Saunders distributions. We have proposed four methods for constructing confidence intervals, namely, the method of variance estimates recovery, the bootstrap confidence interval, the generalized confidence interval based on the variance stabilized transformation, and the generalized confidence interval based on the Wilson score method. The assessment of their performance relies on coverage probabilities and average widths obtained through Monte Carlo simulations. The overall study results reveal that the generalized confidence interval based on the variance stabilized transformation and the generalized confidence interval based on the Wilson score methods provide similar values in both the coverage probabilities and average widths, making them the two most efficient methods. Furthermore, it was found that the method of variance estimates recovery performs well when the shape parameters are small. Finally, all the proposed methods will be applied to wind speed data in Thailand.

Keywords

• Bootstrap confidence interval
• Generalized confidence interval
• Method of variance estimates recovery

References

1. [1] Birnbaum, Z. W., Saunders, S. C., “A new family of life distributions”, Journal of applied probability, 6(2): 319-327, (1969). DOI: https://doi.org/10.2307/3212003
2. [2] Jin, X., Kawczak, J., “Birnbaum-Saunders and lognormal kernel estimators for modelling durations in high frequency financial data”, Annals of Economics and Finance, 4: 103-124, (2003).
3. [3] Lio, Y. L., Tsai, T. R., and Wu, S. J., “Acceptance sampling plans from truncated life tests based on the Birnbaum–Saunders distribution for percentiles”, Communications in Statistics-Simulation and Computation, 39(1): 119-136, (2009). DOI: https://doi.org/10.1080/03610910903350508
4. [4] Saulo, H., Leiva, V., Ziegelmann, F. A., and Marchant, C., “A nonparametric method for estimating asymmetric densities based on skewed Birnbaum–Saunders distributions applied to environmental data”, Stochastic Environmental Research and Risk Assessment, 27: 1479-1491, (2013). DOI: https://doi.org/ 10.1007/s00477-012-0684-8
5. [5] Leão, J., Leiva, V., Saulo, H., and Tomazella, V., “Birnbaum–Saunders frailty regression models: Diagnostics and application to medical data”, Biometrical Journal, 59(2): 291-314, (2017). DOI: https://doi.org/10.1002/bimj.201600008
6. [6] Cordeiro, G. M., De Lima, M. D. C. S., Ortega, E. M. M., and Suzuki, A. K., “A New Extended Birnbaum–Saunders Model: Properties, Regression and Applications”, Stats, 1(1): 32-47, (2018). DOI: https://doi.org/10.3390/stats1010004
7. [7] Martínez-Flórez, G., Barranco-Chamorro, I., Bolfarine, H., and Gómez, H. W., Flexible Birnbaum–Saunders Distribution. Symmetry, 11(10), 1305, (2019). DOI: https://doi.org/10.3390/sym11101305
8. [8] Benkhelifa, L., “The Weibull Birnbaum-Saunders distribution and its applications”, Statistics, Optimization & Information Computing, 9(1): 61-81, (2021). DOI: https://doi.org/10.19139/soic-2310-5070-887

