On bivariate extension of the univariate transmuted distribution family

Abstract

The aim of this study is to examine the bivariate transmuted distributions in the literature and to propose alternative distribution. The method is based on mixing distributions of pairs of order statistics of a sample of size two. Some of proposed distributions allow both negative and positive Pearson correlations with admissible range between pairs of random variates. The results of the study gain importance in terms of eliminating or completing the missing aspects of the bivariate transmuted distributions existing in the literature.

Keywords

• Bivariate extension
• tranmuted distribution
• dependence
• bivariate distribution
• Spearman’s Rho

References

1. Alizadeh, M., Merovci, F., Hamedani, G. G., Generalized transmuted family of distributions: properties and applications, Hacettepe Journal of Mathematics and Statistics, 46 (2017), 645 667.
2. Asadian, N., Amini, M., Bozorgnia, A., Some concepts of negative dependence for bivariate distributions with applications, Journal of Mathematical Extension, 4 (1) (2009), 43–59.
3. Baker, R., An order-statistics-based method for constructing multivariate distributions with fixed marginals, Journal of Multivariate Analysis, 99 (10) (2008), 2312–2327, https://dx.doi.org/10.1016/j.jmva.2008.02.019.
4. Bakouch, H. S., Jamal, F., Chesneau, C., Nasir, A., A new transmuted family of distributions: Properties and estimation with applications, working paper or preprint, Sept. 2017.
5. Barlow, R. E., Proschan, F., Statistical theory of reliability and life testing: probability models, Tech. rep., Florida State Univ Tallahassee, 1975, https://dx.doi.org/10.2307/1268641.
6. Bekçi, M., Yılmaz, M., Construction of bivariate distribution by mixing positively dependent and negatively dependent distributions, International Journal of Statistics and Applications, 9(4) (2015), 122–1127, https://dx.doi.org/10.5923/j.statistics.20190904.04.
7. Bourguignon, M., Ghosh, I., Cordeiro, G. M., General results for the transmuted family of distributions and new models, Journal of Probability and Statistics, 2016 (2016), https://dx.doi.org/10.1155/2016/7208425.
8. Dolati, A., Ubeda-Flores, M., Constructing copulas by means of pairs of order statistics, Kybernetika, 45 (6) (2009), 992–1002.

9. Farlie, D. J. G., The performance of some correlation coefficients for a general bivariate distribution, Biometrika, 47 (3/4) (1960), 307–323, https://dx.doi.org/10.2307/2333302.
10. Gumbel, E. J., Bivariate exponential distributions, Journal of the American Statistical Association, 55 (292) (1960), 698–707, https://dx.doi.org/10.1080/01621459.1960.10483368.
11. Merovci, F., Alizadeh, M., Hamedani, G. G., Another generalized transmuted family of distributions: properties and applications, Austrian Journal of Statistics, 45 (3) (Jun. 2016), 71–93, https://dx.doi.org/10.17713/ajs.v45i3.109.
12. Merovci, F., Alizadeh, M., Yousof, H. M., Hamedani, G. G., The exponentiated transmuted-g family of distributions: theory and applications, Communications in Statistics - Theory and Methods, 46 (21) (2017), 10800–10822, https://dx.doi.org/10.1080/03610926.2016.1248782.
13. Mirhoseini, S., Dolati, A., Amini, M., On a class of distributions generated by stochastic mixture of the extreme order statistics of a sample of size two, arXiv preprint arXiv:1904.04287 (2019).
14. Mirhosseini, S. M., Amini, M., Dolati, A., On a general structure of the bivariate fgm type distributions, Applications of Mathematics, 60 (1) (2015), 91–108, https://dx.doi.org/10.1007/s10492-015-0086-6.
15. Oluyede, B. O., On local dependence and stochastic inequalities with applications to contingency tables, Applied mathematics and computation, 151 (3) (2004), 801–813, https://dx.doi.org/10.1016/s0096-3003(03)00537-x.
16. Rezaei, S., Marvasty, A. K., Nadarajah, S., Alizadeh, M., A new exponentiated class of distributions: Properties and applications, Communications in Statistics - Theory and Methods, 46(12) (2017), 6054–6073, https://dx.doi.org/10.1080/03610926.2015.1116579.
17. Rüschendorf, L., Construction of multivariate distributions with given marginals, Annals of the Institute of Statistical Mathematics, 37 (2) (1985), 225–233, https://dx.doi.org/10.1007/bf02481093.
18. Sarabia, J. M., Raja, A. V., Asha, G., Bivariate distributions with transmuted conditionals: models and applications, Communications in Statistics - Theory and Methods, 49 (1) (2020), 221–242, https://dx.doi.org/10.1080/03610926.2018.1536785.
19. Schweizer, B., Wolff, E. F., On nonparametric measures of dependence for random variables, The Annals of Statistics, 9 (4) (1981), 879 – 885, https://dx.doi.org/10.1214/aos/1176345528.
20. Shaw, W. T., Buckley, I. R. C., The alchemy of probability distributions: beyond Gram-Charlier expansions, and a skew-kurtotic-normal distribution from a rank transmutation map, ArXiv e-prints (Jan. 2009).
21. Ünözkan, H., Yılmaz, M., Construction of continuous bivariate distribution by transmuting dependent distribution, Cumhuriyet Science Journal, 40 (4) (2019), 860–866, https://dx.doi.org/10.17776/csj.618236.