A New Member of T-X Family with Applications in Different Sectors

is paper proposes a member of the T-X family that incorporates heavy-tailed distributions, known as “a new exponential-X family of distribution.” As a special case, the paper studies a submodel of the proposed class named a “new exponential Weibull (NEx-Wei) distribution.” Some mathematical properties including hazard rate function, ordinary moments, moment generating function, and order statistics are discussed. Furthermore, we adopt the method of MLE (maximum likelihood estimation) for estimating its model parameters. A brief Monte Carlo simulation study is conducted to evaluate the performances of the MLEs based on biases and mean square error. Finally, we provide a comprehensive study to illustrate the introduced approach by analyzing three real data sets from dierent disciplines. e analytical goodness of t measure of the proposed distribution is compared with other well-known distributions. We hope that the proposed class may produce many more new distributions for tting monotonic and nonmonotonic data in the eld of reliability analysis and survival analysis as well.


Introduction
In a number of practical areas such as engineering, biomedical, and actuarial sciences, the observations are generally positive in nature and have a unimodal and humpshaped distribution. In such scenarios, extreme values form thick right tails, thus, requiring heavy-tailed distributions to model the data. For instance, in engineering, modeling the unusual phenomena associated with the tails of a statistical distribution is of main interest. Earthquakes, oods, hurricanes, tsunamis, and electrical and power outages market risk are some of the examples of such extreme/rare events [1]. In insurance losses, the data are generally recorded on a positive scale, unimodal, hump-shaped, and positively skewed and have a thick right tail [2]. Also, in health service research, medical expenses that cross a given threshold [3] and the length of stay in a hospital generally represent highly skewed and heavily tailed data [4].
All the above-mentioned scenarios and the rate at which they happen are associated with the distribution in terms of shape and the heaviness of its tails. Classical distributions are not suitable for modeling this type of data [5]. Researchers have observed that the use of gamma, exponential, and Weibull models is discouraged in modeling insurance data because of their ine cient results. Consequently, it has been concluded that it is better to use probability distributions having maximum exibility in order to get higher accuracy in modeling heavy-tailed data than the exponential distribution [6]. To this end, e orts are put on to introduce new "heavy-tailed distributions"; see [7][8][9][10][11].
Distributions where the probabilities on their right tails are greater than the classical exponential models are known as heavy-tailed distributions [12]. For instance, for a cumulative distribution function, we have for any p > 0 ; further details are given in [13,14]. e relevant methods proposed in the literature, and mentioned in the references herein, may be very useful in bringing more flexibility to existing distributions. However, they lack flexibility in terms of inference and computations to derive their distributional properties [8]. Another prominent approach relates to the composition of two or more distributions based on predefined weights, which gives an improved fit for heavy-tailed losses [15][16][17][18]. It is, therefore, important to introduce a new class of models either from the existing classical distributions or from a new family of distributions to model heavy-tailed data from various fields of life.
Motivated by these concerns, this paper proposes a novel family of heavy-tailed distributions using the T-X technique without adding additional parameters. e suggested method, called "a new exponential-X family of distributions" offers a reliable fit for insurance data. e remainder of the paper is arranged as follows: Section 2 discusses the proposed method based on the T-X family; see Alzaatreh et al. [19]. Section 3 presents a new exponential Weibull (NEx-Wei) distribution. Some basic mathematical properties of the proposed family are studied in Section 4. Parameters estimation based on the maximum likelihood estimation method is described in Section 5. In the same section, a Mote Carlo simulation study is also conducted. Applications of the proposed family of distributions on data from vehicle insurance loss, engineering, and medicine are illustrated in Section 6. Finally, Section 7 gives the conclusion of the work based on the proposed distribution.

Proposed Method
In this section, we introduce a new modified method to obtain a new lifetime distribution. e proposed method is introduced by combining the exponential model having PDF (probability density function) m(t) � e − t with the T-X family proposed by Alzaatreh et al. [19].
Consider a random variable, say T, to be a baseline random variable with PDF m(t), where T ∈ [π 1 , π 2 ] for− ∞ ≤ π 1 < π 2 ≤ ∞. Let X be a random variable with CDF (cumulative distribution function) K(x; ω) depending on the parameter vectorω. Let W[K(x; ω)] be a function of CDF of y, satisfying the following three conditions.
According to the Alzaatreh et al. [19] the CDF of the T-X family method is defined by where W[K(x; ω)] satisfies certain conditions presented (I-III). e PDF of T-X distribution, corresponding to equation (1), is given by By using the T-X family of distributions, several novel distribution classes have been proposed in the literature. Table 1 provides some W[K(x; ω)] expressions for some of the widely used members of the T-X family. Now, by using m(t) � e − t and setting (2), we get the CDF of the new Exponential-X family, given by where K(x, ω) is the CDF of the baseline distribution which may depend on ω ∈ R. e PDF of the NEx-X family associated with equation (4) is Similarly, the HF (hazard function) and SF (survival functions) of the NEx-X family are provided by (6) and (7), respectively.
e key motivations of the NEx-X family approach are as follows: (i) A relatively simple approach for extending the available distributions. In addition, the most important motivation is that the proposed method introduces new distributions without inserting extra parameters, which consequently avoids the difficulties of rescaling.

