# Generalized Weibull–Lindley (GWL) Distribution in Modeling Lifetime Data

Authors: Pius Marthin; Gadde Srinivasa Rao
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this manuscript, we have derived a new lifetime distribution named generalized Weibull–Lindley (GWL) distribution based on theT-X family of distribution specifically the generalized Weibull-X family of distribution. We derived and investigated the shapes of its probability density function (pdf), hazard rate function, and survival function. Some statistical properties such as quantile function, mode, median, order statistics, Shannon entropy, Galton skewness, and Moors kurtosis have been derived. Parameter estimation was done through maximum likelihood estimation (MLE) method. Monte Carlo simulation was conducted to check the performance of the parameter estimates. For the inference purpose, two real-life datasets were applied and generalized Weibull–Lindley (GWL) distribution appeared to be superior over its competitors including Lindley distribution, Akash distribution, new Weibull-F distribution, Weibull–Lindley (WL) distribution, and two-parameter Lindley (TPL) distribution.


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## Body

## 1. Introduction

For the past few decades, generation of new distributions has been motivated by the need of fitting complex lifetime data generated from different fields such as engineering, biological and medical sciences, and geology. Probability models are very essential in describing and predicting various real-world problems. Despite the existence of deep literature regarding the development and establishment of new statistical distributions, we are still required to develop new distributions which are more flexible and compatible with the real-world issues to enable generalization [1].An interesting feature of the new families of generalized distributions is based on their flexibility to model real-life data due to possession of many parameters as compared to most classical distributions. Over 24 new families of generalized distribution have been studied in the literature. The pioneer of this field includes many scholars such as Azzalini [2], Azzalini and Capitania [3], Marshall and Olkin [4], and Gupta et al. [5]. After establishment of the beta generator, defined by Eugene et al. [5] followed by Jones [6], paved the way to the development of many other distributions. Cordeiro and de-Castro [7] and Alexander et al. did tremendous work to establish some competing generators which widened the field [1].Alzaatreh et al. [8] have proposed a link function which uses any probability density function as a generator to generate the T-X families of generalized distributions. They also studied generalized Weibull-X as a special case. Generalized Weibull-X has been proposed by fixing T as a Weibull distribution and X is allowed to take any other form. From the T-X family of distributions, we can establish various generalized distributions by either controlling the distribution of T and varying the forms of X or vice versa.In this paper, we consider the pdf of Lindley distribution [9] as the form of X distribution. This distribution has more applications in various areas and plays a prominent role in a wide variety of scientific and technological fields such as reliability engineering, survival analysis, advanced semiconductor technologies, actuarial study, and insurance. For modeling some lifetime data, Weibull distribution may not be sufficient, and therefore, distributions with more flexibility to handle the complexity of a real-life system are of great demand. The proposed distribution could handle such situations in the fields of science and technology due to the generalizations and modifications of the Weibull distribution to provide more flexibility in modeling complicated real-life problems.The rest of the manuscript follows the following order: in the subsequent section, we will give the general form of Weibull-X family of distribution due to Alzaatreh et al. [8] and then use one-parameter Lindley distribution [9]. We will introduce the new distribution called generalized Weibull–Lindley (GWL) distribution. We will then derive and discuss the shapes of its pdf, cdf, survival, and hazard functions. In Section 3, we will study some statistical properties of GWL distribution such as the quantile function, the mode, median, order statistics, Shannon entropy, Galton skewness, and Moors kurtosis. We will then estimate the parameters using maximum likelihood method. In Section 4, we will perform a Monte Carlo simulation to check the performance of parameter estimates, and in Section 5, we will apply GWL distribution along with some other distributions to fit two real lifetime datasets. Finally, in Section 6, we provide a general conclusion to our work and some future recommendations. Model fitting will be done using the AdequacyModel package in R due to Marinho and Bourguignon [10].


## 2. Weibull-X Family of Distribution by Alzaatreh et al. [8]

Letrt be the pdf of the random variable Tϵa,bfor−∞≤a<b≤∞ and let WFx be the function of the cdf Fx of any random variable X such that WFx∈a,b.WFx is monotonically and nondecreasing function.WFx⟶a asx⟶−∞ and WFx⟶b as x⟶∞.From the above, the new distributions were defined as follows:LetX be the random variable with the pdf fx and cdf Fx. Let T be the continuous random variable with pdf rt on a,b. The cdf of the new distribution is given as Gx=∫aWFxrtdt, where WFx satisfies the conditions written above. The cdf Gx can also be written as follows:(1)Gx=RWFx,where Rt is the cdf of the random variable T.The corresponding pdf associated withGx above is given as(2)gx=ddxWFxrWFx,which is the composite function of R.W.Fx. The pdf rt is transformed to the new distribution gx with the help of the link function WFx which acts as the transformer, and thus, T-X family of distribution name has been provided [1]. When the random variable X is discrete, the resulting family of distribution will also be discrete.Different forms ofWFx will give rise to the new family of distribution. The form of WFx depends on the support of random variable T as defined above.When the support ofT is a≤T<∞ and if a≥0 is assumed without loss of generality, then WFx can be defined as −log1−Fx,Fx/1−Fx−log1−FxαandFxα/1−Fxα.We know that lifetime distributions have positive domains, and therefore, using the above link functions, we can obtain several types of lifetime distribution families.Considering Weibull distribution as one of the lifetime distributions, Alzaatreh et al. applied the support ofT as −log1−Fx. By substituting into equation (1), we obtain the cdf of new family of Weibull distribution as(3)Gx=∫a−log1−Fxrtdt=R−log1−Fx=RHx,where −log1−Fx is the cumulative hazard rate function for the random variable X, and here, Rt is the cdf of the random variable T.The corresponding pdf was given as follows:(4)gx=fx1−Fxr−log1−Fx=hxr1−Fx=hxrHx,where hx is the hazard rate function and Hxis the cumulative hazard function for the random variable X with cdf Fx.Now, if the random variableT follows Weibull distribution with parameters c and β, then the pdf of T will be given as rt=c/βt/βc−1exp−t/βc, t≥0,c,β>0. From (4), we have the pdf of Weibull-X family as(5)gx=cfx1−Fx/βc−1β1−Fxexp−−log1−Fxβc.Since for the above Weibull distribution the cdf isRt=1−exp−t/βc, then we can write the cdf of Weibull-X as(6)Gx=1−exp−−log1−Fxβc.We now introduce our new distribution as follows. Let the random variableX follows Lindley distribution with parameter β. According to Lindley (1958) [9], the pdf and cdf will be given as(7)fx=θ21+θ1+xexp−θx,forx≥0,θ>0,(8)Fx=∫0xθ21+θ1+texp−θtdt=1−θ+1+θxθ+1exp−θx.Survival and hazard functions are given as(9)Sx=θ+1+θxθ+1exp−θx,hx=fxSx=θ21+xθ+1+θx.From the relation between cumulative hazard and survival functions, we have cumulative hazard function of Lindley distribution as(10)Hx=−lnSx=−lnθ+1+θxθ+1exp−θx.


