On generating T-X family of distributions using quantile functions

• Published in: Journal of statistical distributions and applications Vol. 1; no. 1; pp. 1 - 2
• Main Authors: Aljarrah, Mohammad A, Lee, Carl, Famoye, Felix
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 11.06.2014; Springer Nature B.V; BioMed Central Ltd
• Subjects: 2013 International Conference on Statistical Distributions and Applications; Humanities and Social Sciences; Mathematics and Statistics; multidisciplinary; Science; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; T-X families; Moments; Generalized distribution; Survival function; Beta-family
• ISSN: 2195-5832; 2195-5832
• DOI: 10.1186/2195-5832-1-2

The cumulative distribution function (CDF) of the T - X family is given by R { W ( F ( x ))}, where R is the CDF of a random variable T , F is the CDF of X and W is an increasing function defined on [0, 1] having the support of T as its range. This family provides a new method of generating univariate distributions. Different choices of the R , F and W functions naturally lead to different families of distributions. This paper proposes the use of quantile functions to define the W function. Some general properties of this T - X system of distributions are studied. It is shown that several existing methods of generating univariate continuous distributions can be derived using this T - X system. Three new distributions of the T-X family are derived, namely, the normal-Weibull based on the quantile of Cauchy distribution, normal-Weibull based on the quantile of logistic distribution, and Weibull-uniform based on the quantile of log-logistic distribution. Two real data sets are applied to illustrate the flexibility of the distributions.