Abstract
In 2020, El-Morshedy et al. introduced a bivariate extension of the Burr type X generator (BBX-G) of distributions, and Muhammed presented a bivariate generalized inverted Kumaraswamy (BGIK) distribution. In this paper, we propose a more flexible generator of bivariate distributions based on the maximization process from an arbitrary three-dimensional baseline distribution vector, which is of interest for maintenance and stress models, and expands the BBX-G and BGIK distributions, among others. This proposed generator allows one to generate new bivariate distributions by combining non-identically distributed baseline components. The bivariate distributions belonging to the proposed family have a singular part due to the latent component which makes them suitable for modeling two-dimensional data sets with ties. Several distributional and stochastic properties are studied for such bivariate models, as well as for its marginals, conditional distributions, and order statistics. Furthermore, we analyze its copula representation and some related association measures. The EM algorithm is proposed to compute the maximum likelihood estimations of the unknown parameters, which is illustrated by using two particular distributions of this bivariate family for modeling two real data sets.
Highlights
Gumbel [1], Freund [2], and Marshall and Olkin [3] in their pioneering papers developed bivariate exponential distributions
Several continuous bivariate distributions can be found in Balakrishnan and Lai [4], and some generalizations and multivariate extensions have been studied by Franco and Vivo [5], Kundu and Gupta [6], Franco et al [7], Gupta et al [8], Kundu et al [9], among others, and recently by Muhammed [10], Franco et al [11], and El-Morshedy et al [12], see the references cited therein
The maximum likelihood estimation (MLE) of the unknown parameters a generalized bivariate distribution (GBD) model cannot be obtained in closed form, and we propose using an EM algorithm to compute the MLEs of such parameters
Summary
Gumbel [1], Freund [2], and Marshall and Olkin [3] in their pioneering papers developed bivariate exponential distributions. Kundu and Gupta [13] introduced a bivariate generalized exponential (BGE) distribution by using the trivariate reduction technique with generalized exponential (GE) random variables, which is based on the maximization process between components with a latent random variable, suitable for modeling of some stress and maintenance models. This procedure has been applied in the Mathematics 2020, 8, 1776; doi:10.3390/math8101776 www.mdpi.com/journal/mathematics. Some of the proofs are relegated to Appendix A for a fluent presentation of the results, and some technical details of the applications can be found in Appendix B
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