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Bivariate Semi-parametric Singular Family of Distributions and its Applications

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  • Debasis Kundu

    (Indian Institute of Technology Kanpur)

Abstract

In this paper we introduce a very general class of bivariate semi-parametric distributions whose marginals belong to the proportional hazard class, and it has a singular component. This model can be used quite effectively to analyze a bivariate data set when there are ties. Note that the Marshall-Olkin bivariate exponential distribution is a special case of the proposed class. We derive several properties of the proposed distribution, and it is observed that it has a very convenient copula structure. Hence, several dependence properties and dependence measures can be obtained based on the copula. The main feature of the proposed distribution is that we do not use any specific parametric form of the base line distribution, instead we have assumed that the base line distribution has piecewise constant hazard function. It makes the proposed family a very flexible family. The maximum likelihood estimators cannot be obtained in explicit form, and we have used a very convenient EM algorithm to compute the maximum likelihood estimators. Simulation experiments have been performed to see the effectiveness of the proposed EM algorithm. Finally we have used this model to analyze a dependent competing risks data. Two data sets have been analyzed and the results are quite satisfactory.

Suggested Citation

  • Debasis Kundu, 2022. "Bivariate Semi-parametric Singular Family of Distributions and its Applications," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 84(2), pages 846-872, November.
    • Handle: RePEc:spr:sankhb:v:84:y:2022:i:2:d:10.1007_s13571-022-00289-y
    DOI: 10.1007/s13571-022-00289-y
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    References listed on IDEAS

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    1. N. Balakrishnan & M. V. Koutras & F. S. Milienos & S. Pal, 2016. "Piecewise Linear Approximations for Cure Rate Models and Associated Inferential Issues," Methodology and Computing in Applied Probability, Springer, vol. 18(4), pages 937-966, December.
    2. Dimitris Karlis, 2003. "ML estimation for multivariate shock models via an EM algorithm," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 55(4), pages 817-830, December.
    3. Johnson, N. L. & Kotz, Samuel, 1975. "A vector multivariate hazard rate," Journal of Multivariate Analysis, Elsevier, vol. 5(1), pages 53-66, March.
    4. Yan Shen & Ancha Xu, 2018. "On the dependent competing risks using Marshall–Olkin bivariate Weibull model: Parameter estimation with different methods," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(22), pages 5558-5572, November.
    5. Hu, Taizhong & Khaledi, Baha-Eldin & Shaked, Moshe, 2003. "Multivariate hazard rate orders," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 173-189, January.
    6. Feizjavadian, S.H. & Hashemi, R., 2015. "Analysis of dependent competing risks in the presence of progressive hybrid censoring using Marshall–Olkin bivariate Weibull distribution," Computational Statistics & Data Analysis, Elsevier, vol. 82(C), pages 19-34.
    7. Kundu, Debasis & Gupta, Rameshwar D., 2009. "Bivariate generalized exponential distribution," Journal of Multivariate Analysis, Elsevier, vol. 100(4), pages 581-593, April.
    8. Jing Cai & Yimin Shi & Bin Liu, 2017. "Analysis of incomplete data in the presence of dependent competing risks from Marshall–Olkin bivariate Weibull distribution under progressive hybrid censoring," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(13), pages 6497-6511, July.
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    Cited by:

    1. Debasis Kundu, 2024. "A Stationary Proportional Hazard Class Process and its Applications," Methodology and Computing in Applied Probability, Springer, vol. 26(4), pages 1-21, December.
    2. Debashis Samanta & Debasis Kundu, 2023. "Bivariate Semi-Parametric Model: Bayesian Inference," Methodology and Computing in Applied Probability, Springer, vol. 25(4), pages 1-23, December.
    3. Ayan Pal & Debashis Samanta & Debasis Kundu, 2025. "A semiparametric approach for simple step-stress model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 34(1), pages 91-124, March.

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