Odds Exponential Log Logistic Distribution: Properties and Estimation

Corresponding author: Gadde Srinivasa Rao Department of Statistics, The University of Dodoma, P.O. Box: 259, Tanzania Email: gaddesrao@yahoo.com Abstract: We propose a distribution called Odds Exponential Log Logistic Distribution (OELLD), which is an odds family of distribution. Its hazard rate is an increasing and decreasing function based on the value of the parameter. Explicit expressions for the ordinary moments, L-moments, quantile, generating functions, Bonferroni Curve, Lorenz Curve, Gini’s index and order statistics are derived. The parameters of the proposed distribution are estimated by using maximum likelihood method and also illustrated by a lifetime data set.


Introduction
In describing the real world phenomena, the distributions are very much useful. In spite of the fact that a number of distributions are developed, always there is a scope for introducing distributions, analysing their properties to use them for fitting real world scenarios. Hence, we are always developing new and not rigid distributions. In recent works, new distributions are defined by introducing one or more parameters to the distribution functions. The addition of parameters to the distribution functions makes the distribution richer and more flexible for modelling life time data. But, at the same time adding of too many parameters to the distribution make inferential aspect more complicated. Proportional Odds Model (POM), Proportional Hazard Model (PHM), Proportional reversed hazard model (PRHM), Power Transformed Model (PTM) are some of the models originated from this idea to add a shape parameter. For such models, a few pioneering works are by Box and Cox (1964;Cox, 1972;Marshall and Olkin, 1997;Gupta and Kundu, 1999;Kantam and Rao, 2002;Ashkar and Mahdi, 2006;Rosaiah et al., 2006;Gupta and Gupta, 2007;Aljarrah et al., 2014;Tahir et al., 2015;El-Damcese et al., 2015) and the references therein. Many distributions have been developed in recent years that involve the logit of the beta distribution. Using the generalized class of beta distribution, the distribution function (df) for this class of distributions for the random variable X is generated by applying the inverse of the df of X to a beta distributed random variable: where, G(x) is the df of other distribution. This class of distributions has not only generalized the beta distribution but also supplemented the parameter(s) to it. Among this class of distributions are, the beta-Normal Eugene et al. (2002); beta-Gumbel Nadarajah and Kotz (2004); beta-Weibull Famoye et al. (2005); beta-Exponential Nadarajah and Kotz (2006); beta-Laplace Kozubowski and Nadarajah (2008) and beta-Pareto Akinsete et al. (2008), beta-Rayleigh Akinsete and Lowe (2009). Number of useful statistical properties arising from these distributions and their applications to real life data has been discussed in the literature. Alzaatreh et al. (2013) has proposed a new generalized exponential family of distributions, known as T-X family and the cumulative distribution function (cdf) is defined as: where, the random variable T, X∈ [a, b], for −∞< a, b < ∞ and W(F θ (x)) be a function of the cdf F θ (x), so that W(F θ (x)) should satisfies the following conditions: ) is differentiable and monotonically nondecreasing • W(F θ (x))→a as x→ ∞ and W(F θ (x)) →b as x→∞ We defined a generalized class of any distribution having positive support. Taking , the odds function, the cdf of the proposed distribution is: The support of the resulting distribution will be that of F θ (⋅).
Here, given in Equation   2. Hence, we name this distribution as Odds Exponential Log Logistic Distribution (OELLD). The development of new distribution is presented in Section 2. A study of statistical properties along with moments, L-moments, quantile, generating functions, Bonferroni Curve, Lorenz Curve, Gini's index and order statistics and reliability of the new distribution is provided in Section 3. In Section 4, ML estimation of parameters is discussed and a real life data set has been analysed and compared the results with other fitted distributions. In Section 5, we present the concluding remarks.

The Odds Exponential Log logistic Distribution (OELLD)
The cumulative distribution function (cdf) of the new Odds Exponential Log Logistic Distribution (OELLD) is defined as: Also the probability density function (pdf) of the OELLD is: with range (0, ∞), the following Fig. 1 shows the pdfs, Fig.  2 shows the distribution function and Fig. 3 hazard rate and reversed hazard rate for different values of λ, σ and θ.

