A New Useful Exponential Model with Applications to Quality Control and Actuarial Data

The compounding approach is used to introduce a new family of distributions called exponentiated Bell G, analogy to exponentiated G Poisson. Several essential properties of the proposed family are obtained. The special model called exponentiated Bell exponential (EBellE) is presented along with properties. Furthermore, the risk theory related measures including value-at-risk and expected-shortfall are also computed for the special model. Group acceptance sampling plan is designed when a lifetime of a product or item follows an EBellE model taking median as a quality parameter. The parameters of the proposed model are estimated by considering maximum likelihood approach along with simulation analysis. The usefulness of the proposed model is illustrated by practical means which yield better fits as compared to several exponential related extended models.


Introduction
Effective implementation of mathematical and statistical models enables the actuarial scientists to know as much as possible about future claims in a portfolio. ese models serve as a guide to achieve better business and risk management decision and policies. Actuaries usually deal with a complex data such as right skewed, unimodal, and having heavy tail. e readers are referred to works of Klugman et al. [1], Cooray and Ananda et al. [2], Lane [3], Vernic [4], and Ibragimov et al. [5]. At the same time, they are eager on some flexible models which are capable of capturing the behaviours of such data to finding along with information when the real development deviates from the expected. e classical models are limited with their tail properties and goodness of fit tests. For instance, Pareto, Lomax, Fisk, and Dagum distribution are excessively used to model statistical size distributions in economics and actuarial sciences but often failed to provide better fits for many application. e Weibull distribution is appropriate for small losses but fail to uncover adequate trend, level, and trajectory for large losses [6]. e reader are referred to [7] for detail discussion on statistical size distributions which can be used in economics and actuarial sciences. To overcome the drawback of classical models, a substantial progress on persistent base related to distribution theory is documented in statistical literature. From the last couple of decades, the emerging trend has been seen in the generalization of the existing classical models. e models are extended by adopting different modes of adding one or more additional shape parameter(s) in the distribution.
e basic aim of this whole exercise is to improve the tail properties as well as goodness of fit test of the classical models. ere are several well-known generators which are documented in the statistical literature; the readers are referred to the works of Tahir and Nadarajah [8], Tahir and Cordeiro [9], Maurya and Nadarajah [10], and Lee et al. [11].
Several new models related to claim data have recently been reported in statistical literature. Ahmad et al. [12] proposed a new method to define heavy-tailed distributions called the exponentiated power Weibull distribution with application to medical care insurance and vehicle insurance. Calderin-Ojeda and Kwok [13] presented a new class of composite model by using the Stoppa distribution and mode matching procedure and modelling the actuarial claims data of mixed sizes. Ahmad et al. [14] suggested nine new methods to define new distributions suitable for modelling heavy right-tail data with application to medical care insurance and vehicle insurance. Afify et al. [15] proposed a new heavy-tailed exponential distribution with application to unemployment claim data. Ahmad et al. [16] introduced a class of claim distributions useful in a number of lifetime analyses. A special submodel of the proposed family, called the Weibull claim model, is considered in detail with claim data application. Among classical discrete distributions, Poisson distribution is a most frequently used distribution for count data. Furthermore, it is extended into G-class and several transformation and family of distributions have been proposed. A detail review study on Poisson generated family of distributions, extensions, and transformation is recently presented by [10]. Castellares et al. [17] introduced a discrete Bell distribution from well-known Bell numbers, as a competitor or counterpart to Poisson distribution which exhibits many interesting properties such as a single parameter distribution, and it belongs to one-parameter exponential family of distributions and the Poisson distributions.
ey investigated that the Poisson model cannot be nested into the Bell model, but small values of the parameter the Bell model tends to Poisson distribution. Furthermore, the Bell model is infinity divisible and has larger variance as compared to the mean, which can be used to overcome the phenomenon of over-dispersion and zerovertex for count data. e characteristics of the Bell model motivated us to develop a generalized class of distributions through compounding and to compare its mathematical and empirical characteristics with compounded Poisson-G class and its special models. e rest of the study is organized as follows. In Section 2, we define the proposed EBell-G family of distributions. Section 3 provides the general mathematical and structural properties of EBell-G family of distributions including linear representation of density, quantile function, rth moments, probability weighted moments, analytical shapes of the density and hazard rate, entropy measures, reversed order statistics, upper record statistics, stochastic ordering, and parameters' estimation by using maximum likelihood estimation. Section 4 illustrates the layout of the special model called EBellE as well as its essential properties, while Section 5 shows the commonly used actuarial measures, specially value-at-risk and expected-shortfall. Section 6 are illustrated group acceptance sampling plane when a lifetime of a certain product or item follows the EBellE model which is presented. e simulation analysis is presented in Section 7, and Section 8 contains the application of real datasets. e concluding remarks are given in Section 9.

