Alpha Power Transformed Extended power Lindley Distribution

The Lindley distribution has been generalized by several researchers in recent years. In this paper, we introduce and study a new generalization of extended power Lindley distribution named alpha power transformed extended power Lindley (APTEPL) distribution that provides better fits than the extended Power Lindley distribution and existing generalizations. It includes the alpha power transformed power Lindley, alpha power transformed extended Lindley, alpha power transformed Lindley, extended power Lindley, power Lindley, extended Lindley and Lindley distribution as a special cases. In this article various properties of the APTEPL distribution such as moments, moment generating function, Characteristic function and cumulant generating function, quantiles and Order Statistics are derived. Method of maximum likelihood estimation is used to obtain the model parameters. A simulation study is performed to examine the performance of the maximum likelihood estimators of the parameters. Two data sets have been utilized to show how the APTEPL distribution works in practice.


Introduction
The power Lindley (PL) distribution and two-parameter Lindley (TPL) distribution were introduced by [8] and [17] respectively, these distributions are an extension to a known Lindley distribution which was proposed by [14]. After two years [2] introduced extended power Lindley distribution whose CDF and PDF are, respectively, given by Several methods of generating new statistical distributions were presented in the literature review such as, [16,4], and [3] for more information can you see [12] and [11]. Another important method for generation a new distribution was proposed by [15] named alpha power transformed (APT). Its CDF and PDF are given as: x −1 (1 + x )e − x , x > 0; , , > 0 and where F(x) and f(x) be the CDF and PDF of a random variable x and x > 0. Several authors utilized APT method to re-extend Lindley distributions in particular cases; for example, [5] introduced the alpha power transformed Lindley (APTL) distribution. Also, [6] presented alpha power transformed inverse Lindley (APTIL) distribution. Whereas alpha power transformed power Lindley (APTPL) distribution was introduced by [9]. Moreover, [7] proposed alpha power transformed power inverse Lindley (APTPIL) distribution.
This study aims to introduce a new lifetime distribution, referred to alpha power transformed extended power Lindley (APTEPL) distribution using the alpha power transformed method to the extended power Lindley (EPL) distribution. This model will be more flexibility in analysing the lifetime data. We are motivated to introduce the (APTEPL) distribution because: (1) It includes a number of well-known lifetime sub-models. (2) It can simulate monotonically increasing, decreasing, constant, bathtub, upside-down bathtub, and increasing -decreasing -increasing hazard rates. and (3) It can be viewed as a suitable model for fitting skewed data that may not be properly fitted by other common distributions, and it can also be used to solve a variety of problems in various fields, such as public health, biomedical studies, and industrial reliability and survival analysis.
The paper is organized as follows: Sect. 1 is introduction. In Sect. 2, the alpha power transformed extended power Lindley (APTEPL) distribution is defined. Some properties of APTEPL distribution are derived in Sect. 3 which includes: r th moment, moment generating function, Characteristic function , cumulant generating function, ordered statistics and quantiles. The estimation of the unknown parameters by using maximum likelihood estimator are studied in Sect. 4. In Sect. 5, a simulation study is conducted to evaluate the performance of the different estimators. Finally, in Sect. 6 comparison the performance of proposed distribution with other distributions is verified using two real data sets, the first data set represent the waiting time (in minutes) of 100 bank customers and second data set consists of 128 bladder cancer patients.

Alpha Power Transformed Extended Power Lindley Distribution
Let X ∈ R + a random variable from extended power Lindley (EPL) distribution [2] with the scale parameter > 0 and shape parameters , > 0 . By substituting the equations (2) and (1) into (3) and (4), then the CDF and PDF for APTEPL distribution given in the following expression: The corresponding survival function S(x) and hazard rate function h(x) respectively, as follow: and A random variable X that follows alpha power transformed extended power Lindley distribution in (6) was denoted by X ∼ APTEPL(x, , , , ). The PDF and HRF plots of X ∼ APTEPL(x, , , , ) are presented in Figs. 1 and 2 respectively, Also, the reversed hazard rate function r(x) and the cumulative hazard rate function H(x) of the APTEPL distribution are, respectively, given as follows: Journal of Statistical Theory and Applications (2023) 22:1-18

Special Cases
Some well-known distributions are special cases of the APTEPL distribution. We present these cases for selected values of parameters.
Note: Some spacial cases for the hazard rate function arising from the APTEPL distribution by assigning relevant values of the parameters.

