An exponentiated XLindley distribution with properties, inference and applications

In this paper, we propose exponentiated XLindley (EXL) distribution. The novel model is adaptable due to the mixt shapes of its density and failure rate functions. The following key statistical properties of EXL distribution are derived: quantile function, moments, hazard function, mean residual life, and Rényi entropy. The parameters are estimated using the maximum likelihood, Anderson Darling, Cramer von Misses, maximum product spacing, ordinary and weighted least square estimation procedures. To examine the behavior of the estimate, Monte Carlo simulation is used. Further Bayesian technique is also utilized to estimate the EXL parameters. The traceplot and Geweke diagnostics are used to track the convergence of simulated processes. The applicability of the EXL distribution is demonstrated by three datasets from different domains such as mortality rate due to COVID-19, precipitation in inches, and failure time for repairable items. The proposed distribution provides efficient results as compared to renowned competitive distributions.


Introduction
Modeling of lifetime data remained a matter of attraction for statisticians to deal with probabilistic reasoning.Lifetime models play an imperative role in fields like engineering, management, biological and health sciences, etc.For the depiction and projection of realworld phenomena, numerous probability models have been devised and utilized for the mentioned purpose.There is always some space to develop new models that are more flexible or have better fitting in special cases related to real life.This goal can be achieved using various generalization approaches such as transmuted approach [1], exponentiated-G [2], Beta-G [3], Weibull-G [4], Alpha power transformed [5], odd Fréchet-G [6], truncated Burr X-G [7], and Teissier-G [8] among others.
Lindley distribution is introduced by Ref. [18] in the context of fiducial and Bayesian inference.It is also used for reliability analysis.Lindley distribution is the combination of two probability (gamma and exponential) models.The probability density function (pdf) is f (z; θ) = δ 2 (1 + δ) 2 (1 + z)e − δz , δ > 0, z > 0 One of the flexible and simplest lifetime models was recently introduced by Ref. [19].It was termed "XLindley distribution" after being derived as a finite combination of exponential and Lindley models.A thorough examination of the salient features of the derived distribution indicated that the XLindley distribution is a more effective model than Lindley and provides a greater basis for real-world applications.The pdf and cumulative distribution function (cdf) of the XLindley distribution are shown. and Some authors further introduced some extended forms of XLindley distribution such as Unit-XLindley distribution [20], Poisson XLindley distribution [21], Power XLindley distribution [22], Quasi-XLindley distribution [23], and new discrete XLindley distribution [24].
The major goal for the work was the development of a novel two-parameter flexible lifetime distribution.This work has the following key goals.
• To provide a new generalization of the XLindley distribution utilizing the exponentiated parameter induction technique.The new model is named "Exponentiated XLindley distribution".The new model has configurable density and hazard functions that can be used to model different types of datasets.• To derive and investigate some of its most important mathematical and reliability aspects.
• To estimate the model parameters using various classical and robust estimation techniques.Bayesian approach is also used.
• The proposed distribution is used to analyze three datasets from different areas.
The rest of the study is systematized as follows: Section 2 is dedicated to the derivation of the new model and its shape analysis.Section 3 derives several key statistical features.Point parameters estimation is achieved in Section 4. Section 5 assesses the selection of an effective estimating technique using a comprehensive simulation study.Three examples from different fields are given in Section 6 to prove the flexibility of the EXL distribution.Bayesian analysis is conferred in Section 7. Some concluding remarks are given in Section 8.

Derivation of new distribution and its shape analysis
A generalized form of XLindley distribution is proposed using power to cdf transformation G(X) = [F(Z)] α , where Z is the random variable that follows the XLindley distribution with parameter δ.A random variable X follows exponentiated XLindley (EXL) model, symbolically it is written as X ∼ EXL(α, δ).The cdf of EXL is as follows: where δ is the scale and α is the shape parameters of EXL distribution.
The pdf of EXL corresponding to equation ( 1) is as follows: The alternative form of pdf is given in equation ( 3) Furthermore, survival and hazard function (HF) of EXL distribution are

Limiting behavior of density function and HF
In this section limiting demonstration of density and the HF of EXL distribution are discussed.The behavior of the pdf at the lower limit (x →0) is given below The density visualizations for different parameter choices are shown in Fig. 1 (a)-(c).The limiting presentation of the HF at the lower and upper limit of variable x is given below: It follows that the parameter α causes various shapes of the pdf and HF.To demonstrate the previously mentioned prerequisites, we plot the HF for various parameter choices in Fig. 2 (a)-(c).