Layout and Formulation of EBell-G Family
A single parameter discrete Bell distribution has been recently introduced by Castellares et al. [17], which is an analogy to discrete Poisson distribution but provides better fits compared to other discrete models including the Poisson model. e following expression given by Bell [18] is where B n denote the Bell numbers and can be derived from the following mathematical expression: Remark 1. e Bell number B n in (2) is the nth moment of the Poisson distribution with parameter equal to 1.
By considering equations (1) and (2), Castellares et al. [17] introduced a single-parameter Bell distribution defined by the following probability mass function (pmf ) as

Proposition 1. LetXfollow a discrete Bell model with parameterλ; then, the following expression represents the pmf of Bell truncated model as
We first give the motivation for the proposed family. Suppose a system is having N subsystems that are working or functioning independently at a given specific time. Here, Y i denotes the life of ith subsystem and θ parallel units constitutes the subsystem. Furthermore, the system will fail or remain functioning if all the subsystem fail; this is for the parallel system. On the contrary, for series system, the failure of any subsystem yields complete destruction of the whole system. Let us have a random variable (rv)N that follows any discrete distribution having pmf P(N � n). Here, we suppose that a component Z i,1 , . . . . . . Z i,θ having failure time for the ith subsystem are i.i.d. with suitable cdf depending upon the vector τ, say for then the conditional cdf of Y given N is as follows: e unconditional cdf of Y corresponding to (5) is given by 2 Computational Intelligence and Neuroscience By using the Bell truncated model given in Eq. (4) and then using Eq. (6), the unconditional cdf of X is defined below as follows.

Properties of the EBell-G Family
is section provides some mathematical properties of the EBell-G family of distributions.
3.1. Quantile Function. Quantile function (qf ) is an important measure for generating random numbers and several other important uses in quality control sampling plans and in risk theory; the two important commonly used measures value-at-risk (VaR) and expected-shortfall (ES) which depend on qf and is given as follows.

Analytic Shapes of the Density and Hazard Rate Function.
e analytical shapes of the density and hrf can be yielded for EBellE, respectively, as follows:

Useful Expansions.
Here, we show the useful expansion for EBell-G density can be used to drive several important properties by taking into account the following two series to obtain the expansion for EBell-G.
Proposition 6. e generalized binomial expansion which holds for any real noninteger b and |t| < 1 is e power series for exponential function is given by Bourguignon et al. [19] and is given as follows: erefore, by using Eq. (11) to Eq. (8), we can deduce pdf and cdf, simultaneously, as Computational Intelligence and Neuroscience 3 where are constants satisfying ∞ v�0 w v � 1. Eq. (12) represents exp-G, that is, h θ(v+1) (x) and the term θ(v + 1) is treated as the power parameter. By using Eq. (12), numerous properties of G-class can be obtained.

Mathematical Properties.
One can derive some important mathematical properties by considering Eq. (12).
e rth raw moment of X is given by where E[X r θ(v+1) ] follows a exp-G with θ(v + 1) treated as the power parameter, and by taking r � 1, in (14), yields the mean for X. e incomplete moments are important and have many practical uses. e expression of sth incomplete moments, denoted by φ s (t), is defined by φ s (t) � t − ∞ x s f(x)dx and can be obtained by using Eq. (12) as e first incomplete moment of the EBell-G family can be obtained as by taking s � 1 in Eq. (15). e sth incomplete moment is an important to compute several measures, namely, mean deviations from mean and median, mean waiting time, conditional moments, and income inequality measures among others.