9. [9] Aitchison, J., “On the distribution of a positive random variable having a discrete probability mass at the origin”, Journal of the merican statistical association, 50(271): 901-908, (1955). DOI: https://doi.org/10.1080/01621459.1955.10501976
10. [10] De la Mare, W. K., “Estimating confidence intervals for fish stock abundance estimates from trawl surveys”, CCAMLR Science, 1: 203-207, (1994).
11. [11] Hasan, M. S., Krishnamoorthy, K., “Confidence intervals for the mean and a percentile based on zero-inflated lognormal data”, Journal of Statistical Computation and Simulation, 88(8): 1499-1514, (2018). DOI: https://doi.org/10.1080/00949655.2018.1439033
12. [12] Zhang, Q., Xu, J., Zhao, J., Liang, H., and Li, X., “Simultaneous confidence intervals for ratios of means of zero-inflated log-normal populations”, Journal of Statistical Computation and Simulation, 92(6): 1113-1132, (2022). DOI: https://doi.org/10.1080/00949655.2021.1986508
13. [13] Muralidharan, K., Kale, B. K., “Modified gamma distribution with singularity at zero”, Communications in Statistics-Simulation and computation, 31(1): 143-158, (2002). DOI: https://doi.org/10.1081/SAC-120002720
14. [14] Kaewprasert, T., Niwitpong, S. A., and Niwitpong, S., “Confidence Interval Estimation for the Common Mean of Several Zero-Inflated Gamma Distributions”, Symmetry, 15(1): 67, (2022). DOI: https://doi.org/10.3390/sym15010067
15. [15] Khooriphan, W., Niwitpong, S. A., and Niwitpong, S., “Confidence Intervals for Mean of Delta Two-Parameter Exponential Distribution”, In International Symposium on Integrated Uncertainty in Knowledge Modelling and Decision Making, Cham: Springer International Publishing, 117-129, (2022). DOI: https://doi.org/10.1007/978-3-030-98018-4_10
16. [16] Kim, Y. Y., Lee, J. T., and Choi, G. H., “An investigation on the causes of cycle variation in direct injection hydrogen fueled engines”, International Journal of Hydrogen Energy, 30(1): 69-76, (2005). DOI: https://doi.org/10.1016/j.ijhydene.2004.03.041
17. [17] Verrill, S., Johnson, R. A., “Confidence bounds and hypothesis tests for normal distribution coefficients of variation”, Communications in Statistics—Theory and Methods, 36(12): 2187-2206, (2007). DOI: https://doi.org/10.1080/03610920701215126
18. [18] Gulhar, M., Kibria, B. G., Albatineh, A. N., and Ahmed, N. U., “A comparison of some confidence intervals for estimating the population coefficient of variation: a simulation study”, SORT-Statistics and Operations Research Transactions, 36(1): 45-68, (2012).
19. [19] Nam, J. M., Kwon, D., “Inference on the ratio of two coefficients of variation of two lognormal distributions”, Communications in Statistics-Theory and Methods, 46(17): 8575-8587, (2017). DOI: https://doi.org/10.1080/03610926.2016.1185118
20. [20] Buntao, N., Niwitpong, S. A., “Confidence intervals for the ratio of coefficients of variation of delta-lognormal distribution”, Applied Mathematical Sciences, 7(77): 3811-3818, (2013). DOI: http://dx.doi.org/10.12988/ams.2013.35248
21. [21] Sangnawakij, P., Niwitpong, S. A., and Niwitpong, S., “Confidence intervals for the ratio of coefficients of variation of the gamma distributions”, In Integrated Uncertainty in Knowledge Modelling and Decision Making: 4th International Symposium, IUKM 2015, Nha Trang, Vietnam, Proceedings 4, Springer International Publishing, 193-203, (2015). DOI: https://doi.org/10.1007/978-3-319-25135-6_19
22. [22] Sangnawakij, P., Niwitpong, S. A., and Niwitpong, S., “Confidence intervals for the ratio of coefficients of variation in the two-parameter exponential distributions”, In Integrated Uncertainty in Knowledge Modelling and Decision Making: 5th International Symposium, IUKM 2016, Da Nang, Vietnam, Proceedings 5, Springer International Publishing, 542-551, (2016). DOI: https://doi.org/10.1007/978-3-319-49046-5_46
23. [23] Hasan, M. S., Krishnamoorthy, K., “Improved Confidence Intervals for the Ratio of Coefficients of Variation of Two Lognormal Distributions”, Journal of Statistical Theory and Applications, 16(3): 345-353, (2017). DOI: https://doi.org/10.2991/jsta.2017.16.3.6
24. [24] Puggard, W., Niwitpong, S. A., and Niwitpong, S., “Generalized confidence interval of the ratio of coefficients of variation of Birnbaum-Saunders distribution”, In Integrated Uncertainty in Knowledge Modelling and Decision Making: 8th International Symposium, IUKM 2020, Phuket, Thailand, Proceedings 8, Springer International Publishing, 396-406, (2020). DOI: https://doi.org/10.1007/978-3-030-62509-2_33
25. [25] Yosboonruang, N., Niwitpong, S., “Statistical inference on the ratio of delta-lognormal coefficients of variation”, Applied Science and Engineering Progress, 14(3): 489-502, (2021). DOI: https://doi.org/10.14416/j.asep.2020.06.003