Statistical properties of EXL distribution
This section contains the mathematical derivation of several important characteristics such as linear presentation, mode, quantile function, moments, and order statistics, as well as entropies.

Mode of the EXL distribution
Taking the log of equation ( 4), differentiating for x, and equating it to zero, we get the following expression The exact solution of mode is not tractable, so the values of mode can be derived numerically by computing the above equation.

Quantile function
The quantile function for the EXL model is .
The proof is given in the Appendix.You can calculate the median of EXL distribution by setting u = 1 2 , that is ) )] .

Moments of EXL distribution
The moments of EXL distribution are Proof: The ordinary moments are demarcated as μ ′ r = E(X r ) = ∫ +∞ − ∞ x r g(x)dx.Now after replacing the pdf given in equation (3) yields Letting (k + 1)δx = y and x = y (k+1)δ .
The mean of the EXL distribution is given under The variance of the EXL distribution is given The skewness and kurtosis of the random variable X which follows EXL distribution can be obtained using the moment ratio 2 ) 3 2 and 2 )2 Table 1 depicts the behavior regarding mean, variance, skewness (γ 1 ), and kurtosis (γ 2 ) for Exponentiated XLindley distribution.It is evident from the table that skewness and kurtosis of EXL decreases for higher values parameters.

Rényi entropy
The Rényi entropy can be derived as Considering the integral part Using the binomial expansions, Using the following binomial expansions.i. ( The last part of equation ( 5) will be Put this in equation ( 4) x m+s e − (k+γ)δx , A.M. Alomair et al. where .
Using integration, we get the final expression )

Mean residual life
The extra lifetime that is expected for the survival of an object of interest is termed mean residual life. Considering Take an integral part and make a transformation Substituting in equation ( 6) ] and

Parameter estimation
In this section, we utilize six methods to estimate the parameters of the EXL distribution.A complete simulation analysis was also carried out to discover the most effective estimating approach.

Maximum likelihood estimation
Let x 1 , x 2 , x 3 , …, x n be a random sample of size n taken from the EXL distribution.The log-likelihood function is given by For MLE, the above equation will be maximized for both parameters ) )] and .

Maximum product spacing estimation
This estimation approach was proposed by Ref. [25] as an alternative to ML estimation.The geometric mean of the differences may be maximized to get the MPSE of parameters.
or by minimizing the log of the geometric mean of sample spacing given by

Anderson-Darling estimation
The ADE of EXL distribution can be obtained by minimizing the following statistic

Cramer von misses estimation
The CVME of EXL distribution can be obtained by minimizing the following distance and

Ordinary and weighted least square estimation
The ordinary least squares (OLSE) of the EXL distribution can be obtained by minimizing the distance between theoretical and empirical cdf.The OLSEs are obtained by minimizing the following function

S(α, δ)
The Weighted Least Square Estimators (WLSE) can be obtained by minimizing the following distance

Simulation study
In the following section, we conduct a simulation study to evaluate the efficiency of EXL estimators.A random sample is generated from the EXL distribution with some selected values of parameters.The following algorithm is used for generating samples from the new model: i.Generation of n values of u from a uniform distribution with parameters (0, 1) ii.Computation of n values of Xu (random numbers of EXLD) using the relation iii.The number of replications is taken as 10000.iv.The different parameter values are taken as (α, δ) = (0.5, 0.25), (0.5, 2.0), (1.0, 0.25), (1.0, 2.0), (2.0, 0.25), (2.0, 1.0).v.The sample sizes used for the study are taken n = 20, 50, 100, and 300.The selected sample sizes reflect small, moderate, and large samples respectively.vi.The performance of estimators was evaluated through the bias, mean relative error (MRE), and mean square errors (MSE).The Bias, MRE, and MSE were calculated from 10,000 samples of each selected sample size.
The simulation results are given in Tables 2-7.
It is evident from the above simulation tables that absolute bias, mean relative error, and MSE reduce with the upsurge in sample size for all estimation methods.For small sample size (n = 20) ADE and MPSE methods show better results regarding bias, MSE, and MSE.While for large samples MLE method performs better than others.