Probability Weighted Moments.
e (s, r)th probability weighted moments (PWM) of X following the EBell-G family, say ρ s,r , is formally defined by By using Eq. (7) and Eq. (8), we can obtain where 3.6. Entropy Measures. e entropy measures are important to underline the randomness or uncertainty or diversity of the system. e most frequently used index of dispersion in ecology as well as in statistics is called the Rényi entropy I δ (x) and is defined by the following expression: where δ > 0 and δ ≠ 1, which then follows where e Shannon entropy say, H q (x), can be obtained by the following expression: where q > 0 and q ≠ 1 and

Order Statistics.
Here, we derived the explicit expression for the ith-order statistics for EBell-G, say f i:n (x). Let a sample of size be n; then, the pdf of ith-order statistics is defined by By using Eq. (7) and Eq. (8), the density for EBell-G can be written as e sth moment of order statistic can be obtained as where μ (s) θ(j+1) is the sth moment of Exp-G distribution with power parameter θ(j + 1).

Reversed Order Statistics.
e reversed order statistics can be used when x 1 , . . . . . . , x n are arranged in the decreasing order; for more detail, see the work of Jamal et al. [20]. e pdf of X r(re): n , represented by f r(re): n (x) � f n− r+1: n (x), is defined by Consider By using Eq. (10), we can obtain en, by using Eq. (11), we can have Let us consider After simplification, we have shapes: Finally, Computational Intelligence and Neuroscience e reduced form will be where h θ(j+1) � θ(j + 1)g(x)G(x) θ(j+1)− 1 and e pth moment of reversed-order statistic can be obtained as where μ (p) θ(j+1) is the pth moment of Exp-G distribution with power parameter θ(j + 1).

Upper Record Statistics.
Record value is an important measure in many practical areas, for instance, economics data and weather and athletic events. Let us consider (X n ) n ≥ 1 a sequence of independent rvs having the same distribution. Let us denote by F(x) and f(x) the related cdf and pdf of EBellE distribution, respectively, and X i: n be the ith-order statistic as described previously. For fixed k ≥ 1, the pdf of kth upper record statistic is defined by where , by using (7), as Considering the last terms, and after using series, we obtain Using power series given in Eq. (11), we obtain Now, the above expression becomes By using Eq. (11) again, we obtain 6 Computational Intelligence and Neuroscience Finally, we have e reduced form becomes A random sample of 50 is generated from the EBellE model using Eq. (23), and then, take k � 3 and α � β � λ � 0.5. Table 1 shows a random sample of 50 from the EBellE model along with upper X U(n) and lower X L(n) records values. e plot of lower and upper record values is illustrated in Figure 1.
e Records package is used in R-Statistical Computing Environment to compute X U(n) and X L(n) records' values.

Stochastic
Ordering. Stochastic ordering is another important tool in statistics to define the comparative behaviour specifically in reliability theory. Suppose the two rvs, say X 1 and X 2 and under specific circumstance; let us consider that rv X 1 is lower than X 2 ; the readers can refer to the work of Khan et al. [21] for detailed illustration on four stochastic ordering and their well-established relationships.
Proof. First, we have the ratio Now, consider Computational Intelligence and Neuroscience (54) After simplification, we obtain If λ 1 < λ 2 , we obtain    Computational Intelligence and Neuroscience us, is decreasing in x, and hence, X 1 ≤ lr X 2 . is completes the proof.

Estimation of Family Parameters.
is section is about estimation of the unknown parameters estimation of the EBell-G model by taking into account the popular estimation method known as maximum likelihood estimation ere are several advantages of MLE over other estimation methods; for instance, the maximum likelihood estimates fulfil the required properties that can be used in constructing confidence intervals as well as maximum likelihood estimates delivering simple approximation very handy while working the finite sample. ℓ(.) represent the vector parameters ϕ � (λ, θ, ξ) ⊤ ; then, where are derivatives of column vectors of the same dimension of ξ, and by setting ϕ λ � 0, ϕ θ � 0, and ϕ ξ � 0, the MLEs can be yielded by solving the above equations simultaneously.