Distribution Function Limits
Since the cdf of OELLD is

Some Statistical Measures of OELLD
The mean and medians are: ( ) The r th order raw moment is: The Skewness (k 1 ) is: The Kurtosis (k 2 ) of the OELLD is: Characteristic Function (C.F.): Cumulant Generating Function (C.G.F):

Mean Deviation
The mean deviation about the mean and the mean deviation about the median are: Therefore, mean deviation about mean is: Therefore, mean deviation about median is:

Conditional Moments
In reliability theory, the residual life and the reversed residual life play an important role.
The r th order raw moment for the residual life is: The r th order raw moment for the reversed residual life is:

L-Moments
Suppose X k:n be the k th smallest moment in a sample of size n, then the L-moments of X are defined by: For OELLD with parameters λ, θ and σ, we have: Therefore, the first four L -Moments OELLD are:

Quantile Function of OELLD
The quantile function, say Q (p), defined by F(Q(p)) = p is the root of the equation: where, µ = E (x) and q = F -1 (p).
The Bonferroni and Gini indices are: 1 1 0 0 1 B(p) and 1 2 L(p) B dp G dp = − = − ∫ ∫ By using quantile function, we calculate the equations given in (13) and: Integrating Equation 15 and 16 with respect to p, we get the Bonferroni and Gini indices as:

Order Statistics
Suppose X 1 ,X 2 ,…X n is a random sample from the p.d.f of OELLD. Let X (1) ,X (2) ,…X (n) be the corresponding order statistics. The probability density function and the cumulative distribution function of the k th order statistics, say Y = X (k) are: and:

Reliability and Related Properties
The reliability function and hazard rate function of OELLD are: ( ) ( ) Reversed hazard rate: Mean Reversed Residual Life (MRRL) function or Expected Inactivity Time (EIT) is defined as:

ML Estimation of the Parameters
Using the method of ML method of Estimation (MLE), we estimate the parameters of the OELLD. The likelihood function is:

Data Analysis
In this section, we study the application of the OELLD (and their sub-models: ELLog, LeLLog and LLog distributions considered by Lemonte (2014)) for a real lifetime data set to illustrate its potentiality. The following real lifetime data set corresponds to an uncensored data set from Nichols and Padgett (2006) on breaking stress of carbon fibres (in Gba): 3. 70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.41, 3.19,3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.53,2.67, 2.93, 3.22, 3.39, 2.81, 4.20, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15,2.35, 2.55, 2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 0.98,2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 1.59, 2.00, 1.22, 1.12,1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68,2.48, 0.85, 1.61, 2.79, 4.70, 2.03 We fitted the proposed OELLD curve for the above data, which is shown in the following graphs: The unknown parameters of OELLD are estimated by the ML method. In order to compare the models considered by Lemonte (2014) with the proposed OELLD model, the Cramér-von Mises (W * ) and Anderson-Darling (A * ) statistics are used. The details of the statistics W * and A * are described in Chen and Balakrishnan (1995). In general, the smaller the values of these statistics, the better the fit to the data. Let H(x; θ) be the c.d.f., where the form of H is known but θ (a k-dimensional parameter vector) is unknown. The statistics W * and A * can be obtained as follows: The Table 1 lists the MLEs (and the corresponding standard errors in parentheses) of the parameters of all the models for the data set (breaking stress of carbon fibres). The statistics W * and A * are also listed in this Table 1 for the models. It can be seen from the table given below, the proposed OELLD has the smallest values for the statistics W * and A * than most of the other models, that is, the proposed model fits the breaking stress of carbon fibres data better than most of the other models considered. More information is provided by a visual comparison in Fig. 4 of the histogram of the data with the fitted OELLD density function. Clearly, the OELLD distribution provides a closer fit to the histogram. The Kaplan-Meier (K-M) estimate and the estimated survival function of the fitted OELLD distribution is shown in Fig. 4. OELLD has three parameters only. From this plot, note that the OELLD model fits the data adequately and hence can be adequate for this data.