Application of EXL distribution
The distribution's use is proven in the manuscript using three real datasets.For comparison purposes following probability distributions are utilized such as exponential (Exp), Lindley (L), XLindley (XL), generalized Lindley (GL), Weibull, power Lindley (PL), and Nadarajah-Haghighi (NH).The MLE method is used to estimate the parameters of models.
Several measures are available in the literature that are used for the selection of fitted distribution.In this study, we will consider six imperative measures: Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), Anderson Darling (AD), Cramer Von Mises (CVM), and Kolmogorov Smirnov (KS), respectively.
The MLEs, SE, and goodness-of-fit measures for this dataset are given in Table 8.We also provide the visual comparison using fitted pdf, cdf, PP, profile log-likelihood, and contour plots given in Fig. 4 (a)-(f).
The MLEs, SE, and goodness-of-fit measures for this dataset are given in Table 9.We also provide the visual comparison using fitted pdf, cdf, PP (probability-probability), profile log-likelihood, and contour plots given in Fig. 6 (a)-(f).
Data III: The third dataset, which has been examined by Ref. [28], corresponds to the period between failures for 30 repairable goods.The data observations are; 1.43, 0.     Further, the visual representation such as the boxplot, TTT plot, and Q-Q plot for the first dataset is presented in Fig. 7 (a)-(c).The MLEs, SE, and goodness-of-fit measures for this dataset are given in Table 10.We also provide the visual comparison using fitted pdf, cdf, PP, profile log-likelihood, and contour plots given in Fig. 8 (a)-(f).
Six estimate methods have been used to achieve one of the main goals of this study, which is to get the best estimators for three data sets.The different estimators for data sets based on various estimating methods are listed in Table 11.

Bayesian analysis
The estimation of parameters using the Bayesian approach is discussed in this section.A prior distribution is a requirement for each parameter to estimate the parameter using the Bayesian approach.Thus, gamma distribution is assumed for both parameters δ and α considering these parameters are real and positive numbers.The joint prior distribution can be written as:

Conclusion
In this study, the exponentiated XLindley distribution is proposed and studied.A new shape parameter is introduced to enhance the   flexibility of the XLindley distribution.There are three subfamilies of distribution which based on the new parameter α (α = 1, α < 1, and α > 1).The proposed distribution has exponentially decreased behavior for α < 1, while for α > 1 the distribution is unimodal, positively skewed with many variations at the start, and becomes flattered as the value of the second parameter δ decreases.The mean and variance decrease with the increase in parameter α.The model parameters were estimated using six different estimation methods.
A comprehensive simulation was carried out for various combinations of parameters and different sample sizes.Estimates improve as the sample size increases.The Bayesian technique with MCMC was utilized for estimation parameters.Traceplots and Geweke diagnostics were used to monitor the convergence of simulated sequences.The application of EXL distribution is illustrated by three datasets from different fields such as mortality rate due to COVID-19, precipitation, and failure time of repairable items.The proposed EXL distribution is compared with the existing seven-lifetime models: exponential, Lindley, XLindley, generalized Lindley, Weibull, power Lindley, and Nadarajah--Haghighi.For all the datasets, the goodness of fit measures and graphical presentations are evident that EXL distribution outperformed all mentioned models by acquiring minimum values of the goodness of fit criteria.
Future research on the new three-parameter distribution may focus on a variety of different topics.Here are a few examples: • Additional examination of the proposed distribution can be explored in different dimensions.One of the most preferable directions is to propose its neutrosophic extension [29][30][31][32] to analyze datasets with indeterminacy.• Bayesian estimation can be used to estimate the model parameters using different loss functions and different approximation techniques.

Delimitations and limitations
The proposed model will be applicable in such situations where the conditions of the model will be satisfied.Unlike the existing XLindley distribution, the newly proposed model will apply to various kinds of data sets.This model will apply to lifetime data ranging from 0 to ∞.As we know that W[ze z ] = z, then the equation becomes For α > 0, δ > 0 and x > 0, (1 + δ) 2 + δx > 0 and it is also checked ) since 0 < u < 1.Thus, by using the properties of the negative branch W − 1 of the Lambert W function.Hence .

Fig. 4 .
Fig. 4. Illustration of the fitted (a) pdf, (b) cdf, (c) PP, (d)-(e) profile log-likelihood, and (f) contour plots of the EXL model for the first data.

A
.M. Alomair et al.The traceplot, and histogram of posterior density are used for the evaluation of the MCMC iterations.The posterior samples for the parameters for the first dataset are shown in Fig. 9(a)-(f), Fig. 10(a)-(f), and Fig. 11(a)-(f), respectively.

Table 1
Some computational statistics of EXL model for various parameters values.

Table 8
Maximum Likelihood Estimate and Goodness of Fit measures for the first data.

Table 11
Estimation and Goodness for all three datasets.It should be noted that all considered estimation approaches analyze failure rate, precipitation, and mortality rate data effectively.However, the OLSE method is best for failure rate and mortality rate data, while the ADE method is best for precipitation data.

Table 12
Bayesian estimates, SE, HPD, and Geweke's score for both datasets.