### 2.1. The pdf of Generalized Weibull–Lindley (GWL) Distribution

We can now substitute equations (7) and (8) into equation (5) to obtain our new distribution as follows:(11)gx=cβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc.Forx≥0,θ,β,c>0 which is the pdf of generalized Weibull–Lindley (GWL) distribution.Figure1 describes various shapes of GWL pdf when different parameter sets are considered.Figure 1
The pdf curves for generalized Weibull–Lindley (GWL) distribution for different parameter values.As displayed in Figures1–3, the pdf of generalized Weibull–Lindley (GWL) distribution has various shapes which can be either increasing or decreasing; it assumes the exponential shape when all parameters are greater or equal to 1 except the case when β=1,c=3,andθ=3 where the curve rises and declines sharply for lower values of x’s. The pdf also has a bell shape similar to that of normal distribution when β=1,c=3,andθ=0.5.Figure 2
The cdf curves for generalized Weibull–Lindley (GWL) distribution for different parameter values.Figure 3
Survival curves for generalized Weibull–Lindley (GWL) distribution for different parameter values.


### 2.2. The cdf of Generalized Weibull–Lindley (GWL) Distribution

We can also obtain the cdf by substituting equation (7) into (6) to obtain(12)Gx=1−exp−−log1−1−θ+1+θx/θ+1exp–θxβc,where x≥0,θ,β,c>0. Figure 2 describes various shapes of generalized Weibull–Lindley (GWL) cumulative distribution function when different sets of parameters are employed.As seen from Figure2, generalized Weibull–Lindley (GWL) cumulative distribution function increases to one under different rates when several sets of parameters are considered.


### 2.3. Survival Function for Generalized Weibull–Lindley (GWL) Distribution

From (12) above, we have the survival function written as a compliment of cdf as(13)1−Gx=Sx=exp−−log1−1−θ+1+θx/θ+1exp–θxβc.Figure3 displays survival distribution for generalized Weibull–Lindley (GWL) distribution when different sets of parameters are considered.As displayed in Figure3, we observe multiple shapes of survival functions which can either decline sharply at lower values of x’s or stay constant when c=3,β=2,andθ=3, we also have constant survival at earlier time with sharp decline and then constant when c=5,β=10,andθ=3, we have an exponential decline of survivals when c=1,β=1,andθ=1andc=1,β=0.5,andθ=1. These properties may be useful in modeling real lifetime data coming from a complicated system or process.


### 2.4. Hazard Function for Generalized Weibull–Lindley (GWL) Distribution

The hazard function will be given using the pdf and survival function in equations (13) and (11):(14)hx=gxSx=cβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1.Figure4 displays various shapes of hazard function for generalized Weibull–Lindley (GWL) distribution when different sets of parameters are considered.From Figure4, we can see that the generalized Weibull–Lindley (GWL) distribution has multiple shapes for hazard function which can be either constantly increasing when c=1,β=1,θ=1, exponentially increasing when c=3,β=5,andθ=2, exponentially decreasing when c=3,β=15,θ=0.5, or “bath tub” shape when c=10,β=1,andθ=0.09. Similar to the survival curves, various shapes for the hazard function provide wide coverage ground in modeling a complex lifetime dataset.Figure 4
Hazard curves for generalized Weibull–Lindley (GWL) distribution for different parameter values.
(a)(b)(c)(d)(e)


### 2.5. The Cumulative Hazard Function for Generalized Weibull–Lindley (GWL) Distribution

From the survival function, we can directly obtain the cumulative hazard function of the GWL like as given below:(15)Hx=−lnSx=−lnexp−−log1−1−θ+1+θx/θ+1exp–θxβc,which is simplified as follows:(16)Hx=−−log1−1−θ+1+θx/θ+1pexp−θxβc.