Some Mathematical Properties of APTEPL Distribution
This section includes some properties of the APTEPL distribution like, r th moment, moment generating function, characteristic function, cumulant generating function, and order statistics. and obtain the mean, standard deviation, skewness, kurtosis, and coefficients of variation.

r th Moment
The r th moment of APTEPL distribution is Using power series , Therefore, (11) can be written as follows: Also, by using series of Taylor, Hence (12) can be expressed as follows: After some steps we found that where : also, The variance, skewness, kurtosis and coefficient of variation can be obtained

Moment Generating Function, Characteristic Function and Cumulant Generating Function
• The moment generating function is given by In case of using the equations (13) and (14) we found that moment generating function of APTEPL distribution can be expressed as follows: • The Characteristic function is given by similarly, after some steps the Characteristic function can be expressed: The Cumulant generating function is given by: In case of using the equations (16) and (17) we found that cumulant generating function of APTEPL distribution can be expressed: where: Journal of Statistical Theory and Applications (2023) 22:1-18

Quantile Function
The quantile function of the APTEPL distribution random variable X is Q X (u) = G −1 X (u) , 0 < u < 1 , and for any , , , > 0 and ≠ 1 . In the following steps, we find out the expression of Q X .
By considering the (5) can be written as By solving G X (x) = u for p, we get By solving p = 1 − 1 + x + e − x for x, we obtain By using negative Lambert W −1 function, from (21) we get From (22) and put Q X (u) = x , we obtain (24)

Order Statistics
Let X (i) , i = 1, 2, ......., n denoted to n independent random variables from any distribution with CDF F X (x) and PDF f X (x) , then the PDF of X (i) is given by : In case of substituting (6) and (5) into equation (25), the PDF of X i according to APTEPL distribution given as the following: Now, the PDF of X (n) and X (1) respectively, given by :

Simulation Study
The simulation study for MLEs of APTEPL distribution is performed by generating N = 1000 samples of sizes n = 20, 40, 80, and 100 from APTEPL distribution and studied the behavior of estimates, based on certain measures, which are mean square errors (MSEs) and absolute biases (ABs). Considering CDF of APTEPL distribution and after performing mathematical calculations it was found that if u is a random number from U(0, 1), then

Maximum Likelihood Estimator
Let X = (x 1 , x 2 , ........, x n ) be the random variables of size n belong to  APTEPL(x, , , , ) , then the Likelihood function can be written as, Substituting (6) into (30) and taking the logarithm function for two side we found that the Loglikelihood function is given by: By taking the partial derivative for Log L in (31) with respected to , , and respectively , we got that log L = n log(log( )) + 2n log + n log − n log( − 1) − n log( + )  The maximum likelihood estimator of , , and can be obtained by solving the above four nonlinear equations when log L = 0 , log L = 0 , log L = 0 and log L = 0 with using the numerical approach.

Application
In this section, two real data sets are used to compare the performance of proposed APTEPL distribution with three existing models: alpha power transformed power Lindley (APTPL) distribution [9] , alpha power transformed extend Lindley (APTEL) which is as spacial case of our proposed distribution and alpha power transformed Lindley (APTL) distribution [5]. To compare the performance of our model with the others models the following criterions are used: Akaike information criterion (AIC), Bayesian information criterion (BIC), Corrected Akaike information criterion (AICc), and Kolmogorov-Smirnov goodness of fit test (KS). The distribution with the smallest values of AIC, BIC, and AICc or maximum p − value for (KS) test is considering as the best model for the given data. In this section the numerical results are obtained by using of R software. First data set taken from Ijaz Muhammad et al. [10]. The data set represent the waiting time (in minutes) of 100 bank customers. The MLE of the parameters and standard errors and goodness of fit statistics of the model parameters are provided in Tables 6 and 7. It is clear from Table 7 that the APTEPL distribution gives better fit than the other distributions. Figure 3 shows the theoretical and empirical PDF and CDF of the APTEPL, APTPL, APTEL, and APTL distributions using Bank customer data.
Second data set consists of 128 bladder cancer patients taken from Lee Elisa et al. [13] and used by Ahmad et al. [1]. The MLEs of the parameters and standard errors Journal of Statistical Theory and Applications (2023) 22:1-18    and goodness of fit statistics of the model parameters are provided in Table 8 and 9,  from Table 9 it can be that the APTEPL distribution gives better fit than the other distributions. Figure 4 shows the theoretical and empirical PDF and CDF of the APTEPL, APTPL, APTEL, and APTL distributions regarding data set 2.

Conclusion
In this article, we have proposed alpha power transformation extended power Lindley distribution which is extension of extended power Lindley distribution using alpha power transformation. Some properties of the proposed distribution are derived such as moments, moment generating function, Characteristic function , cumulant generating function, quantiles and order statistics. The performance of the proposed distribution is verified by using real data sets and performing simulation study. It was observed that the proposed distribution is good w.r.t the comparing criterions AIC, BIC and AICc . Therefore it was better than other models considered in this article. It works well for the real life data to model the waiting time (in minutes) of 100 bank customers data and the data of 128 bladder cancer patients.
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