Layout of the EBellE Model
Due to the closed form solution of many real problems and simplicity, exponential distribution is commonly employed in lifetime testing as well as reliability analysis. However, the exponential distribution failed to yield better fits when hazard rates are nonconstant. However, several studies showed that extended exponential distribution or when it is used as baseline model provides better fits [22][23][24]. In the present study, we used exponential distribution as a baseline model which yielded flexibility in both pdf and hrf shapes given in Figures 2 and 3, respectively. We now define the EBellE distribution by taking the exponential model as baseline, with the following expression of densities g(x) � α exp(− αx) and G(x) � 1 − exp(− αx) for x > 0 and α > 0, by setting these densities in (7) and (8) yielded the following expression for the proposed EBellE distribution. en, the cdf and pdf are of the EBellE distribution, respectively. (60) then, its cdf is given by in Eq. (7): then, its pdf is given by in Eq. (8): e exponential distribution quantile function becomes using (9), . e quantile function of x can be expressed as Computational Intelligence and Neuroscience e sf and the hrf of the EBellE model can be obtained as

Properties of the EBellE Model.
First, we will deduce linear representation of EBellE density to obtain useful properties of that model. By using Eq. (12), 10 Computational Intelligence and Neuroscience After applying Eq. (10), it reduces to where π[x; α(p + 1)] is a exp-exponential density with α(p + 1) parameter and It is obvious from Eq. (25) that the EBellE density is a linear combination of exponential densities, and therefore, one can obtain several properties using Eq. (25).

e Expression of rth Moment
Proposition 11. LetX ∼ EBellE(λ, θ, α), forx > 0andλ, θ, α > 0; then, itssth incomplete moment can be written as by taking into account Eq. (25): By setting s � 1 yielded the first incomplete moment of the EBellE model. Table 2 shows the first four raw moments, central moments, coefficient of variation, coefficient of kurtosis, and Pearson's coefficient of skewness for some parametric values. Six different scenarios of parametric values are used in order to compute different measures of dispersion. S e following relationship is used to obtain the central moments: 4 . e moment-based measure of skewness and kurtosis is obtained by using β 1 � μ 2 3 /μ 3 2 and β 2 � μ 4 /μ 2 2 , respectively. Pearson's coefficient of skewness is simply square root of β 1 , and coefficient of kurtosis is computed as β 2 − 3. Furthermore, we present the mean, Computational Intelligence and Neuroscience variance, skewness, and kurtosis of EBellE in Figures 4 and 5, respectively, utilizing these results. Some plots of Bonferroni and Lorenz curve are also depicted in Figure 6.

e Expression of r th Conditional Moment.
From actuarial prospective, conditional moments are important; let EBellE be (λ, θ, α) for x > 0 and λ, θ, α > 0; then, its r th conditional moment can be written by using Equation (64):

Two Expression of MGF.
Let X ∼ EBellE (λ, θ, α) for x > 0 and λ, θ, α > 0; then, its moment generating function by using Wright generalization hypergeometric function is given as By using (70), Equation (71) yielded as e other representation of mgf is given by

Order Statistics.
e sth moment of order statistic can be obtained by using (41): Simplification yielded the expression of sth moments of order statistics: i: n θ(j + 1)/(p + 1).
To study the distributional behaviour of the set of observation, we can use minimum and maximum (min-max) plot of the order statistics. Min-max plot depends on extreme order statistics, and it is introduced to capture all information not only about the tails of the distribution but also about the whole distribution of the data. Figure 7 shows the min and the max order statistics for some parametric values and depends on E(X 1: n ) and E(X n: n ), respectively.

Stochastic
Ordering. Let X and Y be the two rvs from EBellE distribution with the assumption previously illustrated in Section 3 given that λ 1 < λ 2 , and for X 1 ≤ lr X 2 , f 1 (x)/f 2 (x) shall be decreasing in x if and the only if the following results holds: 4.1.7. Rényi Entropy. e Rényi entropy for the EBellE model by using Eq. (22) given under and δ > 0 and δ ≠ 1: , and the graphical demonstration of Rényi entropy of EBellE at varying of the parameters is given in Figure 8.

