Odds Generalized Exponential – Exponential Distribution

: A new distribution, called Odds Generalized Exponential-Exponential distribution (OGEED) is proposed for modeling lifetime data. A comprehensive account of the mathematical properties of the new distribution including estimation and simulation issues is presented. A data set has been analyzed to illustrate its applicability.


Introduction
There are several ways of adding one or more parameters to a distribution function. Such an addition of parameters makes the resulting distribution richer and more flexible for modeling data. Proportional hazard model (PHM), Proportional reversed hazard model (PRHM), Proportional odds model (POM), Power transformed model (PTM) are few such models originated from this idea to add a shape parameter. In these models, a few pioneering works are by Box and Cox (1964), Cox (1972), Mudholkar and Srivastava (1993), Shaked and Shantikumar (1994), Marshall and Olkin (1997), Gupta and Kundu (1999), Gupta and Gupta (2007) among others.
Many distributions have been developed in recent years that involves the logit of the beta distribution. Under this generalized class of beta distribution scheme, the cumulative distribution function (cdf) for this class of distributions for the random variable is generated by applying the inverse of the cdf of to a beta distributed random variable to obtain, 0, > , ; ) (1 ) , ( where is the cdf of any other distribution. This class has not only generalized the beta distribution but also added parameter(s) to it. Among this class of distributions are, the beta-Normal [Eugene et al. (2002)]; beta-Gumbel [Nadarajah and Kotz (2004)]; beta-Exponential [Nadarajah and Kotz (2006)]; beta-Weibull [Famoye et al. (2005)]; beta-Rayleigh [Akinsete and Lowe (2009)]; beta-Laplace [Kozubowski and Nadarajah (2008)]; and beta-Pareto [Akinsete et al. (2008)], among a few others. Many useful statistical properties arising from these distributions and their applications to real life data have been discussed in the literature.
In the generalized class of beta distribution, since the beta random variable lies between 0 and 1, and the distribution function also lies between 0 and 1, to find out cdf of generalized distribution, the upper limit is replaced by cdf of the generalized distribution. Alzaatreh et al. (2013) has proposed a new generalized family of distributions, called T-X family, and the cumulative distribution function (cdf) is defined as where, the random variable , for and be a function of the cdf so that satisfies the following conditions: is differentiable and monotonically non-decreasing, We have defined a generalized class of any distribution having positive support. Taking , the odds function, the cdf of the proposed generalized class of distribution is given by The support of the resulting distribution will be that of (.)  F . here, The resulting distribution is not only generalized but also added with some parameter(s) to the base distribution. We call this class of distributions as Odds Generalized family of distributions (OGFD).
Throughout this paper we use the following notations. We write upper incomplete gamma function and lower incomplete gamma function as i.e. also the exponential distribution in (1. 2 ). Hence, we call this distribution as Odds Generalized Exponential-Exponential distribution (OGEED). The paper is organized as follows. The distribution is developed in section 2. A comprehensive account of mathematical properties including structural and reliability of the new distribution is provided in section 3. Maximum likelihood method of estimation of parameters of the distribution is discussed in section 4. Simulation study results have been presented and dicussed in section 5.A real life data set has been analyzed and compared with other fitted distributions with respect to Akaike Information Criterion (AIC) in section 6. Section 7 concludes.

The Probability Density Function of the OGEED
The c.d.f. of the OGEED is given by the form as

Limit of the Probability Distribution Function
Since the c.d.f. of this distribution is

Descriptive Statistics of OGEED
The mean of this OGEED is as follows: So mean of the OGEED is The median of the OGEED is given by The mode of the OGEED is given as: The rth order raw moment of this OGEED is as follows: Now putting suitable values of r in the above equation, we get Variance, Skewness, Kurtosis and Coefficients of variation of the Odds Generalized Exponential -Exponential Distribution(OGEED).

Mean Deviation:
The mean deviation about the mean and the mean deviation about the median is defined by denotes the mean and median respectively. Thus (3.9)

Conditional Moments:
The residual life and the reversed residual life play an important role in reliability theory and other branches of statistics. Here, the r-th order raw moment of the residual life is given by The r-th order raw moment of the reversed residual life is given by

Quantile function:
Let X denote a random variable with the probability density function 2. 4

Order Statistics
Suppose n X X X X ,......, , , is a random sample from the distribution in (2.4). Let ,......, , , n X X X X , denote the corresponding order statistics. Hence the probability density function and the cumulative distribution function of the th k order statistic, say

Entropies
An entropy of a random variable X is a measure of variation of the uncertainty. A popular entropy measure is Renyi entropy (Renyi 1961). If X has the probability density function f(x), then Renyi entropy is defined by Shannon measure of entropy is defined as After some algebraic simplification, we have

Loss reserves data for Queensland, Australia
The Reliability function of OGEED is given by the form as: 3.27) and the Hazard rate of OGEED is given by the form as: So, the distribution is log-concave. Therefore, the distribution posses Increasing failure rate (IFR) and Decreasing Mean Residual Life (DMRL) property.

Mean Residual Life (MRL) function is defined as
Reversed Hazard rate:

Stress_Strength Reliability
The Stress-Strength model describes the life of a component which has a random strength X that is subjected to a random stress Y. The component fails at the instant that the stress applied to it exceeds the strength, and the component will function satisfactorily whenever Y X > . So,

Maximum Likelihood Method of Estimation of the Parameters
Using the method of Maximum Likelihood, we estimate the parameter of the OGEED. Since

Simulation Study
Here we use the inversion method for generating random data from the Odds Generalized Exponential-Exponential Distribution. Algorithm: 1. Generate U from Uniform (0, 1)  Table 1 and Table 2 below. In Table  1 we take λ=0.01 for θ = 0.1, 1.0 and 2. In Table 2 we take λ=0.1 for θ = 0.1, 1.0 and 2. The resulting values relating to Odds Generalized Exponential-Exponential Distribution (OGEED) have been presented in first row and that relating to Gamma Distribution (GD), Exponentiated Exponential Distribution (EED), Weibull Distribution (WD) and Pareto Distribution (PD) in second, third, fourth and fifth row.

Observations:
(i) Table 1 and 2 shows that the bias is positive in case Odds Generalized Exponential-Exponential Distribution (OGEED). Table 1 and 2 also shows that bias and MSE decreases as n increases. (ii) In terms of bias and MSE, the parameter λ and θ of the Odds Generalized Exponential -Exponential Distribution is efficiently estimated compared to that of the other distribution.

Data Analysis
In this section, we fit the exponential exponential model to a real data set obtained from Smith and Naylor (1987). The data are the strengths of 1.5 cm glass fibres, measured at the National Physical Laboratory, England and have been shown in Table 3. Histogram shows that the data set is negatively skewed. We have fitted this data set with the Odds Generalized Exponential -Exponential distribution. We have also fitted this data set for some other probability distributions with two parameters like Gamma, Exponentiated Exponential, Weibull and Pareto. The summarized results have been presented Table 4 and it is noticed that the OGEED is the better fit for minimum Akaike Information Criterion (AIC). Histogram and fitted Odds Generalized Exponential-Exponential curve to data set have been shown in Figure 8.

Concluding Remark
In this article, we have studied a new probability distribution called Odds Generalized Exponential -Exponential Distribution. This is a particular case of T-X family of distributions proposed by Alzaatreh et al. (2013). The structural and reliability properties of this distribution have been studied and inference on parameters have also been mentioned. The proposed distribution has been compared with some standard distributions with two parameters through simulation study and the supiriority of the proposed distribution has been established. The appropriateness of fitting the odds generalized exponential -exponential distribution has also been established by analyzing a real life data set.