×

On some mixture models based on the Birnbaum-Saunders distribution and associated inference. (English) Zbl 1214.62014

Summary: We consider three different mixture models based on the Z. W. Birnbaum and S. C. Saunders (BS) distribution [J. Appl. Probab. 6, 319–327 (1969; Zbl 0209.49801)], viz., (1) mixture of two different BS distributions, (2) mixture of a BS distribution and a length-biased version of another BS distribution, and (3) mixture of a BS distribution and its length-biased version. For all these models, we study their characteristics including the shape of their density and hazard rate functions. For the maximum likelihood estimation of the model parameters, we use the EM algorithm. For the purpose of illustration, we analyze two data sets related to enzyme and depressive condition problems. In the case of the enzyme data, it is shown that Model 1 provides the best fit, while for the depressive condition data, it is shown all three models fit well with Model 3 providing the best fit.

MSC:

62E15 Exact distribution theory in statistics
62F10 Point estimation
62N02 Estimation in survival analysis and censored data
65C60 Computational problems in statistics (MSC2010)
62P10 Applications of statistics to biology and medical sciences; meta analysis

Citations:

Zbl 0209.49801

Software:

bs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Balakrishnan, N.; Leiva, V.; López, J., Acceptance sampling plans from truncated life tests from generalized Birnbaum-Saunders distribution, Comm. Statist. Simulation Comput., 36, 643-656 (2007) · Zbl 1121.62089
[2] Balakrishnan, N.; Leiva, V.; Sanhueza, A.; Cabrera, E., Mixture inverse Gaussian distribution and its transformations, moments and applications, Statistics, 43, 91-104 (2009) · Zbl 1278.60024
[3] Balakrishnan, N.; Leiva, V.; Sanhueza, A.; Vilca, F., Estimation in the Birnbaum-Saunders distribution based on scale-mixture of normals and the EM-algorithm, Statist. Oper. Res. Trans., 33, 171-192 (2009) · Zbl 1186.65014
[4] Barros, M.; Paula, G. A.; Leiva, V., A new class of survival regression models with heavy-tailed errors: robustness and diagnostics, Lifetime Data Anal., 14, 316-332 (2008) · Zbl 1356.62198
[5] Bebbington, M.; Lai, C.-D.; Zitikis, R., A proof of the shape of the Birnbaum-Saunders hazard rate function, Math. Sci., 33, 49-56 (2008) · Zbl 1147.62012
[6] Bhattacharyya, G. K.; Fries, A., Fatigue failure models: Birnbaum-Saunders vs. inverse Gaussian, IEEE Trans. Rel., 31, 439-440 (1982) · Zbl 0512.62094
[7] Birnbaum, Z. W.; Saunders, S. C., A new family of life distributions, J. Appl. Probab., 6, 319-327 (1969) · Zbl 0209.49801
[8] Bechtel, Y. C.; Bonaiti-Pellie, C.; Poisson, N.; Magnette, J.; Bechtel, P. R., A population and family study of n-acetyltransferase using caffeine urinary metabolites, Clin. Pharmacol. Therapeut., 54, 134-141 (1993)
[9] Chhikara, R. S.; Folks, J. L., The Inverse Gaussian Distribution (1989), Marcel Dekker: Marcel Dekker New York · Zbl 0408.62011
[10] Cox, D. R., Some sampling problems in technology, (Johnson, N. L.; Smith, H., New Developments in Survey Sampling (1969), John Wiley and Sons: John Wiley and Sons New York), 506-527
[11] Dempster, A. P.; Laird, N. M.; Rubin, D. B., Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Statist. Soc. B, 39, 1-38 (1977) · Zbl 0364.62022
[12] Desmond, A. F., Stochastic models of failure in random environment, Canad. J. Statist., 13, 171-183 (1985) · Zbl 0581.60073
[13] Desmond, A. F., On the relationship between two fatigue-life models, IEEE Trans. Rel., 35, 167-169 (1986) · Zbl 0592.62089
[14] Efron, B.; Hinkley, D., Assessing the accuracy of the maximum likelihood estimator: observed vs. expected Fisher information, Biometrika, 65, 457-487 (1978) · Zbl 0401.62002
[15] Engelhardt, M.; Bain, L. J.; Wright, F. T., Inferences on the parameters of the Birnbaum-Saunders fatigue life distribution basedon maximum likelihood estimation, Technometrics, 23, 251-256 (1981) · Zbl 0462.62077
[16] Fisher, R. A., The effects of methods of ascertainment upon the estimation of frequencies, Ann. Eugenics, 6, 13-25 (1934)
[17] Finch, S. J.; Mendell, N. R.; Thode, H. C., Probabilistic measures of adequacy of a numerical search for a global maximum, J. Amer. Statist. Assoc., 84, 1020-1023 (1989)
[18] Gupta, R. C.; Akman, O., On the reliability studies of a weighted inverse Gaussian distribution, J. Statist. Plann. Inference, 48, 69-83 (1995) · Zbl 0846.62073
[19] Gupta, R. C.; Akman, O., Estimation of critical points in the mixture of inverse Gaussian model, Statist. Papers, 38, 445-452 (1997) · Zbl 0911.62089
[20] Gupta, R. C.; Keating, J. P., Relations for reliability measures under length biased sampling, Scand. J. Statist., 13, 49-56 (1985) · Zbl 0627.62098
[21] Gupta, R. C.; Kirmani, S. N.U. A., The role of weighted distributions in stochastic modeling, Comm. Statist. Theory Methods, 19, 3147-3162 (1990) · Zbl 0734.62093
