Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data

Najwan Alsadat; Muhammad Imran; Muhammad H. Tahir; Farrukh Jamal; Hijaz Ahmad; Mohammed Elgarhy

doi:10.1515/phys-2022-0242

Artikel Open Access

Compounded Bell-G class of statistical models with applications to COVID-19 and actuarial data

Najwan Alsadat , Muhammad Imran , Muhammad H. Tahir , Farrukh Jamal , Hijaz Ahmad und Mohammed Elgarhy

Veröffentlicht/Copyright: 27. April 2023

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Abstract

The compounded Bell generalized class of distributions is proposed in this article as an alternative to the compounded Poisson generalized family of distributions. Some properties and actuarial measures are presented. The properties of a special model named Bell Weibull (BellW) are obtained such as the linear representation of density, rth moment, incomplete moment, moment generating function using Wright generalized hypergeometric function and Meijer’s G function, the pth moment of order statistics, reliability, stochastic ordering, and residual and reversed residual life. Moreover, some commonly used entropy measures, namely, Rényi, Havrda and Charvat, and Arimoto and Tsallis entropy are obtained for the special model. From the inferential side, parameters are estimated using maximum likelihood estimation. The simulation study is performed to highlight the behavior of estimates. Some actuarial measures including expected shortfall, value at risk, tail value at risk, tail variance, and tail variance premium for the BellW model are presented with the numerical illustration. The usefulness of the proposed family is evaluated using insurance claims and COVID-19 datasets. Convincing results are obtained.

Keywords: actuarial measures; Bell numbers; compounded family; entropy measures; Poisson generalized family

1 Introduction

The generalization of models having discrete, continuous, bivariate, or multivariate random variables is a well-established activity in statistical research. The compounding methodology is one of the attractive tools that present new models in the form of a mixture or composition of two or more similar or different models by considering the type of random variable(s) and support(s). The compounding from the count phenomenon was considered for Poisson, geometric, logarithmic, binomial, negative-binomial, and power-series discrete random variables. Here, we discuss Poisson-G class whose structure is more similar and comparable to our proposed class of distributions. The development and motivation of Poisson-G for a system based on a series of components can be read in Gomes et al. [1], Tahir and Cordeiro [2], Alghamdi et al. [3], and Maurya and Nadarajah [4]. The detail on the developments on the generalized classes through compounding for discrete and continuous support is described in Tahir and Cordeiro [2]. The cumulative distribution function (CDF) of a Poisson-G class for series structure by considering truncated random variable is given, where λ is the Poisson parameter and K ( ⋅ ) is the CDF of any baseline or parent model:

(1) H ( x ) = 1 − e − λ K ( x ) 1 − e − λ .

A discrete Bell distribution based on Bell numbers [5] and introduced by ref. [6] as an alternative to discrete Poisson distribution (PD), with the probability mass function (PMF) being as follows:

(2) P ( N = n ) = λ n e − e λ + 1 B n n ! ; n = 0 , 1 , 2 , … ,

where B x are the Bell numbers. A discrete Bell distribution has many intriguing characteristics, including being a single parameter distribution and being a member of the exponential family with one parameter, despite the Poisson model’s inability to nest within the Bell model, the Bell model tends toward the PD for small values of the parameter. In addition, the Bell model is infinitely divisible and has a higher variance than the mean, which can be utilized to combat the over-dispersion problems that arise with count data. The Bell model’s qualities led us to develop its generalized class, which we then compared to the compounded Poisson-G class and its specific models.

This article is structured as follows: Section 2 illustrates the construction of the Bell-G family of distribution with some of its important properties such as the quantile function (QF), analytical shapes, linear functional representation of density, the probability-weighted moments, order statistics, entropy measure, and upper record values. Section 3 presents the BellW distribution along with its properties including the two representations of moment generating function (MGF) using Wright generalized hypergeometric function and Meijer’s G function. Section 4 focuses on some well-established actuarial measures namely the expected shortfall (ES) and value at risk (VaR) in the context of BellW distribution. The detailed simulation study is presented in Section 5. Section 6 shows the empirical investigation of the Bell-G family (through its special model) using insurance claims and COVID-19 datasets. Finally, the concluding remarks are presented in Section 7.

