Sains Malaysiana 51(7)(2022): 2265-2281

http://doi.org/10.17576/jsm-2022-5107-26

Comparison Analysis on the Coefficients of Variation of Two Independent Birnbaum-Saunders Distributions by Constructing Confidence Intervals for the Ratio of Coefficients of Variation

(Analisis Perbandingan Pekali Variasi Dua Taburan Birbaum-Saunders tak Bersandar dengan Membina Selang Keyakinan untuk Nisbah Pekali Variasi)

WISUNEE PUGGARD, SA-AAT NIWITPONG & SUPARAT NIWITPONG^*

Department of Applied Statistics, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand

Diterima: 29 Ogos 2021/Diserahkan: 30 November 2021

Abstract

The fatigue failure of materials can be investigated by applying the Birnbaum-Saunders (BS) distribution to fatigue failure datasets. The coefficient of variation (CV) is an important descriptive statistic that is widely used to measure the dispersion of data. In addition, for two independent datasets following BS distributions, the ratio of their CVs can be used to compare their CVs, especially when the difference is small, and constructing confidence intervals for this scenario is of interest in this study. Hence, we propose new confidence intervals for the ratio of the CVs from two BS distributions by using the bootstrap confidence interval (BCI), the fiducial generalized confidence interval (FGCI), a Bayesian credible interval (BayCI), and the highest posterior density (HPD) interval approaches. The performances of the proposed confidence intervals were compared with the generalized confidence interval (GCI) in terms of their coverage probabilities and average lengths via Monte Carlo simulations. The results indicate that the HPD interval outperformed the others when the coverage probabilities and the average lengths were both considered together. The efficacies of the proposed methods and GCI are illustrated using real datasets of the fatigue life of 6061-T6 aluminum coupons.

Keywords: Bayesian; Birnbaum-Saunders distribution; coefficients of variation; confidence interval; fatigue failure

Abstrak

Kegagalan lesu bahan boleh dikaji dengan menggunakan taburan Birnbaum-Saunders (BS) pada set data kegagalan lesu. Pekali variasi (CV) ialah statistik deskriptif penting yang digunakan secara meluas untuk mengukur serakan data. Di samping itu, untuk dua set data tak bersandar disebabkan taburan BS, nisbah CV mereka boleh digunakan untuk membandingkan CV mereka, terutamanya apabila perbezaannya kecil dan membina selang keyakinan untuk senario ini adalah penting dalam kajian ini. Oleh itu, kami mencadangkan selang keyakinan baharu untuk nisbah CV daripada dua taburan BS dengan menggunakan pendekatan selang keyakinan bootstrap (BCI), selang keyakinan umum fidusial (FGCI), selang boleh percaya Bayesian (BayCI) dan selang ketumpatan posterior tertinggi (HPD). Prestasi selang keyakinan yang dicadangkan telah dibandingkan dengan selang keyakinan umum (GCI) dari segi kebarangkalian liputan dan panjang purata melalui simulasi Monte Carlo. Keputusan menunjukkan bahawa selang HPD mengatasi yang lain apabila kebarangkalian liputan dan panjang purata kedua-duanya diambil kira secara bersama. Keberkesanan kaedah yang dicadangkan dan GCI diilustrasi menggunakan set data sebenar hayat lesu kupon aluminium 6061-T6.

Kata kunci: Bayesian; kegagalan lesu; pekali variasi; selang keyakinan; taburan Birnbaum-Saunders

RUJUKAN

Bayes, T. 1763. An essay towards solving a problem in the doctrine of chances. Philosophical Transactions 53: 370-418.

Birnbaum, Z.W. & Saunders, S.C. 1969a. A new family of life distributions. Journal of Applied Probability 6(2): 319-327.

Birnbaum, Z.W. & Saunders, S.C. 1969b. Estimation for a family of life distributions with applications to fatigue. Journal of Applied Probability 6(2): 328-347.

Box, G.E.P. & Tiao, G.C. 1992. Bayesian Inference in Statistical Analysis. Wiley.

Buntao, N. & Niwitpong, S. 2013. Confidence intervals for the ratio of coefficients of variation of delta-lognormal distribution. Applied Mathematical Sciences 7(77): 3811-3818.

Chou, C., Lin, Y., Chang, C. & Chen, C. 2006. On the bootstrap confidence intervals of the process incapability index Cpp. Reliability Engineering & System Safety 91: 452-459.

Congdon, P. 2001. Bayesian Statistical Modeling. Wiley.

Efron, B. 1979. Bootstrap methods: another look at the jackknife. Annals of Statistics 7(1): 1-26.

Engelhardt, M., Bain, L. & Wright, F. 1981. Inferences on the parameters of the Birnbaum-Saunders fatigue life distribution based on maximum likelihood estimation. Technometrics 23(3): 251-256.

Hannig, J. 2013. Generalized fiducial inference via discretization. Statistica Sinica 23(2): 489-514.