## 2.1. The pdf of Generalized Weibull–Lindley (GWL) Distribution

We can now substitute equations (7) and (8) into equation (5) to obtain our new distribution as follows:(11)gx=cβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc.Forx≥0,θ,β,c>0 which is the pdf of generalized Weibull–Lindley (GWL) distribution.Figure1 describes various shapes of GWL pdf when different parameter sets are considered.Figure 1
The pdf curves for generalized Weibull–Lindley (GWL) distribution for different parameter values.As displayed in Figures1–3, the pdf of generalized Weibull–Lindley (GWL) distribution has various shapes which can be either increasing or decreasing; it assumes the exponential shape when all parameters are greater or equal to 1 except the case when β=1,c=3,andθ=3 where the curve rises and declines sharply for lower values of x’s. The pdf also has a bell shape similar to that of normal distribution when β=1,c=3,andθ=0.5.Figure 2
The cdf curves for generalized Weibull–Lindley (GWL) distribution for different parameter values.Figure 3
Survival curves for generalized Weibull–Lindley (GWL) distribution for different parameter values.


## 2.2. The cdf of Generalized Weibull–Lindley (GWL) Distribution

We can also obtain the cdf by substituting equation (7) into (6) to obtain(12)Gx=1−exp−−log1−1−θ+1+θx/θ+1exp–θxβc,where x≥0,θ,β,c>0. Figure 2 describes various shapes of generalized Weibull–Lindley (GWL) cumulative distribution function when different sets of parameters are employed.As seen from Figure2, generalized Weibull–Lindley (GWL) cumulative distribution function increases to one under different rates when several sets of parameters are considered.


## 2.3. Survival Function for Generalized Weibull–Lindley (GWL) Distribution

From (12) above, we have the survival function written as a compliment of cdf as(13)1−Gx=Sx=exp−−log1−1−θ+1+θx/θ+1exp–θxβc.Figure3 displays survival distribution for generalized Weibull–Lindley (GWL) distribution when different sets of parameters are considered.As displayed in Figure3, we observe multiple shapes of survival functions which can either decline sharply at lower values of x’s or stay constant when c=3,β=2,andθ=3, we also have constant survival at earlier time with sharp decline and then constant when c=5,β=10,andθ=3, we have an exponential decline of survivals when c=1,β=1,andθ=1andc=1,β=0.5,andθ=1. These properties may be useful in modeling real lifetime data coming from a complicated system or process.


## 2.4. Hazard Function for Generalized Weibull–Lindley (GWL) Distribution

The hazard function will be given using the pdf and survival function in equations (13) and (11):(14)hx=gxSx=cβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1.Figure4 displays various shapes of hazard function for generalized Weibull–Lindley (GWL) distribution when different sets of parameters are considered.From Figure4, we can see that the generalized Weibull–Lindley (GWL) distribution has multiple shapes for hazard function which can be either constantly increasing when c=1,β=1,θ=1, exponentially increasing when c=3,β=5,andθ=2, exponentially decreasing when c=3,β=15,θ=0.5, or “bath tub” shape when c=10,β=1,andθ=0.09. Similar to the survival curves, various shapes for the hazard function provide wide coverage ground in modeling a complex lifetime dataset.Figure 4
Hazard curves for generalized Weibull–Lindley (GWL) distribution for different parameter values.
(a)(b)(c)(d)(e)


## 2.5. The Cumulative Hazard Function for Generalized Weibull–Lindley (GWL) Distribution

From the survival function, we can directly obtain the cumulative hazard function of the GWL like as given below:(15)Hx=−lnSx=−lnexp−−log1−1−θ+1+θx/θ+1exp–θxβc,which is simplified as follows:(16)Hx=−−log1−1−θ+1+θx/θ+1pexp−θxβc.


## 3. Some Statistical Properties of Generalized Weibull–Lindley (GWL) Distribution

Here, we will derive and discuss some statistical properties of GWL distribution including quantile function, Galton skewness, Moors kurtosis, mode, median, Shannon entropy, and order statistics.


### 3.1. The Quantile Function of Generalized Weibull–Lindley (GWL) Distribution

From the cdf equation of generalized Weibull–Lindley, we can obtain the quantile function through inversion like as follows:(17)Gx=1−exp−−log1−1−θ+1+θx/θ+1exp–θxβc,y=Gx=1−exp−−log1−1−θ+1+θx/θ+1exp−θxβc,y=1−exp−−log1−Fxβc.The above can be written as(18)x−1=exp−−log1−Fyβc,logx−1=−−log1−Fyβcβclog1−x=−−log1−Fyc,−βlog1−x1/c=−log1−Fy,Fy=1−expβ1−x1/c,y=F−11−exp−β−ln1−x1/c.We know thatFy is the cdf of Lindley distribution whose quantile is given as(19)1−1θ−1θW−11+θp−1exp−1−θ.Therefore, the quantile function of generalized Weibull–Lindley distribution can be written as(20)Qp=1−1θ−1θW−11+θ1−1−exp−β−ln1−x1/cexp−1−θ,which is simplified as follows:(21)Qp=−1−1θ−1θW−11+θ−exp−β−ln1−x1/cexp−1−θ,where β,θ,c>0, and W−1 is the negative branch of the Lambert W function.


### 3.2. The Median of Generalized Weibull–Lindley (GWL) Distribution

We can obtain the median by solving the following equation:(22)Qp=−1−1θ−1θW−11+θ−exp−β−ln1−x1/cexp−1−θ=0.5.For specified values of the parameter, we can obtain the median of this distribution although the computations will be tedious due to the presence of complicated Lambert function.


