A bivariate α-power transformed family: Theory and application

Document Type : Original Article

Authors

1 Department of Statistics, Khansar Campus, University of Isfahan, Iran

2 Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, Iran

3 Department of Statistics, Yazd University, Yazd, Iran

Abstract

In this paper, some general classes of bivariate semi-parametric continuous distributions are introduced. Some important properties of this family of distributions will be illustrated. It is seen that the bivariate distribution corresponds to the known Ali-Mikhail-Haq copula. Hence, some important properties such as the $\hbox{TP}_2$ property are justified. It will be shown that the marginals are kind of heavy tailed weighted distributions whose hazard rate functions can take variety of shapes. The behavior of the hazard rate function is mathematically illustrated. In addition, the $\alpha$-power transformed distributions of a second type, which are introduced for the first time here, can be verified as special cases of the marginals. Some members of the new bivariate classes are studied in details. The estimation of the parameters is illustrated by means of an efficient expectation-maximization algorithm, and some real data sets are also analyzed for illustrative purposes.

Keywords

Main Subjects

References

[1] K. Adamidis, T. Dimitrakopoulou, and S. Loukas, On an extension of the exponential-geometric distri- bution, Statist. Probab. Lett., 73 (2005), pp. 259–269.
[2] K. Adamidis and S. Loukas, A lifetime distribution with decreasing failure rate, Statist. Probab. Lett., 39 (1998), pp. 35–42.
[3] G. Asha, J. K. K. M., and D. Kundu, An extension of the freund’s bivariate distribution to model load- sharing systems, American Journal of Mathematical and Management Sciences, 35 (2016), pp. 207–226.
[4] N. Balakrishnan and C.-D. Lai, Continuous bivariate distributions, Springer, Dordrecht, second ed., 2009.
[5] W. Barreto-Souza, A. L. de Morais, and G. M. Cordeiro, The Weibull-geometric distribution, J. Stat. Comput. Simul., 81 (2011), pp. 645–657.
[6] A. P. Basu, Bivariate failure rate, Journal of the American Statistical Association, 66 (1971), pp. 103–104.
[7] J. V. Deshpande, I. Dewan, and U. V. Naik-Nimbalkar, Two component load sharing systems with applications to Biology, tech. rep., Indian Statistical Institute, New Delhi, India, 2007. Available at https: //www.isid.ac.in/~statmath/eprints/2007/isid200706.pdf.
[8] F. Hemmati, E. Khorram, and S. Rezakhah, A new three-parameter ageing distribution, J. Statist. Plann. Inference, 141 (2011), pp. 2266–2275.
[9] N. L. Johnson and S. Kotz, A vector multivariate hazard rate, J. Multivariate Anal., 5 (1975), pp. 53–66.
[10] C. Kus¸, A new lifetime distribution, Comput. Statist. Data Anal., 51 (2007), pp. 4497–4509.
[11] D. Kundu, Bivariate geometric (maximum) generalized exponential distribution, Journal of Data Science, 13 (2015), pp. 693–712.
[12] D. Kundu and R. D. Gupta, Modified Sarhan-Balakrishnan singular bivariate distribution, J. Statist. Plann. Inference, 140 (2010), pp. 526–538.
[13] , Power-normal distribution, Statistics, 47 (2013), pp. 110–125.
[14] P. H. Kvam and E. A. Pe˜na, Estimating load-sharing properties in a dynamic reliability system, Journal of the American Statistical Association, 100 (2005), pp. 262–272.
[15] A. Mahdavi and D. Kundu, A new method for generating distributions with an application to exponential distribution, Comm. Statist. Theory Methods, 46 (2017), pp. 6543–6557.
[16] E. Mahmoudi and A. A. Jafari, Generalized exponential-power series distributions, Comput. Statist. Data Anal., 56 (2012), pp. 4047–4066.
[17] A. W. Marshall and I. Olkin, Life Distributions: Structure of Nonparametric, Semiparametric, and Para- metric Families, Springer New York, NY, 1 ed., 2007.
[18] V. Nekoukhou, A. Khalifeh, and H. Bidram, Univariate and bivariate extensions of the generalized exponential distributions, Math. Slovaca, 71 (2021), pp. 1581–1598.
[19] V. Nekoukhou, A. Khalifeh, and E. a. Mahmoudi, Bivariate rayleigh-geometric distribution, Journal of Statistical Sciences, 13 (2020).
[20] R. B. Nelsen, An introduction to copulas, Springer Series in Statistics, Springer, New York, second ed., 2006.
[21] P. S. Puri and H. Rubin, On a characterization of the family of distributions with constant multivariate failure rates, Ann. Probability, 2 (1974), pp. 738–740.
[22] R. B. Silva, M. Bourguignon, C. R. B. Dias, and G. M. Cordeiro, The compound class of extended Weibull power series distributions, Comput. Statist. Data Anal., 58 (2013), pp. 352–367
AUT Journal of Mathematics and Computing
Volume 7, Issue 1
January 2026
Pages 1-18

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