2 The layout of the Bell-G family

2.1 Construction

If a system comprises N independent subsystems that are all working at a given specific time. Suppose that the Y i ( i = 1 , 2 , … , ) represents the failure time of the i th subsystem. With parallel structures, the system fails if even one subsystem malfunctions. On the other side, a series system would completely fail if any one of its components stopped working. Assuming that each subsystem’s failure time is followed by the zero truncated Bell distribution with PMF P ( N = n ) :

(3) P ( N = n ) = λ n e 1 − e λ B n n ! ( 1 − e 1 − e λ ) ; n = 1 , 2 , … ,

it is the Bell-distributed random variable’s conditional probability distribution given that the random variable does not take zero values. Let the minimum time of system is denoted X = min { Y 1 , Y 2 , … , Y N } . Then the conditional CDF of X given N is given as follows:

(4) H ( x ∣ N = n ) = 1 − P ( X > x ∣ N = n ) = 1 − [ 1 − K ( x ) ] N ,

and then the unconditional CDF based on Eq. (4) is as follows:

(5) H ( x ) = 1 − ∑ n = 1 ∞ [ 1 − K ( x ) ] n P ( N = n ) ,

where P ( N = n ) denotes the PMF of a zero truncated Bell distribution that is given in Eq. (3).

Proposition 2.1

The expression of CDF of the Bell-G family using Eq. (5) is given by

(6) H ( x ) = 1 − exp ( − e λ [ 1 − e − λ K ( x ) ] ) 1 − exp ( 1 − e λ ) ,

where K ( x ) represents the baseline CDF.

Proof

If N is a zero truncated Bell random variable with PMF given in Eq. (3), using Eq. (5), the CDF of the Bell-G family is given as follows:

(7) H ( x ) = 1 − ∑ n = 1 ∞ [ 1 − K ( x ) ] n λ n e 1 − e λ B n n ! ( 1 − e 1 − e λ ) ,

and the aforementioned expression can also be expressed as follows:

(8) H ( x ) = 1 − e 1 − e λ ( 1 − e 1 − e λ ) ∑ n = 1 ∞ [ λ ( 1 − K ( x ) ) ] n n ! B n .

The aforementioned expression is rewritten as follows:

(9) H ( x ) = 1 − e 1 − e λ ( 1 − e 1 − e λ ) ∑ n = 0 ∞ [ λ ( 1 − K ( x ) ) ] n n ! B n − 1 .

The following series express the functional relationship of Bell numbers, and for more details, readers are referred to Castellares et al. [6]

(10) e e x − 1 = ∑ n = 0 ∞ x n n ! B n .

Comparing Eqs. (9) and (10), we obtain the desire results of Proposition 2.1. This completes the proof.□

The PDF corresponding to Eq. (6) is given by

(11) h ( x ) = λ k ( x ) exp ( λ [ 1 − K ( x ) ] ) exp ( − e λ [ 1 − e − λ K ( x ) ] ) 1 − exp ( 1 − e λ ) ,

where K ( x ) and k ( x ) are the baseline distribution’s CDF and PDF, respectively. It is of interest for the reasons using the Bell-G family listed below:

It has an original functional structure and contained the feature of Bell numbers.
The proposed family adds only one additional parameter. Moreover, the proposed family yields a better fit compared to the counterpart well-established commonly used the Poisson-G.
Further, the PDF of the special BellW model can be expressed as a linear combination of the Weibull PDFs, and this property helps to easily obtain several properties directly from the Weibull distribution.
In addition, very few three-parameter distributions possess flexible shapes of hazard rate function (HRF) that are frequently encountered in various practical domains.
The proposed family provides better goodness-of-fit measures for highly skewed data, particularly those from the actuarial side, and is fitted successfully (with improved P -value as in data-2).

The survival function (SF) and HRF, which are the two important properties commonly used in reliability and survival analysis, are given, respectively,

(12) SF ( x ) = exp ( − e λ [ 1 − e − λ K ( x ) ] ) − exp ( 1 − e λ ) 1 − exp ( 1 − e λ )

and

(13) HRF ( x ) = λ k ( x ) exp ( λ [ 1 − K ( x ) ] ) exp ( − e λ [ 1 − e − λ K ( x ) ] ) exp ( − e λ [ 1 − e − λ K ( x ) ] ) − exp ( 1 − e λ ) .