### 3.3. Skewness and Kurtosis

Like in many generalized families of distributions, moments of generalized Weibull–Lindley (GWL) is quite complicated to obtain since it involves the expansion of the pdf. In this case, we suggest the use of Moors kurtosis and Galton skewness which use quantile functions [11, 12]:(23)Galton skewnessS=Q6/8−2Q4/8+Q2/8Q6/8+Q2/8,Moors kurtosisK=Q7/8−2Q5/8+Q3/8−Q1/8Q6/8+Q2/8.We have computed both quantities for different sets of parameters and results are displayed in Table1. From Table 1, we can see that the generalized Weibull–Lindley (GWL) is a positively skewed distribution. Further, GWL has a normal approximation with lower values of parameter combinations (beta and c) while highly skewed to the right with higher values of parameter combinations. On the other hand, we observe high values of Moors kurtosis with the decrease in parameter values.Table 1
Moors kurtosis and Galton skewness for GWL distribution at different parameter combinations.
βcθGalton skewnessMoors kurtosis5517.00−0.04804413.33−0.03203314.22−0.00622212.85−0.04401111.650.18705527.00−0.04854423.33−0.03233324.22−0.00622222.850.04411121.650.18705537.00−0.04844433.33−0.03233334.22−0.00622232.850.04411131.650.18705547.00−0.04844443.33−0.03233344.22−0.00612242.850.04411141.650.18705557.00−0.04844453.33−0.03203354.22−0.00622252.850.04421151.650.1870For better visualization of the above properties, respective mesh plots were obtained. We also employed a bivariate standard Gaussian distribution as a reference. Figure5 displays the skewness and kurtosis of the GWL distribution with respect to a standard normal distribution.Figure 5
Galton skewness and Moors kurtosis of GWL for different sets of parameter combination.
(a)(b)(c)(d)From Figure5, we see that as compared to standard Gaussian distribution, GWL distribution is skewed to the right with larger values of parameter combinations. We also observe spikes due to various shapes of its pdf when different parameter combinations are used. These properties may be suitable to model lifetime data of a complicated real-life system.


### 3.4. The Mode of Generalized Weibull–Lindley Distribution

From the pdf(24)gx=cβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc.We can obtain the mode of this distribution by solving the equationd/dxgx=0.For simplification, apply natural logarithm both sides, and we have(25)lngx=lnc+lnθ2+ln1+x−lnβ−lnθ+1+θx+ln−log1−1−θ+1+θx/θ+1exp−θx/βc−1−ln1−1−θ+1+θxθ+1exp−θx+−log1−1−θ+1+θx/θ+1exp−θxβc.By taking derivative with respect tox both sides of the above equation and simplifying, we obtain(26)ddxgx=11+x−θθ+1+x−c+1logθ+1+θx/θ+1exp−θx−θθ+1+θx+θ+c−1x−θ1+θ+θx−θcβ−1βc−θ+1+θxθ+1exp−θxc1−1cgx.Therefore, we can obtain the mode of generalized Weibull–Lindley (GWL) distribution by solving the following equation:(27)11+x−θθ+1+θx−c+1−logθ+1+θx/θ+1exp−θx−θθ+1+θx+θ+c−1x−θ1+θ+θx−θcβ−1βc−θ+1+θx/θ+1exp−θxc1−1cgx=0.


### 3.5. Shannon Entropy of Generalized Weibull–Lindley (GWL) Distribution

As a measure of variation of uncertainty, Alzaatreh et al. [8] have given the general formula of obtaining Shannon entropy of a random variable X which follows the Weibull-X family of distribution with pdf:(28)gx=fx1−Fxr−log1−Fx,as(29)ηX=−ElogfF−11−e−T−μT+ηT,where μTandηT are the mean and Shannon entropy of the random variable T with pdf rt.For the case of generalized Weibull-X, we obtain Shannon entropy by substituting mean and Shannon entropy of the Weibull distribution in the above equation:(30)ηX=−ElogfF−11−e−T−βΓ1+1c+γ1−1c−logcβ+1,where candβ are the parameters of Weibull distribution, and for our case, F.andf. will represent the cdf and pdf of Lindley distribution.From the previous equations, we had thepdf and cdf of Lindley distribution as(31)fx=θ21+θ1+xexp−θx,Fx=1−θ+1+θxθ+1exp−θxforx≥0,θ>0.The inverse ofFx is given as(32)F−1x=−1−1θ−1θW−11+θp−1exp−1−θ.where W−1 is the lower part of the Lambert W function.Therefore,(33)F−11−e−T=−1−1θ−1θW−11+θ−e−Texp−1−θ,fF−11−e−T=θ21+θ1+−1−1θ−1θW−11+θ−e−Texp−1−θexp−θ−1−1θ−1θW−11+θ−e−Texp−1−θ.This gives(34)logfF−11−e−T=2logθ−log1+θ+log1+−1−1θ−1θW−11+θ−e−Texp−1−θ+−θ−1−1θ−1θW−11+θ−e−Texp−1−θ.From the above equation, we can obtain the expected value oflogfF−11−e−T as(35)E2logθ−log1+θ+log1+−1−1θ−1θW−11+θ−e−Texp−1−θ+−θ−1−1θ−1θW−11+θ−e−Texp−1−θ.And hence, we have the Shannon entropy of generalized Weibull–Lindley (GWL) distribution given as(36)ηX=−E2logθ−log1+θ+log1+−1−1θ−1θW−11+θ−e−Texp−1−θ+−θ−1−1θ−1θW−11+θ−e−Texp−1−θ−βΓ1+1c+γ1−1c−logcβ+1forβ,θ,c>0.


