Digital Commons@Georgia Southern Digital Commons@Georgia Southern

: A new distribution called the log generalized Lindley-Weibull (LGLW) distribution for modeling lifetime data is proposed. This model further generalizes the Lindley distribution and allows for hazard rate functions that are monotonically decreasing, monotonically increasing and bathtub shaped. A comprehensive investigation and account of the mathematical and statistical properties including moments, moment generating function, simulation issues and entropy are presented. Estimates of model parameters via the method of maximum likelihood are given. Real data examples are presented to illustrate the usefulness and applicability of this new distribution.


Introduction
The continuous one parameter Lindley distribution was introduced by Lindley (1958). Lindley used the distribution named after him to illustrate a difference between fiducial distribution and posterior distribution. Lindley distribution with the probability density function is a two-component mixture of an exponential distribution with scale parameter θ and gamma distribution with shape parameter 2 and scale parameter θ. The mixing proportion is p = θ/(θ + 1). Sankaran (1970) derived the Poisson-Lindley distribution. In this case, Lindley distribution was chosen as the mixing distribution when the parameter of the Poisson distribution is considered random. The resulting Poisson-Lindley distribution provided a better fit to the empirical set of data considered than the negative binomial and Hermite distributions. Recently, Ghitany et al. (2008Ghitany et al. ( , 2011) studied various properties of Lindley distribution and the twoparameter weighted Lindley distribution with applications to survival data. Bakouch et al. (2012) introduced an extension of the Lindley distribution that offers more flexibility in the modeling of lifetime data. Ghitany et al. (2013) presented results on the two-parameter generalization referred to as the power Lindley distribution. See Krishna and Kumar (2011) for additional results on reliability estimation of the Lindley distribution with progressive type II censored sample.
Because of having only one parameter, the Lindley distribution does not provide enough flexibility for analyzing different types of lifetime data. To increase the flexibility for modeling purposes it will be useful to consider further generalizations of this distribution. This paper offers a five-parameter family of distributions which generalizes the Lindley distribution.
There are several ways of generalizing a continuous distributionG(x), and they include Kumaraswamy-G, beta-G, McDonald-G, and gamma-G to mention a few. Kumaraswamy (1980) distribution is given by We consider a further generalization of the generalized Lindley distribution via the T-X family of distributions proposed by Alzaatreh et al. (2013) to obtain the cumulative distribution function (cdf) of the log generalized Lindley-Weibull distribution. The generalization (Alzaatreh et al., 2013) is given by the following cdf: where 0 < W(F(x)) < ∞ , is a nondecreasing function of x , k(. ) is taken to be the generalized Lindley distribution of Zakerzadeh and Dolati (2009) and F(x)is the Weibull cdf. The corresponding pdf g, is given by where W(F(x)) = − ln(1 − F(x)).
The main objective of this article is to construct and explore the properties of the fiveparameter log generalized Lindley-Weibull (LGLW) distribution. The beauty of this model is the fact that it not only generalizes the generalized Lindley distribution but also exhibits the desirable properties of increasing, decreasing, and bathtub shaped hazard function.
The model provides a better fit to data in the sense that it leads to more accurate results and prediction, which should facilitate better public policy in a wide range of areas including but not limited to medicine and environmental health, genetics, reliability, survival analysis and time-to event data analysis.
The outline of this paper is as follows: In section 2 some generalized Lindley distributions including the new LGLW distribution are introduced. This section also includes some properties such as the behavior of the hazard function, reverse hazard function and sub-models of the log generalized Lindley-Weibull distribution. Section 3 contains the moment generating function, moments, distribution of functions of log generalized Lindley-Weibull random variables and simulation. Measures of uncertainty are given in section 4. Section 5 contains the estimation of parameters via the maximum likelihood estimation technique. Fisher information and asymptotic confidence intervals are also presented in section 5. We end with applications in section 6 and concluding remarks in section 7.

Generalizations of the Lindley Distribution
In this section, we present further generalizations of the Lindley distribution. First, we discuss some generalizations that are in the literature, or in preparation.

Exponentiated Lindley Distribution
A generalization of the Lindley distribution due to Nadarajah et al. (2011) is the two parameter Exponentiated Lindley distribution with cumulative distribution function (cdf) and probability density function (pdf) given by and for x > 0, α > 0, and θ > 0, respectively.

Beta-Generalized Lindley Distribution
A further generalization of the Lindley distribution, although not studied in this paper is the beta-generalized Lindley (BGL) distribution, . The four parameter beta-generalized Lindley (BGL) cdf is given by for x ≥ 0, α > 0, θ > 0, a > 0, b > 0 . If α = 1 , we obtained the beta-Lindley (BL) distribution. If a = b = α = 1, we obtain the Lindley distribution. See Yang and Oluyede (2014) for additional details on the Exponentiated Kumaraswamy Lindley distribution.

The Log Generalized Lindley-Weibull Distribution
In this section, we introduce a new generalization of the Lindley distribution via the Weibull model and study its mathematical and statistical properties.

