SOME CLASS OF CHI-SQUARE MIXTURE OF THE TRANSFORMED BETA FAMILY DISTRIBUTIONS A.B. OBEJAS AND J.P. ARCEDE Abstract. We introduce a class of chi-square mixtured of the transformed Beta family distributions which are the Chi-square mixtured distributions of Pareto, Loglogistic and Burr. The main results in this study are as follows: (i) the definition of the Chi-square mixtured of Pareto, Loglogistic and Burr distributions, (ii) derivation of some properties of each distributions, that is, moments and the shape characteristics, also (iii) relationships of Chi-square mixture of Pareto distribution, Chi-square mixture of Loglogistic distribution and Chi-square mixture of Burr distributions were given. 1. Introduction Mixture of distributions were studied by a number of authors since 1894. Ac- cording to Blischke [1] a mixture of distribution is a weighted average of probability distribution with positive weights that sum to one. The distributions thus mixed are called components of mixture. The weights themselves incorporate a proba- bility distribution called the mixing distribution. For this property of weights, a mixture is again a probability distribution. Pearson [10] was the pioneer in the field of mixture of distributions who reflected on the mixture of two normal distribu- tions. After a long gap, some basic properties of mixture distribution were studied by Robin [12], Mendhandall and Haider [8]. Rider [11] published a paper on the method of moment applied to a mixture of two exponential distributions. Mixture of two geometric distributions was studied by Daniele [5]. Haselblad [7] studied in greater detail the finite mixture of distribution from the exponential family. Beta mixture of binomial, gamma mixture of poisson and poisson mixture of binomial distribution were considered by David [6]. Roy. et al. [13], [15] defined and studied poisson mixture of distribution and negative binomial mixture of distri- bution. Other authors [2, 3, 4], considers mixtured distributions which they called Laplace mixture, Rayleigh mixture, F, and Dual mixture of distributions. In this study, we defined the Chi-squre mixture of distributions of some class of the trans- formed beta family distribution, that is, Chi-square mixtures of Pareto, Loglogistic and Burr and studied some of their properties. 2. Preliminaries We shall now present how distributions are being mixed. Roy et al [16], defined mixture distribution by taking θ, the parameter of a family of distributions, given by the density function f (x; θ), is itself subject to the change variation. Date : November 20, 2012. 2010 Mathematics Subject Classification. 60E05, 60F99. Key words and phrases. Chi-square mixtured distribution; Pareto; Loglogistic; Burr. 1