Testing the Number of Components in a Birnbaum-Saunders Mixture Model under a Random Censoring Scheme

  • Walaa EL-Sharkawy Department of Mathematics, Faculty of Science, Cairo University, Egypt
  • Moshira A. Ismail Department of Statistics, Faculty of Economics and Political Science, Cairo University, Egypt
Keywords: Birnbaum-Saunders Mixture Model, Bootstrap Test, Censoring, Modified Likelihood Ratio Test, Monte Carlo Method, Power

Abstract

This paper deals with testing the number of components in a Birnbaum-Saunders mixture model under randomly right censored data. We focus on two methods, one based on the modified likelihood ratio test and the other based on the shortcut of bootstrap test. Based on extensive Monte Carlo simulation studies, we evaluate and compare the performance of the proposed tests through their size and power. A power analysis provides guidance for researchers to examine the factors that affect the power of the proposed tests used in detecting the correct number of components in a Birnbaum-Saunders mixture model. Finally an example of aircraft Windshield data is used to illustrate the testing procedure.

References

J. Aza¨ıs, E. Gassiat, and C. Mercadier, The Likelihood ratio test for general mixture models with or without structural parameter, ESAIM: Probability and Statistics, vol. 13, pp. 301-327, 2009.

N. Balakrishnan, R. Gupta, D. Kundu, V. Leiva, and A. Sanhueza, On some mixture models based on the Birnbaum-Saunders distribution and associated inference, Journal of Statistical Planning and Inference, vol. 141, no. 7, pp. 2175-2190, 2011.

N. Balakrishnan, and D. Kundu, Birnbaum-Saunders distribution: A review of models, analysis, and applications, Applied Stochastic Models in Business and Industry, vol. 35, no. 1, pp. 4-49, 2019.

L. Benites, R. Maehara, F. Vilca, and F. Marmolejo-Ramos, Finite Mixture of Birnbaum-Saunders distributions using the k bumps algorithm, 2017. https://arxiv.org/abs/1708.00476

W. Blischke, and D. Murthy, Reliability: Modeling, Prediction, and Optimization, Wiley, New York, 2000.

D. Chauveau, B. Garel, and S. Mercier, Testing for univariate two-component Gaussian mixture in practice, 2018.

https://hal.archives-ouvertes.fr/hal-01659771v2

J. Chen, Penalized likelihood ratio test for finite mixture models with multinomial observations, Canadian Journal of Statistics, vol. 26, no. 4 pp. 583-599, 1998.

J. Chen, On finite mixture models, Statistical Theory and Related Fields, vol. 1, no. 1, pp. 15-27, 2017.

H. Chen, and J. Chen, The likelihood ratio test for homogeneity in the finite mixture models, Canadian Journal of Statistics, vol. 29, no. 2 pp. 201-215, 2001.

H. Chen, J. Chen, and J. Kalbfleisch, A modified likelihood ratio test for homogeneity in finite mixture models, Journal of the Royal Statistical Society, Series B, vol. 63, no. 1, pp. 19-29, 2001.

H. Chen, J. Chen, and J. Kalbfleisch, Testing for a finite mixture model with two components, Journal of the Royal Statistical Society, Series B, vol. 66, no. 1 pp. 95-115, 2004.

J. Chen, and P. Li, Hypothesis test for normal mixture models: the EM approach, The Annals of Statistics, vol. 37, no. 5A, pp. 2523-2542, 2009.

J. Chen, and P. Li, Tuning the em-test for finite mixture models, Canadian Journal of Statistics, vol. 39, no. 3, pp. 389-404, 2011.

J. Chen, and P. Li, Testing the order of a normal mixture in mean, Communications in Mathematics and Statistics, vol. 4, no. 1, pp. 21-38, 2016.

J. Chen, P. Li, and Y. Fu, Inference on the order of a normal mixture, Journal of the American Statistical Association, vol. 107, no. 499, pp. 1096-1105, 2012.

D. Dacunha-Castelle, and E. Gassiat, Testing the order of a model using locally conic parametrization: population mixtures and stationary ARMA processes, The Annals of Statistics, vol. 27, no. 4, pp. 1178-1209, 1999.

A. Dempster, N. Laird, and D. Rubin, Maximum likelihood from incomplete data via EM algorithm Journal of the Royal Statistical Society, Series B, vol. 39, no. 1, pp. 1-38, 1977.