2.2 Quantile function

The QF is an important statistical measure that is used to obtain the median and the random numbers generation of the distribution. It has several other uses in various empirical and theoretical domains including quality control (acceptance sampling), finance, and actuarial sciences. The QF of the Bell-G family is given by

(14) Q ( u ) = K − 1 1 − 1 λ { ln [ ln ( 1 − u { Q ∗ } ) + e λ ] } ,

where 0 ≤ u ≤ 1 , Q ∗ = 1 − exp ( 1 − e λ ) , and K − 1 ( x ) is a inverse function of K ( x ) , that is, the QF of the baseline distribution. Moreover, the L-moments can be obtained by the following expressions: L 1 = ∫ 0 1 Q ( u ) d u , L 2 = ∫ 0 1 Q ( u ) ( 2 u − 1 ) d u , L 3 = ∫ 0 1 Q ( u ) ( 6 u 2 − 6 u + 1 ) d u and L 4 = ∫ 0 1 Q ( u ) ( 20 u 3 − 30 u 2 + 12 u − 1 ) d u .

2.3 Analytic shapes

Here, we provide analytical information regarding the PDF and HRF of the Bell-G family. The critical points are the solution to the following equations:

∂ log h ( x ) ∂ x = ∂ k ( x ) / ∂ x k ( x ) − λ k ( x ) − λ k ( x ) exp ( λ [ 1 − K ( x ) ] ) = 0

and

∂ log HRF ( x ) ∂ x = ∂ k ( x ) / ∂ x k ( x ) − λ k ( x ) = 0 ,

where k ( x ) and K ( x ) are the baseline PDF and CDF, respectively.

2.4 Useful expansions

Here, we develop a helpful expansion of the Bell-G densities that can be utilized to drive several significant features of X .

Proposition 2.2

A linear functional representation of the PDF and CDF is given by

(15) h ( x ) = ∑ ω = 0 ∞ v ω f ω + 1 ( x )

and

(16) H ( x ) = ∑ ω = 0 ∞ v ω F ω + 1 ( x ) ,

respectively, where

v ω = λ 1 + ω ( ω + 1 ) − 1 [ 1 − exp ( 1 − e λ ) ] ω ! × ∑ σ , Φ = 0 ∞ 1 σ ! ( − 1 ) σ + Φ + ω σ Φ ( 1 + Φ ) ω e λ ( 1 + σ ) ,

f ω + 1 ( x ) = ( ω + 1 ) k ( x ) K ( ω + 1 ) − 1 ( x ) and F ω + 1 ( x ) = K ( ω + 1 ) ( x ) , respectively, are the PDF and CDF of the exp-G family with power parameter ( ω + 1 ) .

Proof

Here, we use the generalized binomial expansion, which is valid for any real noninteger z and ∣ y ∣ < 1 and is given as follows:

(17) ( 1 − y ) z = ∑ v = 0 ∞ ( − 1 ) v z v y v .

As employed by Bourguignon et al. [7], the power series for the exponential functions is given as follows:

(18) exp ( − c x m ) = ∑ r = 0 ∞ ( − 1 ) r c r x r m r ! ,

for any real numbers c , m , and x . Using Eq. (18) to the last term of Eq. (11) yields

(19) exp { − e λ [ 1 − exp ( − λ K ( x ) ) ] } = ∑ σ = 0 ∞ ( − 1 ) σ σ ! { e λ [ 1 − exp ( − λ K ( x ) ) ] } σ .

Using Eq. (17), Eq. (11) becomes

h ( x ) = λ k ( x , ϖ ) e λ { 1 − exp [ 1 − e λ ] } ∑ σ = 0 ∞ ∑ r = 0 ∞ × ( − 1 ) σ + r σ ! σ r e λ σ exp { − λ K ( x ) [ 1 + r ] } ,

after simplification, Eq. (11) reduces as follows:

h ( x ) = λ 1 + ω ( 1 + ω ) − 1 { 1 − exp [ 1 − e λ ] } ω ! ∑ ω = 0 ∞ ∑ σ , Φ = 0 ∞ × ( − 1 ) σ + Φ + ω σ ! σ Φ e λ ( σ + 1 ) ( 1 + Φ ) ω × ( ω + 1 ) k ( x ) K ( ω + 1 ) − 1 ( x ) .

