International Journal of Advanced and Applied Sciences, 5(6) 2018, Pages: 64-69 Contents lists available at Science-Gate International Journal of Advanced and Applied Sciences Journal homepage: http://www.science-gate.com/IJAAS.html 64 Univariate and bivariate Burr x-type distributions Mervat K. Abd Elaala *, Lamya A. Baharith Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia ARTICLE INFO ABSTRACT Article history: Received 24 January 2018 Received in revised form 30 March 2018 Accepted 9 April 2018 The Burr X distribution has been extensively studied by many researchers. It has many applications in medical, biological, agriculture and other fields. In this paper, a new family of Burr X-type distributions is introduced; the univariate Burr X-type distribution and the bivariate Burr X-type distribution. The bivariate Burr X-type distribution is constructed based on Gaussian copula with univariate Burr X-type distribution as marginals. This type distribution is more flexible and provides easier implementation and extension to bivariate form. A Gibbs sampler procedure is used to obtain Bayesian estimates of the unknown parameters. A simulation study is carried out to illustrate the efficiency of the proposed bivariate Burr X-type distribution. Finally, the proposed bivariate distribution is applied on real data to demonstrate its usefulness for real life applications. Keywords: Burr X distribution M mixture representation Copula Bivariate Burr X type distribution Gibbs sampler © 2018 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction *Burr type X distribution is a member of the family of Burr distributions which was appeared since 1942 (Burr, 1942). It is known also as generalized Rayleigh distribution. This distribution has increasing importance in several areas of applications such as lifetime tests, health, agriculture, biology, and other sciences. In recent years, Burr X distribution has been extensively used in medical, biological, agriculture, lifetime tests, and other sciences applications. The Burr X distribution was first introduced by Burr (1942) and later a generalized form of this distribution is introduced by Mudholkar and Srivastava (1993). Several characteristics and inferences of this distribution were studied by many researchers, see for example Abd et al. (2015), Ali Mousa (2001), Aludaat et al. (2008), Jaheen and Al-Matrafi (2002), Kjelsberg (1962), Kundu and Raqab (2005) and Raqab (1998), among others. The probability density function (Pdf) of the Burr type X distribution with shape parameter β and scale parameter α is given by f(t) = 2  2 exp (− (   ) 2 ) [1 − exp (− (   ) 2 )] −1 , (1) where t, α, β >0. * Corresponding Author. Email Address: mkabdelaal@kau.edu.sa (M. K. Abd Elaala) https://doi.org/10.21833/ijaas.2018.06.010 2313-626X/© 2018 The Authors. Published by IASE. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Diverse methods have been studied in previous research for establishing new multivariate or bivariate distributions. Among these, the study of Al-Hussaini and Ateya (2005), Johnson et al. (2002), Marshall and Olkin (1997) and Walker and Stephens (1999). Recently, the copula method has received special attention for constructing multivariate or bivariate distributions due to its simplicity and useful dependency properties. Some of these studies combined the mixture and copula ideas to establish new family of distributions which concluded that the resulting multivariate or bivariate distribution is easy to analyze and has a full dependence structures. These studies include Adham and Walker (2001), AL Dayian et al. (2008), Abd Elaal et al. (2016), and Adham et al. (2009). According to Adham and Walker (2001) and Walker and Stephens (1999), the mixture representation for a Pdf of a random variable Ton [0,∞) can be written in the following form f(t) = ∫ f(t|u) Ω f(u) du, for all u ∈ Ω, (2) where u is a non-negtive latent variable that follows a gamma distribution with shape parameter 2 and scale parameter 1. Then, the mixture representation for any lifetime distribution can be written as f(t) = ∫ f(t|u) Ω f(u) du, for all u ∈ Ω, (3) where ℎ(t) and (t) are the hazard rate function and the cumulative hazard rate function of T, respectively. The studies of Walker and Stephens (1999), Agarwal and Al-Saleh (2001), and Arslan (2005)