Reliability.
Reliability is an important measure, and several applications are documented in the field of economics, physical science, and engineering. Reliability enables us to determine the failure probability at certain point in a time. Let X 1 and X 2 be the two random variable following the EBellE distribution. e component fails if the applied stress exceeds its strength; if X 1 > X 2 , the component will perform satisfactory. Reliability is defined by the following expression. Here, we derive the reliability of the EBellE model when X 1 and X 2 have independent f 1 (x; λ 1 , θ, α) and F 2 (x; λ 2 , θ, α) with identical scale (α) and shape (θ) parameters. e reliability is given by By using equations (14) and (15), we get the pdf and the cdf of EBellE, respectively, as follows: Hence, erefore, By using gamma function, the above expression is reduced to where

Estimation.
e log-likelihood function for the vector of parameters ϕ � (λ, θ, α) ⊤ for model given in (60) is given by 14 Computational Intelligence and Neuroscience

Computational Intelligence and Neuroscience
By setting U λ � 0, U θ � 0 and U α � 0, the MLEs can be yielded by solving the above equations simultaneously.

Value at Risk. Value-at-risk or quantile risk or simply
VaR is the extensively used as a standard finial market risk measure. It plays an important role in many business decisions; the uncertainty regarding foreign market, commodity price, and government policies can affect significantly firm earnings. e loss portfolio value is specified by the certain degree of confidence say q(90%, 95%or99%). VaR of random variable X is simply the qth quantile of its cdf. If X follows the EBellE model; then, its VaR is defined by the following expression:  16 Computational Intelligence and Neuroscience

Expected Shortfall.
e other important financial risk measure is called an expected-shortfall (ES) introduced in [25] and generally considered a better measure than valueat-risk. It is defined by the following expression: For 0 < q < 1, using Eq. (85) in Eq. (86), yielded ES for EBellE. In Figure 9, the graphical representation of VaR and ES measures for some parameter combinations is presented.

Tail Value at Risk.
e problem of risk measurement is one of the most important problems in the risk management. From finance and insurance prospective, Tail value-atrisk (TVaR) or tail conditional expectation or conditional tail expectation is an important measure and define as the expected value of the loss, given the loss is greater than the VaR: By using (25) in (35) yielded tail value-at-risk as

Tail Variance.
Tail variance (TV) has yet another important risk measure because it considers the variability of the risk along the tail of distribution; it is defined as from the following expression: Consider I � E[X 2 |X > x q ]: (90) Using (88) and (90) in (89), we obtain the expression for tail variance for the EBellW model.

Tail Variance Premium.
e TVP is the mixture of both central tendency as well as dispersion statistics. It is defined by the following: where 0 < δ < 1. Using expressions (89) and (88) in (91), we obtain the tail variance premium for the EBellW model.

Numerical Illustration of VaR and ES.
Here, we demonstrate the numerical as well as graphical presentation of the two important risk measures ES and VaR. e comparative study of ES and VaR of the proposed EBellE model with their counterpart exponentiated exponential Poisson (EEP) and exponentiated exponential (EE) model is performed by taking MLEs' estimates of the parameters for the models in all datasets. It is worth emphasising that a model with higher values of the risk measures is said to have a heavier tail. Table 3 provides the numerical illustration of the ES and VaR for EBellE and EEP and EE model of all three datasets and yielded that the EBellE model has higher values of both the risk measures as compared to their counterpart EEP and EE model. e graphical demonstration of the models from Figures 10-12 also revealed that the proposed model has slightly heavier tail than EEP and EE model. e reader should refer to Chan et al. [26] for detail discussion of VaR and ES and their computation by using an R-Programming Language. A sample of 100 is randomly drawn, and the effect of shape and scale parameters of the proposed models is underlined for both risk measures. Various combinations of the scale and shape parameters are executed

