Topp–Leone Family of Distributions: Some Properties and Application

In this paper we have proposed a new family of distributions; the Topp–Leone family of distributions. We have given general expression for density and distribution function of the new family. Expression for moments and hazard rate has also been given. We have also given an example of the proposed family.


Introduction
Topp-Leone distribution is a simple bounded J-shaped distribution that has attracted various statisticians as an alternative to Beta distribution. The density and distribution function of Topp-Leone distribution is given by Nadarajah and Kotz (2003)  Since its emergence the distribution has been studied by number of authors, see for example Ghitany et al. (2005), van Dorp and Kotz (2006), Zhou et al. (2006), Kotz and Seier (2007), Nadarajah (2009), and Genç (2012).
Generalizing probability distributions has been area of study by number of authors. Various families of distributions have also been proposed by using different bounded and unbounded distributions. Eugene et al. (2002) proposed Beta family of distributions by using distribution function of a Beta random variable. The distribution function of the proposed family is is the Gamma function. The Gamma family of distribution has also been studied by number of authors.
We propose a new family of distributions by using distribution function of Topp-Leone distribution in the following section.

Topp-Leone Family of Distributions
The distribution function of Topp-Leone distribution is given in (1.2) as Using above distribution function we propose the Topp-Leone family of distributions as one having the distribution function given as The Topp-Leone family of distributions reduces to the baseline distribution   The density function of Topp-Leone family of distributions is readily written as . Various choices of   Gx lead to various members of Topp-Leone family. We will denote Topp-Leone family by TL-G distributions.

Expansion for Density and Distribution Function
The distribution function of Topp-Leone family of distributions is given in (2.1) as Now using the following identity given in Prudnikov et al. (1986) the distribution function of TL-G family is written as Again from (3.1), the density function of TL-G family is (3.4) Using series expansion the density can be written as The distribution function and density function of TL-G family given in (3.3) and (3.6) respectively immediately show that the TL-G family can be viewed as weighted sum of exponentiated class of distributions.
Using ( The moments for TL-G class can be computed from (3.8) for certain special cases.
The quantile function of TL-G family can be readily written from quantile function of Topp-Leone distribution as   (3.9) The quantile function can be used to generate the random data for TL-G family of distributions for certain special cases.

Parameter Estimation
The density function of TL-G family of distribution is given in (3.4) as The log of likelihood function for TL -G family is hence The MLE of  for special cases can be readily obtained from (4.1).

The Topp-Leone Exponential Distribution
In this section we have given an illustrated example of TL-G family of distributions by using , that is the Exponential distribution and we will call the overall distribution the Topp-Leone-Exponential distribution or TL-E distribution for short. The distribution function of TL-E distribution is readily written from (3.1) as   These entries can be obtained numerically.

Application
In this section we have given a numerical application of the TL -E distribution. We have used data from Nigm et al. (2003) and is about ordered failure of components. The data is given below The graph also shows that the data is adequately fitted by TL-E distribution.

Conclusion and Further Work
New family of distributions are proposed; the Topp-Leone family of distributions. The density function and distribution function for new family are given as weighted sum of exponentiated base distribution. In addition, expression for moments and hazard rate are given. Exponential distribution is applied as special case of TL-G family of distributions. Numerical application of the TL-E distribution is discussed. This example indicate that the data is adequately fitted by TL-E distribution.