### 3.6. Order Statistic for Generalized Weibull–Lindley (GWL) Distribution

Order statistic of the random variable has multiple applications such as modeling extreme datasets, estimations of various statistical quantities, and many more. We can define a distribution of an order statistic for the generalized Weibull–Lindley distribution like as follows:Let us assume we haveX1,X2,X3,…,Xn be iid generalized Weibull–Lindley (GWL) random variables with pdf given as(37)gx=cβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc,and cdf given as(38)Gx=1−exp−−log1−1−θ+1+θx/θ+1exp−θxβc.LetX1,X2,X3,…,Xn be n-order statistics for the above random variables. The rth order statistic has the pdf given as follows:(39)fxr=n!r−1!n−r!∑i=1n−r−1in−rigxGxi+r−1,where Gx and gx are the cdf and pdf of any continuous distribution in our case being generalized Weibull–Lindley distribution.We can now write the pdf ofrth order statistic for generalized Weibull–Lindley as(40)fxr=n!r−1!n−r!∑i=1n−r−1in−ricβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc1−exp−−log1−1−θ+1+θx/θ+1exp–θxβci+r−1.Now, the smallest1st order statistic has the pdf given as(41)fx1=n∑i=1n−1−1in−1icβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβcexp1−exp−−log1−1−θ+1+θx/θ+1exp−θxβcexp−−log1−1−θ+1+θx/θ+1exp−θxβci.We also have the pdf of the largestnth order statistic written as(42)fxn=ncβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc1−exp−−log1−1−θ+1+θx/θ+1exp−θxβcn−1.Therefore, the extremes pdf for generalized Weibull–Lindley are given in the above equations.


### 3.7. Parameter Estimation through Maximum Likelihood Method

To employ the maximum likelihood, we need to establish the likelihood function of the sample observations. This is the probability of observing a particular set of data given the chosen probability distribution model for the data [13].Let us assume we haveX1,X2,X3,…,Xn as random sample of size ″n″ drawn from the generalized Weibull–Lindley (GWL) distribution with pdf given as(43)gx=cβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc.The likelihood function can be written as(44)Lx,ψ=∏i=1ncβθ21+xi1+θ+θxi−logexp–θxi1+θ+θxi/θ+1βc−1exp−−logθ+1+θxiexp−θxi/1+θβc,where ψ=θ,β,c.The log-likelihood function can be written as(45)lx,ψ=2nlnθ+nlnc−nln1+θ+∑i=1nln1+xi+exp−θ∑i=1nxi+c−1ln−log1−1−θ+1+θ∑i=1nxi/θ+1exp−θ∑i=1nxi−nc−1lnβ−nlnβ−ln∑i=1n1−1−θ+1+θxi/θ+1exp−θxi+−log1−1−θ+1+θ∑i=1nxi/θ+1exp−θ∑i=1nxiβc.To obtain the normal equations, we take partial derivatives with respect toψ=θ,β,c which results in the following equations after simplifications:(46)dldθ=0⟹2nθ+∑i=1nθ−xi1+θ+xi+c−1∑i=1nxiθ+1+θxi−1θ+1−xi−1βc∑i=1ncxiθ+1+θxi−cxi−cθ+1=0,dldc=0⟹nc−nlnβ+ln−∑i=1nlogθ+1+θxi/θ+1exp−θxiβ−ln−∑i=1nlogθ+1+θxi/θ+1exp−θxiβ=0,dldβ=0⟹−ncβ−c−1β−2ln−∑i=1nlogθ+1+θxi/θ+1exp−θxi+cβc+1−∑i=1nlogθ+1+θxi/θ+1exp−θxic.=0.Analytical solutions of the above normal equations may be difficult to obtain through the usual computation methods. Therefore, we suggest the use of embedded computer software such as R or MATLAB to obtain numerical solutions for the parameters.


## 3.1. The Quantile Function of Generalized Weibull–Lindley (GWL) Distribution

From the cdf equation of generalized Weibull–Lindley, we can obtain the quantile function through inversion like as follows:(17)Gx=1−exp−−log1−1−θ+1+θx/θ+1exp–θxβc,y=Gx=1−exp−−log1−1−θ+1+θx/θ+1exp−θxβc,y=1−exp−−log1−Fxβc.The above can be written as(18)x−1=exp−−log1−Fyβc,logx−1=−−log1−Fyβcβclog1−x=−−log1−Fyc,−βlog1−x1/c=−log1−Fy,Fy=1−expβ1−x1/c,y=F−11−exp−β−ln1−x1/c.We know thatFy is the cdf of Lindley distribution whose quantile is given as(19)1−1θ−1θW−11+θp−1exp−1−θ.Therefore, the quantile function of generalized Weibull–Lindley distribution can be written as(20)Qp=1−1θ−1θW−11+θ1−1−exp−β−ln1−x1/cexp−1−θ,which is simplified as follows:(21)Qp=−1−1θ−1θW−11+θ−exp−β−ln1−x1/cexp−1−θ,where β,θ,c>0, and W−1 is the negative branch of the Lambert W function.


## 3.2. The Median of Generalized Weibull–Lindley (GWL) Distribution

We can obtain the median by solving the following equation:(22)Qp=−1−1θ−1θW−11+θ−exp−β−ln1−x1/cexp−1−θ=0.5.For specified values of the parameter, we can obtain the median of this distribution although the computations will be tedious due to the presence of complicated Lambert function.


