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

In this manuscript, we have derived a new lifetime distribution named generalized Weibull–Lindley (GWL) distribution based on the T-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.


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. e 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. ey 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. is 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. e 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. e 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].

Weibull-X Family of Distribution by Alzaatreh et al. [8]
Let r(t) be the pdf of the random variable Tϵ[a, b] for − ∞ ≤ a < b ≤ ∞ and let W(F(x)) be the function of the cdf F(x) of any random variable X such that W(F(x)) ∈ [a, b]. W(F(x)) is monotonically and nondecreasing function.
From the above, the new distributions were defined as follows: Let X be the random variable with the pdf f(x) and cdf F(x). Let T be the continuous random variable with pdf r(t) on [a, b]. e cdf of the new distribution is given as ) satisfies the conditions written above. e cdf G(x) can also be written as follows: where R(t) is the cdf of the random variable T. e corresponding pdf associated with G(x) above is given as which is the composite function of R.W. F(x). e pdf r(t) is transformed to the new distribution g(x) with the help of the link function W(F(x)) 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 of W(F(x)) will give rise to the new family of distribution. e form of W(F(x)) depends on the support of random variable T as defined above.
When the support of T is a ≤ T < ∞ and if a ≥ 0 is assumed without loss of generality, then W(F(x)) can be defined as 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 of T as − log (1 − F(x)) . By substituting into equation (1), we obtain the cdf of new family of Weibull distribution as where − log [1 − F(x)] is the cumulative hazard rate function for the random variable X, and here, R(t) is the cdf of the random variable T.
e corresponding pdf was given as follows: where h(x) is the hazard rate function and H(x) is the cumulative hazard function for the random variable X with cdf F(x). Now, if the random variable T follows Weibull distribution with parameters c and β, then the pdf of T will be given as r(t) � (c/β)(t/β) c− 1 exp(− (t/β) c ), t ≥ 0, (c, β) > 0. From (4), we have the pdf of Weibull-X family as Since for the above Weibull distribution the cdf is R(t) � 1 − exp(− (t/β) c ), then we can write the cdf of Weibull-X as We now introduce our new distribution as follows. Let the random variable X follows Lindley distribution with parameter β. According to Lindley (1958) [9], the pdf and cdf will be given as 2

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Survival and hazard functions are given as From the relation between cumulative hazard and survival functions, we have cumulative hazard function of Lindley distribution as

e 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: For x ≥ 0, (θ, β, c) > 0 which is the pdf of generalized Weibull-Lindley (GWL) distribution. Figure 1 describes various shapes of GWL pdf when different parameter sets are considered.
As displayed in Figures 1-3, 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, and θ � 3 where the curve rises and declines sharply for lower values of x's. e pdf also has a bell shape similar to that of normal distribution when β � 1, c � 3, and θ � 0.5.

2.2.
e cdf of Generalized Weibull-Lindley (GWL) Distribution. We can also obtain the cdf by substituting equation (7) into (6) to obtain where Figure 2 describes various shapes of generalized Weibull-Lindley (GWL) cumulative distribution function when different sets of parameters are employed. As seen from Figure 2, generalized Weibull-Lindley (GWL) cumulative distribution function increases to one under different rates when several sets of parameters are considered.

Survival Function for Generalized Weibull-Lindley
(GWL) Distribution. From (12) above, we have the survival function written as a compliment of cdf as  we also have constant survival at earlier time with sharp decline and then constant when c � 5, β � 10, and θ � 3, we have an exponential decline of survivals when c � 1, β � 1, and θ � 1 and c � 1, β � 0.5, and θ � 1 . ese properties may be useful in modeling real lifetime data coming from a complicated system or process.
e hazard function will be given using the pdf and survival function in equations (13) and (11): From Figure 4, 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, and θ � 2, exponentially decreasing when (c � 3, β � 15, θ � 0.5), or "bath tub" shape when c � 10, β � 1, and θ � 0.09. Similar to the survival curves, various shapes for the hazard function provide wide coverage ground in modeling a complex lifetime dataset.

e 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: which is simplified as follows:

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.

e 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: e above can be written as We know that F(y) is the cdf of Lindley distribution whose quantile is given as erefore, the quantile function of generalized Weibull-Lindley distribution can be written as which is simplified as follows: where (β, θ, c) > 0, and W − 1 is the negative branch of the Lambert W function.

e Median of Generalized Weibull-Lindley (GWL) Distribution.
We can obtain the median by solving the following equation: 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.

Skewness and
We have computed both quantities for different sets of parameters and results are displayed in Table 1 For better visualization of the above properties, respective mesh plots were obtained. We also employed a bivariate standard Gaussian distribution as a reference. Figure 5 displays the skewness and kurtosis of the GWL distribution with respect to a standard normal distribution.
From Figure 5, 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. ese properties may be suitable to model lifetime data of a complicated real-life system.

e Mode of Generalized Weibull-Lindley Distribution.
From the pdf We can obtain the mode of this distribution by solving the equation (d/dx)(g(x)) � 0.
For simplification, apply natural logarithm both sides, and we have ln(g(x)) � ln(c) + ln θ 2

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By taking derivative with respect to x both sides of the above equation and simplifying, we obtain  Journal of Mathematics erefore, we can obtain the mode of generalized Weibull-Lindley (GWL) distribution by solving the following equation:

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: as where μ T an d η T are the mean and Shannon entropy of the random variable T with pdf r(t).
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: where c and β are the parameters of Weibull distribution, and for our case, F(.) and f(.) will represent the cdf and pdf of Lindley distribution. From the previous equations, we had the pdf and cdf of Lindley distribution as e inverse of F(x) is given as where W − 1 is the lower part of the Lambert W function. erefore, where ψ � (θ, β, c).
e log-likelihood function can be written as To obtain the normal equations, we take partial derivatives with respect to ψ � (θ, β, c) which results in the following equations after simplifications: Analytical solutions of the above normal equations may be difficult to obtain through the usual computation methods.
erefore, we suggest the use of embedded computer software such as R or MATLAB to obtain numerical solutions for the parameters.

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 sizes n � 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. e following points are observed from Table 2. e 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. e simulation results also show that there is no considerable difference in the average bias for different choices of the parameters.

Application of Generalized Weibull-Lindley (GWL) in Modeling Real-Life Survival Data
We applied this new generated distribution to model two real survival datasets. e 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]. e data are as follows: 1.  [15]. e data are shown in Table 3.
By examining Table 4, 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.

Conclusion
e 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.
Data Availability e used datasets are given in the manuscript.