Special Submodel of the Proposed Novel Family
In this section of the article, a special submodel based on the proposed family called the NEx-Wei distribution is introduced. Let K(x; ω) and k(x; ω) be the corresponding CDF and PDF of the Weibull distribution given by K(x; ω) . en the CDF of NEx-Wei model is defined by Expressions for PDF, SF (survival function), and function for HF (hazard rate function) are given in equations (8)- (10), respectively.
Different shapes for the f(x; α, β) of NEx-Wei distribution for various parameter values are sketched in Figure 1. Figure 2 graphically displays the h(x; α, β) of the NEx-Wei model for different combinations of the model parameters. From Figure 2, we can see that the h(x; α, β) of the NEx-Wei distribution have six different patterns including (i) increasing, (ii) decreasing, (iii) reverse-J shaped, (iv) unimodal, and (vi) slightly bathtub shaped. Hence, the proposed model is capable and becomes an important model to fit several lifetime data in applied areas, particularly in reliability engineering, biomedical, economics, and finance analysis.

Basic Mathematical Properties
is section presents some mathematical properties of the NEx-X family, such as the quantile function and ordinary moments, which can further be used to obtain some important characteristics of the model. In addition to these properties, the moment generating function is also derived.

Quantile Function.
e quantile function (QF), also called inverse distribution function (IDF), is an important statistical terminology used to generate random numbers (RNs). ese RNs can be used for simulation purposes to evaluate the performance of the estimators. Later in Section 4, the IDF method has been implemented to generate RNs from the NEx-Wei model. For the proposed model, the QF is given by where t is the solution of equation (1 − u)(e − 1) + 1 + (1 − u)t − e 1− t � 0 and u has the uniform distribution on interval (0, 1). e expression can be used to generate RNs from any subcase of the NEx-X family of distributions.

r th Moment.
e r th moment is an important and a useful ST (statistical tool) to obtain certain characteristics and features of a model. ese characteristics are known as (i) central tendency: which deals with the mean point of any distribution, (ii) dispersion: which measures the variance of a model, (iii) skewness: which describe the tail behavior of the model, and (iv) kurtosis: which helps in studying the

W[K(x, ω)]
Range of X T-X family member peakedness of the distribution. For the proposed NEx-X family, the r th moment expressed by μ r / is derived as By (5), we have Using the series expansion When replacing x by ((1 − K(x, ω))/e) in (15), we get Also, using Taylor series representation By replacing x by K(x, ω) in (17), we get By (16) and (18), we get Furthermore, incorporating the binomial expansion When replacing x by K(x, ω) and p by n − 1 and i, respectively, in (20), we arrive at Using (21) and (22), in (19), we obtain where Furthermore, a simple general expression for the MGF (moment generating function) of the NEx-X random variable X, say M X (t), is given by By using (23) and (24), we get the MGF of the NEx-X family of distributions.

Order Statistics.
In distribution theory, OS (order statistic) has great importance. ey make their appearance in the reliability analysis, problems of estimation theory, and life testing in a number of ways. ey can characterize the lifetimes of elements or components of a reliability system. Let X 1 , X 2 , . . . , X q be a random sample of q chosen from NEx-X with CDF and PDF given by (5) and (6), respectively. en the density function of g r: q is given by We express the 1 st order statistic as X 1: q � min(X 1 , X 2 , . . . , X q ) and the q th order statistic as X q: q � max(X 1 , X 2 , . . . , X q ). en, 0 < K(x; ω) < 1 for x > 0. We may utilize the binomial expansion of [1 − K(x; ω)] q− r as follows: On using equation (25) into equation (26), we get Using equations (5) and (6), in equation (27), we obtain the DF (density function) of g r: q .

Residual and Reverse Residual Lifetime.
e RL (residual lifetime) of the NEx-X random variable X, expressed by R (X) (t), is derived as In addition to the RL, we obtain the RRL (reverse residual lifetime) of the NEx-X distributions denoted by

Estimation and Simulation
is section is divided into two subsections. e first subsection provides a detailed description of the maximum Journal of Mathematics likelihood estimation implemented for estimating the parameters (α, β) of the NEx-Wei model, while the second subsection provides a comprehensive Monte Carlo simulation study for assessing the performance of the MLEs of the proposed method.

Maximum Likelihood Estimation.
Several methods for estimating the parameters of any distribution have been introduced in the literature. e MLE (maximum likelihood estimation) is one of the most frequently used of such methods.
is method furnishes estimators with several important properties and can be used in the construction of confidence intervals as well as other tests for checking statistical significance. For further details about MLEs, see [30].
is subsection provides a discussion on the MLEs approach for estimating the model parameters of the NEx-Wei distribution.
Suppose x 1 , x 2 , ..., x n are the observed values from the pdf given in equation (9) with α andβ as the associated parameters. Corresponding to equation (9), the Log-likelihood function is Taking derivatives of equation (30) with respect to the desired parameters and setting it equal to zero give Numerical solutions of (31) and (32) simultaneously yield the MLEs of α and β.