Designing of GASP under the EBellE Model
Saving time and cost is attributed to the sampling method. Certain quality control checks are implemented either accepting or rejecting a lot under various sampling plans. is section based on the illustration of GASP under the assumption when the lifetime distribution of an item followed a EBellE model with known parameter λ and θ having cdf in Eq. (96). In a GASP, let a random sample of size n be taken and distributed in such a way; that is, n � r × g and r items for each group are kept on life testing under predefined time. If the number of failures in each group are higher than the acceptance number c, the performed experiment is truncated. e reader is referred to the work of Aslam et al. [27] and Khan and Alqarni [28] for simple illustration of GASP and application to real data. Designing the GASP reduced both the time and cost. Several lifetime traditional and extended models are used [27,[29][30][31][32] in designing the GASP by taking into account the quality parameter as mean or median; usually, for skewed distribution, median is preferable [27]. e GASP is simply the extension of ordinary sampling plan (OSP), i.e., the GASP tends to OSP by replacing r � 1, and thus, n � g [33].
GASP is based on the following form; first of all, select g and allocate predefine r items to each group so that the sample of size of the lot will be n � r × g. Secondly, select c and t 0 representing the acceptance number and the experiment time, respectively. irdly, do experiment simultaneously for g groups and record the number of failure for each group. Finally, conclusion is drawn either accepting or rejecting the lot; the lot is accepted if there is no more than c failure occurring in each and every group; otherwise, reject a Computational Intelligence and Neuroscience lot. e accepting probability of a lot yielded by the following expression: where the probability that an item in a group fail before t 0 is denoted by p and yielded by inserting (61) in (96). Let the lifetime of an item or product follow a EBellE with known parameters θ and λ, with cdf given for t > 0: qf of the EBellE model using (61) is given by, and if p � 0.5 yielded median lifetime distribution for a product or item,

Computational Intelligence and Neuroscience
By taking η as follows, Eq. (94) becomes by replacing η; henceforth, α � − η/m and t � m 0 a 1 , m � − 1/αη. e ratio of a of product mean lifetime ti and the specified life time m/m 0 can be used to express the quality level of product. By replacing α � − η/m and t � m 0 a 1 in Eq. (96) yielded the probability of failure given by From Eq. (96), for chosen θ and λ, p can be determined when a 1 and r 2 are specified, where r 2 � m/m 0 . Here, we define the two failure probabilities say p 1 and p 2 corresponding to the consumer risk and producer risk, respectively. For a given specific values of the parameters θ and λ, r 2 , a 1 , β, and c, we need to evaluate the value of c and g that satisfy the following two equation simultaneously: where the mean ratio at consumer's risk and at producer's risk, respectively, is denoted by r 1 and r 2 and the probability of failure to be used in the above expression is as follows: From Tables 4 and 5, with β � 0.25, a 1 � 0.5, and r 2 � 4 and taking r � 5, there are 40 groups or 200 (40 × 5 � 200); total units are needed for lifetime testing. While on the contrary, significant reduction can be observed in groups or number of units to be tested under the identical circumstances when r � 10; a total of 3 groups or 30 (3 × 10 � 30) item are needed for life testing. Here, we prefer the group size 10. Similarly, when β � 0.25, a 1 � 1, and r 2 � 4 and taking r � 5, there are 7 groups or 35 (7 × 5 � 35); total units are needed for life testing. While, on the contrary, in the number of units to be tested under the identical circumstances when r � 10, a total of 2 groups or 20 (2 × 10 � 20), items are needed for life testing. Here, we prefer the group size 10.