Hannig, J. 2009. On generalized fiducial inference. Statistica Sinica 19(2): 491-544.

Hasan, M.S. & Krishnamoorthy, K. 2017. Improved confidence intervals for the ratio of coefficients of variation of two lognormal distributions. Journal of Statistical Theory and Applications 16(3): 345-353.

Johnson, N., Kotz, S. & Balakrishnan, N. 1994. Continuous Univariate Distributions. Wiley.

Kashif, M., Aslam, M., Rao, G.S., Marshadi, A.H.A. & Jun, C.H. 2017. Bootstrap confidence intervals of the modified process capability index for Weibull distribution. Arabian Journal for Science and Engineering 42: 4565-4573.

Leiva, V., Ruggeri, F., Saulo, H. & Vivanco, J. 2017. A methodology based on the Birnbaum-Saunders distribution for reliability analysis applied to nano-materials. Reliability Engineering & System Safety 157: 192-201.

Leiva, V., Athayde, E., Azevedo, C. & Marchant, C. 2011. Modeling wind energy flux by a Birnbaum-Saunders distribution with an unknown shift parameter. Journal of Applied Statistics 38(12): 2819-2838.

Lemonte, A., Simas, A. & Cribari-Neto, F. 2008. Bootstrap-based improved estimators for the two-parameter Birnbaum-Saunders distribution. Journal of Statistical Computation and Simulation 78(1): 37-49.

Li, Y. & Xu, A. 2016. Fiducial inference for Birnbaum-Saunders distribution. Journal of Statistical Computation and Simulation 86(9): 1673-1685.

MacKinnon, J.G. & Smith, J.A.A. 1998. Approximate bias correction in econometrics. Journal of Econometrics 85(2): 205-230.

Mahmoudvand, R. & Hassani, H. 2009. Two new confidence intervals for the coefficient of variation in a normal distribution. Journal of Applied Statistics36: 429-442.

Marshall, A.W. & Olkin, I. 2007. Life Distributions. Structure of Nonparametric, Semiparametric and Parametric Families. Springer.

Moslim, N.H., Zubairi, Y.Z., Hussin, A.G., Hassan, S.F. & Mokhtar, N.A. 2019. A comparison of asymptotic and bootstrapping approach in constructing confidence interval of the concentration parameter in von Mises distribution. Sains Malaysiana 48(5): 1151-1156.

Nam, J. & Kwon, D. 2017. Inference on the ratio of two coefficients of variation of two lognormal distributions. Communications in Statistics-Theory and Methods 46(17): 8575-8587.

Ng, H., Kundu, D. & Balakrishnan, N. 2003. Modified moment estimation for the two-parameter Birnbaum-Saunders distribution. Computational Statistics & Data Analysis 43(3): 283-298.

Niwitpong, S-A. 2013. Confidence interval for coefficient of variation of lognormal distribution with restricted parameter space. Applied Mathematical Sciences 7(77): 3805-3810.

Niwitpong, S. & Wongkhao, A. 2016. Confidence intervals for the difference and the ratio of coefficients of variation of normal distribution with a known ratio of variances. International Journal of Mathematics Trends and Technology 29(1): 13-20.

Puggard, W., Niwitpong, S.A. & Niwitpong, S. 2020. Generalized confidence interval of the ratio of coefficients of variation of Birnbaum-Saunders distribution. Integrated Uncertainty in Knowledge Modelling and Decision Making. IUKM 2020. Lecture Notes in Artificial Intelligence 12482: 396-406.

Sangnawakij, P., Niwitpong, S.A. & Niwitpong, S. 2015. Confidence intervals for the ratio of coefficients of variation of the gamma distributions. Integrated Uncertainty in Knowledge Modelling and Decision Making. IUKM 2015. Lecture Notes in Computer Science 9376: 542-551.

Sun, Z.L. 2009. The confidence intervals for the scale parameter of the Birnbaum-Saunders fatigue life distribution. Acta Armamentarii 30(11): 1558-1561.

Thangjai, W., Niwitpong, S-A. & Niwitpong, S. 2021. A Bayesian approach for estimation of coefficients of variation of normal distributions. Sains Malaysiana 50(1): 261-278.

Wakefield, J., Gelfand, A. & Smith, A. 1991. Efficient generation of random variates via the ratio-of-uniforms method. Statistics and Computing 1: 129-133.

Wang, B.X. 2012. Generalized interval estimation for the Birnbaum-Saunders distribution. Computational Statistics & Data Analysis 56(12): 4320-4326.

Wang, M., Sun, X. & Park, C. 2016. Bayesian analysis of Birnbaum-Saunders distribution via the generalized ratio-of-uniforms method. Computational Statistics 31: 207-225.

Weerahandi, S. 1993. Generalized confidence intervals. Journal of the American Statistical Association 88: 899-905.

^*Pengarang untuk surat-menyurat; email: suparat.n@sci.kmutnb.ac.th