## 3.3. Skewness and Kurtosis

Like in many generalized families of distributions, moments of generalized Weibull–Lindley (GWL) is quite complicated to obtain since it involves the expansion of the pdf. In this case, we suggest the use of Moors kurtosis and Galton skewness which use quantile functions [11, 12]:(23)Galton skewnessS=Q6/8−2Q4/8+Q2/8Q6/8+Q2/8,Moors kurtosisK=Q7/8−2Q5/8+Q3/8−Q1/8Q6/8+Q2/8.We have computed both quantities for different sets of parameters and results are displayed in Table1. From Table 1, we can see that the generalized Weibull–Lindley (GWL) is a positively skewed distribution. Further, GWL has a normal approximation with lower values of parameter combinations (beta and c) while highly skewed to the right with higher values of parameter combinations. On the other hand, we observe high values of Moors kurtosis with the decrease in parameter values.Table 1
Moors kurtosis and Galton skewness for GWL distribution at different parameter combinations.
βcθGalton skewnessMoors kurtosis5517.00−0.04804413.33−0.03203314.22−0.00622212.85−0.04401111.650.18705527.00−0.04854423.33−0.03233324.22−0.00622222.850.04411121.650.18705537.00−0.04844433.33−0.03233334.22−0.00622232.850.04411131.650.18705547.00−0.04844443.33−0.03233344.22−0.00612242.850.04411141.650.18705557.00−0.04844453.33−0.03203354.22−0.00622252.850.04421151.650.1870For better visualization of the above properties, respective mesh plots were obtained. We also employed a bivariate standard Gaussian distribution as a reference. Figure5 displays the skewness and kurtosis of the GWL distribution with respect to a standard normal distribution.Figure 5
Galton skewness and Moors kurtosis of GWL for different sets of parameter combination.
(a)(b)(c)(d)From Figure5, we see that as compared to standard Gaussian distribution, GWL distribution is skewed to the right with larger values of parameter combinations. We also observe spikes due to various shapes of its pdf when different parameter combinations are used. These properties may be suitable to model lifetime data of a complicated real-life system.


## 3.4. The Mode of Generalized Weibull–Lindley Distribution

From the pdf(24)gx=cβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc.We can obtain the mode of this distribution by solving the equationd/dxgx=0.For simplification, apply natural logarithm both sides, and we have(25)lngx=lnc+lnθ2+ln1+x−lnβ−lnθ+1+θx+ln−log1−1−θ+1+θx/θ+1exp−θx/βc−1−ln1−1−θ+1+θxθ+1exp−θx+−log1−1−θ+1+θx/θ+1exp−θxβc.By taking derivative with respect tox both sides of the above equation and simplifying, we obtain(26)ddxgx=11+x−θθ+1+x−c+1logθ+1+θx/θ+1exp−θx−θθ+1+θx+θ+c−1x−θ1+θ+θx−θcβ−1βc−θ+1+θxθ+1exp−θxc1−1cgx.Therefore, we can obtain the mode of generalized Weibull–Lindley (GWL) distribution by solving the following equation:(27)11+x−θθ+1+θx−c+1−logθ+1+θx/θ+1exp−θx−θθ+1+θx+θ+c−1x−θ1+θ+θx−θcβ−1βc−θ+1+θx/θ+1exp−θxc1−1cgx=0.


## 3.5. Shannon Entropy of Generalized Weibull–Lindley (GWL) Distribution

As a measure of variation of uncertainty, Alzaatreh et al. [8] have given the general formula of obtaining Shannon entropy of a random variable X which follows the Weibull-X family of distribution with pdf:(28)gx=fx1−Fxr−log1−Fx,as(29)ηX=−ElogfF−11−e−T−μT+ηT,where μTandηT are the mean and Shannon entropy of the random variable T with pdf rt.For the case of generalized Weibull-X, we obtain Shannon entropy by substituting mean and Shannon entropy of the Weibull distribution in the above equation:(30)ηX=−ElogfF−11−e−T−βΓ1+1c+γ1−1c−logcβ+1,where candβ are the parameters of Weibull distribution, and for our case, F.andf. will represent the cdf and pdf of Lindley distribution.From the previous equations, we had thepdf and cdf of Lindley distribution as(31)fx=θ21+θ1+xexp−θx,Fx=1−θ+1+θxθ+1exp−θxforx≥0,θ>0.The inverse ofFx is given as(32)F−1x=−1−1θ−1θW−11+θp−1exp−1−θ.where W−1 is the lower part of the Lambert W function.Therefore,(33)F−11−e−T=−1−1θ−1θW−11+θ−e−Texp−1−θ,fF−11−e−T=θ21+θ1+−1−1θ−1θW−11+θ−e−Texp−1−θexp−θ−1−1θ−1θW−11+θ−e−Texp−1−θ.This gives(34)logfF−11−e−T=2logθ−log1+θ+log1+−1−1θ−1θW−11+θ−e−Texp−1−θ+−θ−1−1θ−1θW−11+θ−e−Texp−1−θ.From the above equation, we can obtain the expected value oflogfF−11−e−T as(35)E2logθ−log1+θ+log1+−1−1θ−1θW−11+θ−e−Texp−1−θ+−θ−1−1θ−1θW−11+θ−e−Texp−1−θ.And hence, we have the Shannon entropy of generalized Weibull–Lindley (GWL) distribution given as(36)ηX=−E2logθ−log1+θ+log1+−1−1θ−1θW−11+θ−e−Texp−1−θ+−θ−1−1θ−1θW−11+θ−e−Texp−1−θ−βΓ1+1c+γ1−1c−logcβ+1forβ,θ,c>0.