Simulations.
e behaviors of the MLEs of the parameters of the suggested distribution are evaluated in this section based on simulated data. ree sets of parameters of the NEx-Wei model are assessed in the simulation. e process is described below: Simulation results on estimated parameters in terms of MSEs and biases values are provided in Table 2 and also graphically displayed in Figures 3-5. From the simulation results in Table 2, we conclude that the biases for all parameters are positive and the estimated biases and MSEs decrease as the sample size increases.

Applications
is section assesses the applicability of the NEx-Wei model in applied areas that include financial, engineering, and medical sciences. In all these areas, the fits of the NEx-Wei model are compared with other familiar distributions.
For checking the goodness of the distributions, we consider different goodness of fits measures in order to examine which competitor provides the best fit to the considered data sets. e goodness of fit measures include CM (Cramer-von-Misses) test statistic, AD (Anderson-Darling) test statistic, KS (Kolmogorov-Smirnov), AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), corrected Akaike information criterion (CAIC), and HQIC (Hannan-Quinn Information Criterion) as well as P-values.
In general, a distribution with smaller values for these analytical measures and a greater p-value could be considered a good candidate for the underlying data set. Based on the considered analytical measures, the results reveal that the NEx-Wei distribution produces greater distributional flexibility among all the other applied distributions.

Application in Vehicle Insurance Loss Data.
e first case study is that of insurance, where vehicle insurance losses are considered. e data are taken from the website: http://www. businessandeconomics.mq.edu.au/our-depatments-/Apllied -Finance-and-Acturial-Studies/research/books/GLMs-for-in surance-Data. Some basic measures for the dataset are given by minimum � 1.0, 1st quartile � 23. 25 Corresponding to this dataset, the comparison of the NEx-Wei distribution is made with other well-known distributions including APT-Wei (Alpha Power transformed Weibull) [31], Degum [32], Lomax distribution, Burr-XII (B-XII) distribution [33], MO-Wei distribution [34], and Kumaraswamy Weibull (Ku-Wei) distribution [35]. e reason for considering these distributions for comparison purposes is their frequent application in modeling financial and financial risk management problems. Furthermore, for the analyzed data, the maximum likelihood estimates of the fitted models are presented in Table 3. e numerical values of the analytical measures of   Based on these analytical measures, the proposed model fits better than the other competing models to the considered data. In the support of the numerical illustration in Tables 4  and 5, the estimated PDF and CDF plots of the NEx-Wei distribution are presented in Figure 6. Moreover, the PP plot and Kaplan-Meier survival plot are presented in Figure 7, whereas Figure 8 shows the box and QQ plots. Obviously, these plots reveal the closer fit of the NEx-Wei model.

Application in Reliability Engineering.
e second case study is from reliability engineering regarding the failure time of cutting layers machine [36]. Basic measures for the  e performance of the proposed model is evaluated by comparing it with other well-known models such as Kumaraswamy Weibull (Ku-Wei) [35], two parameters' Weibull, extended alpha power Weibull (EAP-Wei) [37], Beta Weibull (B-Wei) [38], and new alpha power Weibull (NAP-Wei) [39] models. Furthermore, the Ku-Wei, EAP-Wei, and NAP-Wei models are widely used in the literature for modeling failure time data.
Corresponding to the second data set, the values of MLEs of the parameters are presented in Table 6, whereas the analytical results of the proposed and other competitive models are reported in Tables 7 and 8 Figure 9 gives the corresponding estimated plots of PDF and CDF. Furthermore, Figure 10 gives the PP and Kaplan-Meier survival plots, whereas Figure 11 shows the box and QQ plots. e results demonstrate, given the positively skewed data (see box plot), that the newly suggested model fits the data better than the other methods.

Application in Biomedical Science Data.
e third case study is from biomedical science, where the dataset consists of forty-four observations reported in [40].
is data set represents the survival time of a group of patients suffering from     Corresponding to the third data set, we applied the NEx-Wei model with several other competitive models, namely, the two parameters' classical Weibull, FRL-Wei [41], APT-Wei [31], and MO-Wei [34] distributions.

Conclusion
is article presented the idea of a new family of distribution, called the new exponential-X family or NEx-X. is family of distributions has a wide range of applications without adding additional parameters to the already available distributions. A special submodel of the proposed method called a NEx-Wei (new exponential Weibull) is derived and studied in detail. Besides, general expressions for different statistical properties of the proposed family have been derived including quantile function, moments, moments generating function, and order statistics. MLE (maximum likelihood estimation) method has been used for estimating the unknown parameters, and in addition, a Monte Carlo simulation study is carried out to assess the performance of the proposed model estimators. In the field of reliability engineering, insurance, and medicine, we have analyzed three data sets and the proposed class provides a very good fit for all data sets. We hope that this novel improvement in the theory of the distribution will give more attractive applications in reliability engineering, medical, and other related fields.
Data Availability e references of the data sets are given within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.

Supplementary Materials
To replicate the results of the simulation study in Table 2, the simulation codes are provided as a supplementary file. (Supplementary Materials)