26. [26] Yosboonruang, N., Niwitpong, S. A., and Niwitpong, S., “Confidence intervals for rainfall dispersions using the ratio of two coefficients of variation of lognormal distributions with excess zeros”, Plos one, 17(3): e0265875, (2022). DOI: https://doi.org/10.1371/journal.pone.0265875
27. [27] Thangjai, W., Niwitpong, S. A., Niwitpong, S., “Bayesian confidence interval for ratio of the coefficients of variation of normal distributions: A practical approach in civil engineering”, Civil Engineering Journal, 7: 135-147, (2022). DOI: https://doi.org/10.28991/CEJ-SP2021-07-010
28. [28] Chankham, W., Niwitpong, S. A., and Niwitpong, S., “Confidence intervals for ratio of coefficients of variation of Inverse Gaussian distribution”, In Proceedings of the 2022 International Conference on Big Data, IoT, and Cloud Computing, 1-5, (2022). DOI: https://doi.org/10.1145/3588340.3588492
29. [29] La-ongkaew, M., Niwitpong, S. A., and Niwitpong, S., “Estimation of the Confidence Interval for the Ratio of the Coefficients of Variation of Two Weibull Distributions and Its Application to Wind Speed Data”, Symmetry, 15(1): 46, (2022). DOI: https://doi.org/10.3390/sym15010046
30. [30] Zou, G. Y., Donner, A., “Construction of confidence limits about effect measures: a general approach”, Statistics in medicine, 27(10): 1693-1702, (2008). DOI: https://doi.org/10.1002/sim.3095
31. [31] Zou, G. Y., Taleban, J., and Huo, C. Y., “Confidence interval estimation for lognormal data with application to health economics”, Computational Statistics & Data Analysis, 53(11): 3755-3764, (2009). DOI: https://doi.org/10.1016/j.csda.2009.03.016
32. [32] Donner, A., Zou, G. Y., “Closed-form confidence intervals for functions of the normal mean and standard deviation”, Statistical Methods in Medical Research, 21(4): 347-359, (2010). DOI: https://doi.org/10.1177/09622802103830
33. [33] Ng, H. K. T., Kundu, D., Balakrishnan, N., “Modified moment estimation for the two-parameter Birnbaum–Saunders distribution”, Computational Statistics & Data Analysis, 43(3): 283-298, (2003). DOI: https://doi.org/10.1016/S0167-9473(02)00254-2
34. [34] Efron, B., “Bootstrap methods: another look at the jackknife”, The Annals of Statistics, 7: 1-26, (1979).
35. [35] MacKinnon, J. G., Smith Jr, A. A., “Approximate bias correction in econometrics”, Journal of Econometrics, 85(2): 205-230, (1998). DOI: https://doi.org/10.1016/S0304-4076(97)00099-7
36. [36] Brown, L. D., Cai, T. T., DasGupta, A., “Interval estimation for a binomial proportion”. Statistical science, 16(2): 101-133, (2001). DOI: https://doi.org/10.1214/ss/1009213286
37. [37] Weerahandi, S., “Generalized confidence intervals”, Journal of the American Statistical Association, 88 (423): 899-905, (1993). DOI: https://doi.org/10.1080/01621459.1993.10476355
38. [38] Sun, Z. I., “The confidence intervals for the scale parameter of the Birnbaum-Saunders fatigue life distribution”, Acta Armamentarii, 30(11): 1558, (2009).
39. [39] Wang, B. X., “Generalized interval estimation for the Birnbaum–Saunders distribution”, Computational Statistics & Data Analysis, 56(12): 4320-4326, (2012). DOI: https://doi.org/10.1016/j.csda.2012.03.023
40. [40] DasGupta, A., “Asymptotic theory of statistics and probability”, New York: Springer. (2008).
41. [41] Wu, W. H., Hsieh, H. N., “Generalized confidence interval estimation for the mean of delta-lognormal distribution: an application to New Zealand trawl survey data”, Journal of Applied Statistics, 41(7): 1471-1485, (2014). DOI: https://doi.org/10.1080/02664763.2014.881780
42. [42] Li, X., Zhou, X., and Tian, L., “Interval estimation for the mean of lognormal data with excess zeros”, Statistics & Probability Letters, 83(11): 2447-2453, (2013). DOI: https://doi.org/10.1016/j.spl.2013.07.004
43. [43] Wilson, E. B., “Probable inference, the law of succession, and statistical inference”, Journal of the American Statistical Association, 22(158): 209-212, (1927). DOI: https://doi.org/10.1080/01621459.1927.10502953
44. [44] Puggard, W., Niwitpong, S. A., and Niwitpong, S., “Comparison analysis on the coefficients of variation of two independent Birnbaum-Saunders distributions by constructing confidence intervals for the ratio of coefficients of variation”, Sains Malaysiana, 51(7): 2265-2281, (2022). DOI: http://doi.org/10.17576/jsm-2022-5107-26
45. [45] Thai Meteorological Department. Automatic Weather System. [cited 2023, November 28]; available from: http://www.aws-observation.tmd.go.th
46. [46] Burnham, K. P., Anderson, D. R., “Model Selection and Multimodel Inference”, Springer. (2002).