Abstract
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.
1. Introduction
Effective implementation of mathematical and statistical models enables the actuarial scientists to know as much as possible about future claims in a portfolio. These 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. The 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. The 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. The Weibull distribution is appropriate for small losses but fail to uncover adequate trend, level, and trajectory for large losses [6]. The 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. The models are extended by adopting different modes of adding one or more additional shape parameter(s) in the distribution. The basic aim of this whole exercise is to improve the tail properties as well as goodness of fit test of the classical models. There 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. They 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 zero-vertex for count data. The 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.
The 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, th 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. The simulation analysis is presented in Section 7, and Section 8 contains the application of real datasets. The concluding remarks are given in Section 9.
2. 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. The following expression given by Bell [18] iswhere denote the Bell numbers and can be derived from the following mathematical expression:
Remark 1. The Bell number in (2) is the th 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. Letfollow a discrete Bell model with parameter; 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, denotes the life of subsystem and 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 that follows any discrete distribution having pmf . Here, we suppose that a component having failure time for the subsystem are i.i.d. with suitable cdf depending upon the vector , say for , . If we define , then the conditional cdf of given is as follows:
The unconditional cdf of corresponding to (5) is given by
By using the Bell truncated model given in Eq. (4) and then using Eq. (6), the unconditional cdf of is defined below as follows.
Proposition 2. LetEBell-G, forand; then, its cumulative distribution function (cdf) having baseline pdf and cdf respectivelyandis given by
Proposition 3. LetEBell-Gforand; then, its probability distribution function (pdf) having Eq. (8), with baseline pdf and cdf respectivelyand, is given by
Proposition 4. LetEBell-Gforand; then, its survival function (sf) and hazard rate function (hrf) are, respectively, given by
3. Properties of the EBell-G Family
This 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.
Proposition 5. LetEBell-Gforand; then, the expression of qf is given below, where, and by replacing, it yields the median of the EBell-G:
3.2. Analytic Shapes of the Density and Hazard Rate Function
The analytical shapes of the density and hrf can be yielded for EBellE, respectively, as follows:
3.3. 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. The generalized binomial expansion which holds for any real noninteger b and is
The power series for exponential function is given by Bourguignon et al. [19] and is given as follows:
Therefore, by using Eq. (11) to Eq. (8), we can deduce pdf and cdf, simultaneously, aswhereare constants satisfying . Eq. (12) represents exp-G, that is, and the term is treated as the power parameter. By using Eq. (12), numerous properties of -class can be obtained.
3.4. Mathematical Properties
One can derive some important mathematical properties by considering Eq. (12). The th raw moment of is given bywhere follows a exp-G with treated as the power parameter, and by taking , in (14), yields the mean for .
The incomplete moments are important and have many practical uses. The expression of th incomplete moments, denoted by , is defined by and can be obtained by using Eq. (12) as
The first incomplete moment of the EBell-G family can be obtained as by taking in Eq. (15). The th 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.
3.5. Probability Weighted Moments
The th probability weighted moments (PWM) of X following the EBell-G family, say , is formally defined by
By using Eq. (7) and Eq. (8), we can obtainwhere
3.6. Entropy Measures
The entropy measures are important to underline the randomness or uncertainty or diversity of the system. The most frequently used index of dispersion in ecology as well as in statistics is called the Rényi entropy and is defined by the following expression:where and , which then followswhere
The Shannon entropy say, , can be obtained by the following expression:where and and
3.7. Order Statistics
Here, we derived the explicit expression for the th-order statistics for EBell-G, say . Let a sample of size be ; then, the pdf of th-order statistics is defined by
By using Eq. (7) and Eq. (8), the density for EBell-G can be written aswhere
The sth moment of order statistic can be obtained aswhere is the sth moment of Exp-G distribution with power parameter .
3.8. Reversed Order Statistics
The reversed order statistics can be used when are arranged in the decreasing order; for more detail, see the work of Jamal et al. [20]. The pdf of , represented by , is defined byand
Consider
By using Eq. (10), we can obtain
Then, by using Eq. (11), we can have
Let us consider
After simplification, we have shapes:
Finally,
The reduced form will bewhere and
The th moment of reversed-order statistic can be obtained aswhere is the th moment of Exp-G distribution with power parameter .