Generalization-The Model
Based on a continuous baseline cdf F(x) and survival function ̅ ( ) = 1 − ( ), with pdf f(x), Zografos and Balakrishnan (2009) defined the cdf Along the same lines, Ristić and Balakrishnan (2011) proposed an alternative gammagenerator given by the cdf and pdf and Now, we consider a generalizations of the generalized Lindley distribution given by Zakerzadeh and Dolati (2009) via the Weibull distribution. The generalization is given by the following cdf (Alzaatreh et al., 2013): . It follows therefore that the five-parameter LGLW cdf is given by The corresponding pdf is given by for x > 0, and θ, α, c, γ, β > 0. The graphs of the LGLW pdf, g LGLW are given in Figure 1 for selected values of the parameters α, θ, β, γ, and c. Note that the parameters β, γ, and θ are scale parameters, and α, c are shape parameters. The graphs show that the pdf of the LGLW distribution can be right skewed or decreasing for the selected values of the model parameters.

Some LGLW Sub-models
In this subsection, we present some sub-models of the LGLW distribution for selected values of the parameters c, α, γ, βand θ.
. This is the generalized

Hazard and Reverse Hazard Functions
In this section, we present the hazard and reverse hazard functions of the LGLW distribution.

Moments and Distribution of Functions of Random Variables
This section deals with the moment generating function, moments and related functions of LGLW distribution. The mean, standard deviation, coefficients of variation, skewness and kurtosis can be readily computed. Distributions of functions of the LGLW random variables are also presented.

Moments
In this section, we obtain the moments of the LGLW distribution and its sub-models. The ℎ non-central moment for the LGLW distribution is Let u = θy, then du dy = θ and dy = du θ , so that The mean of LGLW distribution is

Moment Generating Function
Let X denote a random variable with pdf g LGLW ( ). The moment generating function (MGF) of X, M(t) = E(exp (tX)), is given by , where E(X j ) is given by equation (18).

Uncertainty Measures
The concept of entropy plays a vital role in information theory. The entropy of a random variable is defined in terms of its probability distribution and can be shown to be a good measure of randomness or uncertainty. In this section, we present Renyi entropy, generalized entropy and s-entropy for the LGLW distribution.

Generalized Entropy
Generalized entropy (GE) is widely used to measure inequality trends and differences. It is primarily used in income distributions. Kleiber and Kotz (2003) derived Theil index for GB2 distribution and Singh-Maddala model. The generalized entropy (GE) I(α * ) is defined as: The bottom-sensitive index is I(−1) , and the top-sensitive index is I(2) . The mean logarithmic deviation (MLD) index is given by: and Theil index is: (23) The generalized entropy for the sub-models can be readily obtained as well.

Renyi Entropy
An entropy of a random variable X is a measure of variation of the uncertainty. A popular entropy measure is Rényi entropy (1961). If X has the pdf f(. ), then Rényi entropy is defined by I R ( ) = Now, for any real number b > 0, and b ≠ 1, Rényi entropy is given by for α, β, γ, c > 0. By taking the limit as b ↑ 1 and using L'Hospital's rule, we obtain Shannon entropy (1948). Rényi entropy for the sub-models can be readily obtained.

s-Entropy
The s-entropy is a one parameter generalization of the Shannon entropy and is defined by Now, if s ∈ R + and s ≠ 1, The integral in equation (26)

Maximum Likelihood Estimation in the LGLW Distribution
In this section, we obtain estimates of the parameters of the LGLW distribution. Methods of maximum likelihood (ML) estimation and asymptotic confidence intervals for the model parameters are presented.
The second and mixed partial derivatives of the log-likelihood function used to obtain the observed Fisher information matrix can be readily computed. where L(̂) = L is the value of the likelihood function evaluated at the estimated parameters, n is the number of observations, and p is the number of estimated parameters are given in Tables  1, 2 and 3.
The third example consists of prices (× 10 4 dollars) of 428 new vehicles for the 2004 year (Kiplinger's Personal Finance, Dec 2003). The data are given in Table 3.
The results and plots are given in Table 4 and Figure 6. The estimated covariance matrix for the LGLW distribution is given by: Plots of the fitted densities and the histogram, observed probability vs predicted probability, and empirical survival function are given in Figure 6.

Concluding Remarks
In line with results on generalized distributions and following the contents of the T-X class of distributions (Alzaatreh et al., 2013), we derive and present the mathematical and statistical properties of a new generalized Lindley distribution called log generalized Lindley-Weibull (LGLW) distribution. This distribution contains several sub-models including Lindley distribution and the generalized Lindley distribution of Zakerzadeh and Dolati (2009). The hazard rate function of the LGLW distribution can be decreasing, decreasing or bathtub shaped. Moments and distributions of functions of random variables from the LGLW distribution are derived. Uncertainty measures including generalized entropy, Rényi and Shannon entropies are obtained. We discuss maximum likelihood estimation and hypotheses tests of the model parameters. The LGLW distribution permits testing the goodness-of-fit of Lindley and generalized Lindley distribution by taking these distributions as sub-models. Asymptotic confidence intervals for the parameters of the LGLW distribution are given. We fit the LGLW distribution and its sub-models to three real data sets to demonstrate the potential importance, practical relevance and applicability of this model in lifetime analysis and other areas.