## 3.6. Order Statistic for Generalized Weibull–Lindley (GWL) Distribution

Order statistic of the random variable has multiple applications such as modeling extreme datasets, estimations of various statistical quantities, and many more. We can define a distribution of an order statistic for the generalized Weibull–Lindley distribution like as follows:Let us assume we haveX1,X2,X3,…,Xn be iid generalized Weibull–Lindley (GWL) random variables with pdf given as(37)gx=cβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc,and cdf given as(38)Gx=1−exp−−log1−1−θ+1+θx/θ+1exp−θxβc.LetX1,X2,X3,…,Xn be n-order statistics for the above random variables. The rth order statistic has the pdf given as follows:(39)fxr=n!r−1!n−r!∑i=1n−r−1in−rigxGxi+r−1,where Gx and gx are the cdf and pdf of any continuous distribution in our case being generalized Weibull–Lindley distribution.We can now write the pdf ofrth order statistic for generalized Weibull–Lindley as(40)fxr=n!r−1!n−r!∑i=1n−r−1in−ricβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc1−exp−−log1−1−θ+1+θx/θ+1exp–θxβci+r−1.Now, the smallest1st order statistic has the pdf given as(41)fx1=n∑i=1n−1−1in−1icβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβcexp1−exp−−log1−1−θ+1+θx/θ+1exp−θxβcexp−−log1−1−θ+1+θx/θ+1exp−θxβci.We also have the pdf of the largestnth order statistic written as(42)fxn=ncβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc1−exp−−log1−1−θ+1+θx/θ+1exp−θxβcn−1.Therefore, the extremes pdf for generalized Weibull–Lindley are given in the above equations.


## 3.7. Parameter Estimation through Maximum Likelihood Method

To employ the maximum likelihood, we need to establish the likelihood function of the sample observations. This is the probability of observing a particular set of data given the chosen probability distribution model for the data [13].Let us assume we haveX1,X2,X3,…,Xn as random sample of size ″n″ drawn from the generalized Weibull–Lindley (GWL) distribution with pdf given as(43)gx=cβθ21+x1+θ+θx−logexp−θx1+θ+θx/θ+1βc−1exp−−logθ+1+θxexp−θx/1+θβc.The likelihood function can be written as(44)Lx,ψ=∏i=1ncβθ21+xi1+θ+θxi−logexp–θxi1+θ+θxi/θ+1βc−1exp−−logθ+1+θxiexp−θxi/1+θβc,where ψ=θ,β,c.The log-likelihood function can be written as(45)lx,ψ=2nlnθ+nlnc−nln1+θ+∑i=1nln1+xi+exp−θ∑i=1nxi+c−1ln−log1−1−θ+1+θ∑i=1nxi/θ+1exp−θ∑i=1nxi−nc−1lnβ−nlnβ−ln∑i=1n1−1−θ+1+θxi/θ+1exp−θxi+−log1−1−θ+1+θ∑i=1nxi/θ+1exp−θ∑i=1nxiβc.To obtain the normal equations, we take partial derivatives with respect toψ=θ,β,c which results in the following equations after simplifications:(46)dldθ=0⟹2nθ+∑i=1nθ−xi1+θ+xi+c−1∑i=1nxiθ+1+θxi−1θ+1−xi−1βc∑i=1ncxiθ+1+θxi−cxi−cθ+1=0,dldc=0⟹nc−nlnβ+ln−∑i=1nlogθ+1+θxi/θ+1exp−θxiβ−ln−∑i=1nlogθ+1+θxi/θ+1exp−θxiβ=0,dldβ=0⟹−ncβ−c−1β−2ln−∑i=1nlogθ+1+θxi/θ+1exp−θxi+cβc+1−∑i=1nlogθ+1+θxi/θ+1exp−θxic.=0.Analytical solutions of the above normal equations may be difficult to obtain through the usual computation methods. Therefore, we suggest the use of embedded computer software such as R or MATLAB to obtain numerical solutions for the parameters.


## 4. Simulation Study

In this section, we demonstrate some results based on Monte Carlo simulations to compare the performance of the parameter estimates when the MLE method is applied. In our simulation study, we employed samples of sizesn = 10, 20, 30, 40, and 50, parameter combinations as β = 1, θ  = 0.5, 1.0, and 3.0, and c = 1.0 and 3.0 with 5000 iterations. Using the GWL distribution with the above sets of parameters, we generated random samples repeatedly 5000 times. Parameter estimates were computed as well as their biases and mean squared errors (MSEs). Finally, the average bias and average mean square error (MSE) of the parameter estimates were computed over the 5000 replications. Without loss of generality, β is fixed as a unit to avoid convergence issues. Simulation outputs are displayed in Table 2. The following points are observed from Table 2. The average bias and average MSE decrease as the sample size increases for both parameters estimated in this study as expected. It can be further observed that both estimates are positively biased. The simulation results also show that there is no considerable difference in the average bias for different choices of the parameters.Table 2
Average biases and MSEs of the ML estimates of the parameters of GWL distribution.
True valuesBiasMSEncθc^θ^c^θ^1010.50.16410.03130.14470.01972010.50.07320.01710.04500.00833010.50.04810.01150.02640.00544010.50.03590.00810.01820.00385010.50.02890.00560.01420.003010010.50.01530.00300.00650.00141011.00.16330.06920.14480.09012011.00.07280.03730.04530.03733011.00.04800.02490.02660.02384011.00.03590.01770.01830.01705011.00.02900.01240.01430.013210011.00.01540.00640.00650.00631013.00.16400.25730.14651.10132013.00.07290.13580.04620.43703013.00.04810.09040.02720.27414013.00.03600.06420.01870.19425013.00.02910.04560.01460.150510013.00.01540.02370.00660.07161030.50.49560.00681.32360.00182030.50.22050.00400.41540.00083030.50.14490.00270.24420.00054030.50.10820.00190.16790.00045030.50.08730.00120.13140.000310030.50.04600.00070.05980.00021031.00.49480.01451.32540.00782031.00.22010.00850.41690.00363031.00.14460.00580.24530.00244031.00.10800.00400.16860.00175031.00.08720.00260.13200.001410031.00.04590.00140.06010.00071033.00.49420.05151.33060.08922033.00.21970.02980.42010.04113033.00.14430.02020.24730.02724033.00.10780.01420.17000.01985033.00.08710.00940.13320.015610033.00.04590.00510.06060.0077