Simulation Analysis
Simulation analysis is very important tools in statistics and used to determine the performance of estimates over predefine replication at varying sample sizes. So, this section is primarily based on simulation analysis in order to underline the performance parameter estimates of the proposed EBellE model. A simulation process is replicated 1000 times with at varying sample sizes, n � 25, 50, 100, and 500. In Table 6, various combinations of the parameter α, θ, and λ are considered, say scenario I � [α � e finding of the simulation analysis yielded that bias, mean square error (MSE), and average width (AW) of the confidence interval of the parameters reduced as sample size increases. On the contrary, the coverage probabilities (CPs) touch 95% nominal level. So, therefore, the MLEs and their asymptotic results can be used for estimating and constructing confidence intervals for proposed EBellE model parameters. Readers are referred to the work of Sigal et al. [34] for simple but comprehensive way in designing Monte Carlo simulation study by using R-programming language:  β   [15]. e third data deal with upheld most frequent complaints such as nonrenewal of insurance, and no fault claims commonly against vehicle insurance company over two-year period as a proportion of their overall business. e dataset was also used by Khan et al. [21]. e descriptive summary of all three datasets is shown in Table 7 and consists of sample size n, minimum claim x 0 , maximum claim x n , lower Q 1 and upper Q 3 , quartile deviations, mean x, median x, standard deviation σ, measures of skewness S k , and kurtosis K. Total time on test (TTT) plots of the datasets is illustrated in Figure 13, revealing that the first two datasets have increasing hazard rate function, while the third dataset has decreasing (increasing) hazard rate function. e comparative study is carried out with several modified well-established exponential models, namely, exponentiated exponential Poisson (EEP) [35], alpha power exponentiated exponential (APEE) [15], Transmuted generalized exponential (TGE) [36], gamma exponentiated exponential (GEE) [37], exponential (E), exponentiated exponential (EE), Marshal Olkin exponential (MOE) [38], exponentiated Weibull (EW) [39], odd Weibull exponential (OWE) [19], Weibull (W), Kumaraswamy exponential (KE) [40], beta exponential (BE) [41], Tope Leone exponential (TLE) [42], and Nadarajah Haghigh (NH) [43] distributions.
All statistical computational work is carried out using Rprogramming language. Table 8 shows the MLEs and standard errors (S.E) of the estimates of the fitted models of the data sets. Table 9 demonstrated the commonly used wellknown model selection information criterion, namely, AIC, CAIC, BIC, and HQIC with important measures including Anderson-Darling (A * ), Cramér-von Mises (W * ), and Kolmogrov-Smirnov (K-S) test and p value of all three datasets. e analysis of the datasets revealed the proposed three-parameter EBellE model, outperforming compared to several well established models. A model having higher p values and least information criterion and A * and W * , and the K-S value is considered as best models among all other comparative models. TTT plots of the respective datasets are shown in Figure 13. Likewise, plots of the estimated pdf, cdf, hrf, and sf for the four datasets are provided in Figures 14-17. Additionally, PP-plots are presented in Figure 18.

GASP.
Recently, Almarashi et al. [29] designed a GASP under Marshall-Olkin-Kumaraswamy exponential distribution by using the data of breaking strength of carbon fibers. e data consist the 50 observed values with mean (1.975) and median (1.190) breaking strength of carbon fibers, respectively. Under the K-S test, the maximum distance between actual and fitted yielded as 0.0681 with p value 0.9743 under Marshall-Olkin-Kumaraswamy exponential distribution. We used the same data as data-4, and our proposed three-parameter EBellE model is slightly better fit compared to four-parameter Marshall-Olkin-Kumaraswamy exponential distribution [29] as K-S test as 0.0680 with improved p value as 0.9749. e estimated parameters (SEs), namely, α � 0.3913 (0.1308), θ � 0.9088 (0.2114), and λ � 0.3431 (0.5766). Table 10 shows the GAPS under the EBellE model at MLEs' values showing minimum g and c when r � 5 and r � 10, with a 1 � 0.5 and 1.
e analysis of the data yielded from Table 10, with β � 0.25, a 1 � 1, and r 2 � 4 and taking r � 5, there are 7 groups or 35 (7 × 5 � 35); total units are needed for lifetime testing. While, on the contrary, significant reduction can be

Concluding remarks.
We introduced and documented the new flexible family of distributions called exponentiated Bell-G family. We also derived general mathematical properties of the proposed family, namely, linear representation of the density, random variable generation, reliability properties, ordinary moments, generating function, probability weighted moment, entropies, order statistics, reverse order statistics, entropies measures, upper records values, stochastic ordering, and estimation of parameters. We also illustrated the important actuarial measures and design of GASP. We also implemented the new proposed generator to the four real datasets by taking exponential distribution as a special case. e analysis of the data yielded that the new generator is found to be superior compared to their counterparts. [44].
Data Availability e data used in the findings of the study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.