[22] Jain, K.; Singh, H.; Bagai, I., Relations for reliability measures of weighted distributions, Comm. Statist. Theory Methods, 18, 4393-4412 (1989) · Zbl 0707.62197
[23] Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous Univariate Distributions—Vol. 1 (1994), John Wiley and Sons: John Wiley and Sons New York · Zbl 0811.62001
[24] Johnson, N. L.; Kotz, S.; Balakrishnan, N., Continuous Univariate Distributions—Vol. 2 (1995), John Wiley and Sons: John Wiley and Sons New York · Zbl 0821.62001
[25] Jörgensen, B., Statistical Properties of the Generalized Inverse Gaussian Distribution (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0486.62022
[26] Jörgensen, B.; Seshadri, V.; Whitmore, G. A., On the mixture of the inverse Gaussian distribution with its complementary reciprocal, Scand. J. Statist., 18, 77-89 (1991) · Zbl 0723.62006
[27] Jörgensen, B., The Theory of Dispersion Models (1997), Chapman and Hall: Chapman and Hall London · Zbl 0928.62052
[28] Khattree, R., Characterization of inverse Gaussian and gamma distributions through their length biased versions, IEEE Trans. Rel., 38, 610-611 (1989) · Zbl 0695.62014
[29] Kundu, D.; Kannan, N.; Balakrishnan, N., On the hazard function of Birnbaum-Saunders distribution and associated inference, Comput. Statist. Data Anal., 52, 2692-2702 (2008) · Zbl 1452.62729
[30] Leiva, V.; Hernández, H.; Riquelme, M., A new package for the Birnbaum-Saunders distribution, R Journal, 6, 35-40 (2006)
[31] Leiva, V.; Barros, M.; Paula, G. A.; Galea, M., Influence diagnostics in log-Birnbaum-Saunders regression models with censored data, Comput. Statist. Data Anal., 51, 5694-5707 (2007) · Zbl 1445.62199
[32] Leiva, V.; Barros, M.; Paula, G.; Sanhueza, D., Generalized Birnbaum-Saunders distributions applied to air pollutant concentration, Environmetrics, 19, 235-249 (2008)
[33] Leiva, V.; Sanhueza, A.; Angulo, J. M., A length-biased version of the Birnbaum-Saunders distribution with application in water quality, Stochastic Environ. Res. Risk. Assess., 23, 299-307 (2009) · Zbl 1411.62339
[34] Leiva, V.; Sanhueza, A.; Kotz, S.; Araneda, N., A unified mixture model based on the inverse Gaussian distribution, Pakistan J. Statist., 26, 445-460 (2010) · Zbl 1509.62121
[35] Leiva, V.; Vilca, F.; Balakrishnan, N.; Sanhueza, A., A skewed sinh-normal distribution and its properties and application to air pollution, Comm. Statist. Theory Methods, 39, 426-443 (2010) · Zbl 1187.62027
[36] Louis, T. A., Finding the observed information matrix when using the EM algorithm, J. Roy. Statist. Soc. B, 44, 226-233 (1982) · Zbl 0488.62018
[37] Marshall, A. W.; Olkin, I., Life Distributions (2007), Springer-Verlag: Springer-Verlag New York
[38] McLachlan, G. J.; Peel, D., Finite Mixture Models (2000), John Wiley and Sons: John Wiley and Sons New York · Zbl 0963.62061
[39] Ng, H. K.T.; Kundu, D.; Balakrishnan, N., Modified moment estimation for the two-parameter Birnbaum-Saunders distribution, Comput. Statist. Data Anal., 43, 283-298 (2003) · Zbl 1429.62451
[40] Ng, H. K.T.; Kundu, D.; Balakrishnan, N., Point and interval estimations for the two-parameter Birnbaum-Saunders distribution based on type-II censored samples, Comput. Statist. Data Anal., 50, 3222-3242 (2006) · Zbl 1161.62418
[41] Olyede, B. O.; George, E. O., On stochastic inequalities and comparisons of reliability measures for weighted distributions, Math. Probab. Eng., 8, 1-13 (2002) · Zbl 1047.60010
[42] Patil, G. P., Weighted distributions, (El-Shaarawi, A.; Piegorsch, W. W., Encyclopedia of Environmetrics, vol. 4 (2002), John Wiley and Sons: John Wiley and Sons New York), 2369-2377
[43] Patil, G. P.; Rao, C. R., Weighted distributions and size-biased sampling with applications to wildlife populations and human families, Biometrics, 34, 179-189 (1978) · Zbl 0384.62014
[44] Rao, C. R., On discrete distributions arising out of methods of ascertainment, Sankhya A, 27, 311-324 (1965) · Zbl 0212.21903
[45] Sanhueza, A.; Leiva, V.; Balakrishnan, N., The generalized Birnbaum-Saunders distribution and its theory, methodology and application, Comm. Statist. Theory Methods, 37, 645-670 (2008) · Zbl 1136.62016
[46] Sansgiry, P. S.; Akman, O., Reliability estimation via length-biased transformation, Comm. Statist. Theory Methods, 30, 2473-2479 (2001) · Zbl 1009.62589
[47] Seshadri, V., The Inverse Gaussian Distribution: A Case Study in Exponential Families (1993), Oxford University Press: Oxford University Press Oxford, England
[48] Seshadri, V., The Inverse Gaussian Distribution: Statistical Theory and Applications (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0942.62011
[49] Xiao, Q.; Lu, Z.; Balakrishnan, N.; Lu, X., Estimation of the Birnbaum-Saunders regression model with current status data, Comput. Statist. Data Anal., 54, 326-332 (2010) · Zbl 1464.62186
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.