## 5. Application of Generalized Weibull–Lindley (GWL) in Modeling Real-Life Survival Data

We applied this new generated distribution to model two real survival datasets. The first dataset is related to relief times (in minutes) of 20 patients receiving an analgesic as reported by Gross and Clark (1975, p. 105) [14]. The data are as follows: 1.1, 1.4, 1.3, 1.7, 1.9, 1.8, 1.6, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.2, 1.4, 3.0, 1.7, 2.3, 1.6, 2.0.The second dataset represents the remission times (in months) of a random sample of 128 cancer patients obtained from Lee and Wang (2003, p. 231) [15]. The data are shown in Table 3.Table 3
Relief times in months of cancer patients from Lee and Wang [15].
4.50, 32.15, 19.13, 4.87, 14.24, 5.71, 7.87, 7.59, 5.49, 3.02, 2.02, 4.51, 9.22, 1.05, 3.82, 9.47, 26.31, 79.05, 2.02, 2.62, 4.26,0.90,11.25,21.73, 10.34, 10.66,0.51, 12.03,3.36, 2.64, 43.01, 14.76, 0.81, 1.19, 3.36, 8.66 1.46, 14.83, 5.62, 18.10, 17.14, 25.74, 15.96, 17.36, 1.35, 4.33, 9.02, 22.69, 6.94, 2.46, 7.26, 3.48, 4.23, 3.70, 6.54, 3.64, 8.65, 3.57, 5.41, 11.64, 2.09, 2.23, 6.25, 7.93, 4.34, 25.82, 12.02, 3.88, 13.80, 5.85, 7.09, 20.28, 5.32, 46.12,5.17, 2.8, 0.20, 8.37, 36.66, 14.77, 10.06, 8.53, 4.98, 11.98, 5.06, 1.76, 16.62, 4.40, 12.07, 34.26, 6.97, 2.07, 0.08, 17.12, 1.40, 12.63, 2.75, 7.66, 7.32, 4.18, 1.26, 13.29, 6.76, 23.63, 3.25, 7.62, 7.63, 3.52, 2.87, 9.74, 3.31, 0.40, 2.26, 5.41, 2.69, 2.54, 11.79, 2.69, 5.34, 8.26, 6.93, 0.50, 10.75, 5.32,13.11, 5.09, 7.39.For comparison purposes, four additional models were also fitted including Weibull–Lindley (WL) [16], new Weibull-F [17], two-parameter Lindley (TPL) [18], Akash distribution [19], and Lindley distribution [9]. Comparisons were made using AIC, CAIC, BIC, HQIC, and minus log-likelihood values with the criteria of selecting the model with smaller values. Results are displayed in Tables 4 and 5.Table 4
Analysis results of remission dataset by Gross and Clark [14].
DistributionParameters−2logLAICBICCAICHQICGWLβ=0.869, c=1.973, θ=0.69621.38248.76451.75150.26449.347Akashδ = 1.15759.50061.70062.50061.70060.645Lindleyθ=0.81430.25062.49963.49562.72262.694TPLθ=1.110, α=0.00426.25856.51858.50957.22456.907WLα=1.887, λ=0.964, β=0.02430.83867.67770.66469.17768.260Table 5
Analysis results of remission dataset by Lee and Wang [15].
DistributionParameters−2logLAICBICCAICHQICGWLβ=1.489,c=0.838,θ=0.278412.655831.309839.865831.503834.786NW-Fα = 5.355, β = 0.222, Θ = 0.600409.387831.520840.080831.720835.000Lindleyθ=0.201448.225841.201844.0531841.233842.360TPLθ=0.203, α=1.466418.993841.986847.690842.082844.304WLα=0.407, λ=0.143, β=1.470521.82211049.6441058.21049.8381053.121By examining Table4, the GWL distribution provides the best fit for survival time of 20 patients receiving an analgesic as reported by Gross and Clark based on the values of log-likelihood, AIC, BIC, CAIC, and HQIC. Similarly, results displayed in Table 5 provide information about the superiority of GWL distribution in modeling remission time of cancer patients after receiving treatments based on the values of log-likelihood, AIC, BIC, CAIC, and HQIC which both are in favor to generalized Weibull–Lindley (GWL) distribution.


## 6. Conclusion

The new lifetime distribution named as generalized Weibull–Lindley distribution is developed in this article. Some statistical properties of the developed distribution are discussed. A simulation study is also given to learn the performance of the shape parameters. Two real datasets are employed to fit the proposed distribution, and the results demonstrate high performance for the newly established generalized Weibull–Lindley (GWL) distribution. Future research may be considered as the other versions of Weibull distribution or Lindley distribution or both to generate other forms of GWL distributions and several more lifetime datasets with complicated structure employed to evaluate their performance.



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