W. El-Sharkawy, and M. Ismail, Mixture of Birnbaum-Saunders Distributions: Identifiability, Estimation and Testing

Homogeneity with Randomly Censored Data, American Journal of Mathematical and Management Sciences, 2020.

https://doi.org/10.1080/01966324.2020.1837041.

Z. Feng, and C. McCulloch, Using bootstrap likelihood ratios in finite mixture models, Journal of the Royal Statistical Society, Series B, vol. 58, no. 3, pp. 609-617, 1996.

D. Gudicha, V. Schmittmann, F. Tekle, and J. Vermunt, Power Analysis for the Likelihood-Ratio Test in Latent Markov Models: Shortcutting the Bootstrap p-Value-Based Method, Multivariate Behavioral Research, vol. 51, no. 5, pp. 649-660, 2016.

M. Khosravi, D. Kundu, and A. Jamalizadeh, n bivariate and mixture of bivariate Birnbaum-Saunders distributions, Statistical Methodology, vol. 23, pp. 1-17, 2015.

J. Lawless, Statistical models and methods for lifetime data, Wiley, New York, 1982.

V. Leiva, The Birnbaum-Saunders Distribution, Amsterdam, Elsevier, 2016.

P. Li, Hypothesis testing in finite mixture models, PhD thesis, University of Waterloo, Canada, 2007.

P. Li, and J. Chen, Testing the order of a finite mixture, Journal of the American Statistical Association, vol. 105, no. 491, pp. 1084-1092, 2010.

P. Li, J. Chen, and P. Marriot, Non-finite Fisher information and homogeneity: an EM approach, Biometrika, vol. 96, no. 2, pp. 411-426, 2009.

X. Liu, and Y. Shao, Asymptotics for likelihood ratio tests under loss of identifiability, The Annals of Statistics, vol. 31, no.3, pp. 807-832, 2003.

G. McLachlan, On bootstrapping the likelihood ratio test stastistic for the number of components in a normal mixture, Applied Statistics, vol. 36, no. 3, pp. 318-324, 1987.

G. McLachlan, and T. Krishnan, The EM algorithm and extensions, 2nd edn. Wiley, New York, 2008.

G. McLachlan, and D. Peel, Finite mixture models, John Wiley and Sons, New York, 2000.

X. Niu, Homogeneity test in finite mixture models using EM-test, PhD thesis, University of Alberta , Canada, 2014.

X. Niu, P. L., and P. Zhang, Testing homogeneity in a multivariate mixture model, Canadian Journal of Statistics, vol. 39, no. 2, pp. 218-238, 2011.

K. Nylund, T. Asparouhov, and B. Muth´en, Deciding on the number of classes in latent class analysis and growth mixture modeling: A Monte Carlo simulation study, Structural Equation Modeling: A Multidisciplinary Journal, vol. 14, no. 4, pp. 535-569, 2007.

S. Ruhi, S. Sarker, and M. Karim, Mixture models for analyzing product reliability data: a case study, SpringerPlus 4:634, 2015.

J. Shen, and X. He, Inference for subgroup analysis with a structured logistic-normal mixture model, Journal of the American Statistical Association, vol. 110, no. 509, pp. 303-312, 2015.

K. Sultan, A. Ismail, and A. AL-Moisheer, Testing the number of components of the mixture of two inverse Weibull distributions, International Journal of Computer Mathematics, vol. 86, no. 4, pp. 693-702, 2009.

D. Titterington, A. Smith, and U. Makov, Statistical analysis of finite mixture distributions, New York: Wiley, 1985.

F. Tekle, D. Gudicha, and J. Vermunt, Power analysis for the bootstrap likelihood ratio test for the number of classes in latent class models, Advances in Data Analysis and Classification, vol. 10, pp. 209-224, 2016.

T. Wong, K. Lam,, and V. Zhao, Asymptotic null distribution of the modified likelihood ratio test for homogeneity in finite mixture models, Computational Statistics and Data Analysis, vol. 127, no. 4, pp. 248-257, 2018.

Y. Yu,and J. Harvill, Bootstrap likelihood ratio test for Weibull mixture models fitted to grouped data, Communications in Statistics - Theory and Methods, vol. 48, no. 18, pp. 4550-4568, 2019.

Published
2021-01-15
How to Cite
EL-Sharkawy, W., & Ismail, M. A. (2021). Testing the Number of Components in a Birnbaum-Saunders Mixture Model under a Random Censoring Scheme. Statistics, Optimization & Information Computing, 9(1), 157-175. https://doi.org/10.19139/soic-2310-5070-919
Section
Research Articles