3.9. 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 a sequence of independent having the same distribution. Let us denote by and the related cdf and pdf of EBellE distribution, respectively, and be the -order statistic as described previously. For fixed , the pdf of th upper record statistic is defined bywhere correspond to the cumulative hazard rate function related to . Eq. (20) can also be expressed for , 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
Finally, we have
The reduced form becomeswhere and
A random sample of 50 is generated from the EBellE model using Eq. (23), and then, take and . Table 1 shows a random sample of 50 from the EBellE model along with upper and lower records values. The plot of lower and upper record values is illustrated in Figure 1. The Records package is used in R-Statistical Computing Environment to compute and records’ values.
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3.10. Stochastic Ordering
Stochastic ordering is another important tool in statistics to define the comparative behaviour specifically in reliability theory. Suppose the two , say and and under specific circumstance; let us consider that is lower than ; the readers can refer to the work of Khan et al. [21] for detailed illustration on four stochastic ordering and their well-established relationships.
Theorem 1. LetEBell-GandEBell-G. If, then:
Proof. First, we have the ratio
Now, consider
After simplification, we obtain
If , we obtain
Thus, is decreasing in , and hence, . This completes the proof.
3.11. Estimation of Family Parameters
This 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 (MLE). There 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. represent the vector parameters ; then,where and are derivatives of column vectors of the same dimension of , and by setting , and , the MLEs can be yielded by solving the above equations simultaneously.
Proposition 7. A randomly selected sample of sizeis under EBell-G; then, the score vectoris given by
4. 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 [22–24]. 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 and for and , by setting these densities in (7) and (8) yielded the following expression for the proposed EBellE distribution. Then, the cdf and pdf are of the EBellE distribution, respectively.
Proposition 8. LetEBellE, forand; then, its cdf is given by in Eq. (7):
Proposition 9. LetEBellE, forand; then, its pdf is given by in Eq. (8):
The exponential distribution quantile function becomes ; using (9), . The quantile function of x can be expressed as
The sf and the hrf of the EBellE model can be obtained as
4.1. 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),
After applying Eq. (10), it reduces towhere is a exp-exponential density with 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).
4.1.1. The Expression of th Moment
Proposition 10. LetEBellE, forand; then, itsth moment can be written as by taking into account Eq. (25):
By setting = 1 yielded the mean of the EBellE model.
4.1.2. The Expression of th Incomplete Moment
Proposition 11. LetEBellE, forand; then, itsth incomplete moment can be written as by taking into account Eq. (25):
By setting = 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-1 = , S-2 = , S-3 = , S-4 = , S-5 = , and . The following relationship is used to obtain the central moments: , , and . The moment-based measure of skewness and kurtosis is obtained by using and , respectively. Pearson’s coefficient of skewness is simply square root of , and coefficient of kurtosis is computed as . Furthermore, we present the mean, 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.
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4.1.3. The Expression of th Conditional Moment
From actuarial prospective, conditional moments are important; let EBellE be for and ; then, its th conditional moment can be written by using Equation (64):
4.1.4. Two Expression of MGF
Let EBellE for and ; then, its moment generating function by using Wright generalization hypergeometric function is given as
Consider and ; equation (70) is reduced to
By using (70), Equation (71) yielded as
The other representation of mgf is given by
4.1.5. Order Statistics
The sth moment of order statistic can be obtained by using (41):
Simplification yielded the expression of th moments of order statistics:where .
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 and , respectively.
4.1.6. Stochastic Ordering
Let and be the two from EBellE distribution with the assumption previously illustrated in Section 3 given that , and for , shall be decreasing in if and the only if the following results holds:
4.1.7. Rényi Entropy
The Rényi entropy for the EBellE model by using Eq. (22) given under and and :where , and the graphical demonstration of Rényi entropy of EBellE at varying of the parameters is given in Figure 8.
4.1.8. 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 and be the two random variable following the EBellE distribution. The component fails if the applied stress exceeds its strength; if , the component will perform satisfactory. Reliability is defined by the following expression. Here, we derive the reliability of the EBellE model when and have independent and with identical scale and shape parameters. The reliability is given by
By using equations (14) and (15), we get the pdf and the cdf of EBellE, respectively, as follows:
Hence,
Therefore,
By using gamma function, the above expression is reduced towhere .
4.2. Estimation
The log-likelihood function for the vector of parameters for model given in (60) is given by
The components of the score vector are