For h ( x ) , the desired expansion is obtained. The expression for H ( x ) is obtained upon integral. This completes the proof of proposition 2.2.□

The constant term ∑ ω = 0 ∞ v ω = 1 , and by using Eq. (15), numerous properties of X that those from exp-G can be derived. Most computational tools such as Mathematica, Maple, Matlab, and MathCad can correctly deal with the formulas derived in this article.

2.5 Mathematical properties

We present here some mathematical properties of the Bell-G family from Eq. (15) and those properties of the exp-G distribution. The μ r ′ represents the r th ordinary or raw moment and as follows:

(20) μ r ′ = E ( X r ) = ∑ ω = 0 ∞ v ω E ( X ω + 1 r ) ,

where E ( X ω + 1 r ) = ( ω + 1 ) ∫ − ∞ + ∞ x r k ( x ) K ω ( x ) d x and the first four raw moments of the Bell-G can be derived by taking r = 1 , 2 , 3 , 4 , respectively, in Eq. (20). In statistics, important distributional properties can be assessed directly through moments. It is used to underline the measures of tendency and dispersion, asymmetric (skewness), and kurtosis of the distribution. The s th incomplete moment of a random variable X is given by

(21) μ s ( x ) = ∑ ω = 0 ∞ v ω J ω + 1 ( x ) ,

where the expression of s th incomplete moment J ω + 1 ( x ) = ∫ − ∞ t x s f ω + 1 ( x ) d x . It has important uses in the computation of Bonferroni and Lorenz curves, conditional moments, and residual and reversed residual life.

2.6 Probability weighted moments (PWMs)

The concept of PWMs was first presented by Greenwood et al. [8]. It is the expectation of function with existing mean for any random variable and can be expressed as follows:

ρ s , r = ∫ − ∞ + ∞ X s h ( x ) H ( x ) r d x .

Proposition 2.3

The linear representation of PWMs is given by

ρ s , r = ∑ d = 0 ∞ w d E ( X ( 1 + d ) s ) ,

where E [ X ( 1 + d ) s ] = ( 1 + d ) ∫ − ∞ + ∞ x s k ( x , ϖ ) K ( 1 + d ) − 1 ( x , ϖ ) d x , and

w d = λ exp ( λ ) { 1 − exp [ 1 − e λ ] } 1 + r ∑ Θ , v , p = 0 ∞ ( − 1 ) Θ + p + v + d × [ ( 1 + Θ ) e λ ] v v ! ( 1 + d ) d ! r Θ v p { [ p + 1 ] λ } d .

Proof

Consider I = H ( x ) r h ( x ) and using Eq. (17), I becomes

I = h ( x ) { 1 − exp [ 1 − e λ ] } r ∑ Θ = 0 ∞ ( − 1 ) Θ r Θ × exp { − Θ e λ [ 1 − e − λ K ( x ) ] } .

By using Eq. (18), we obtain

I = λ k ( x ) exp { λ [ 1 − K ( x ) ] } { 1 − exp [ 1 − e λ ] } 1 + r ∑ Θ = 0 ∞ ( − 1 ) Θ r Θ × exp { − e λ [ 1 − e − λ K ( x ) ] ( 1 + Θ ) } .

By using Eqs. (17) and (18), I reduces as follows:

I = λ exp ( λ ) { 1 − exp [ 1 − e λ ] } 1 + r ∑ Θ , v , p = 0 ∞ ∑ d = 0 ∞ ( − 1 ) Θ + v + p + d × [ ( 1 + Θ ) e λ ] v v ! ( 1 + d ) d ! × r Θ v p { λ [ 1 + p ] } d ( 1 + d ) k ( x ) [ K ( x ) ] ( 1 + d ) − 1 .

Hence,

I = ∑ d = 0 ∞ w d f ( 1 + d ) ( x ) .

This ends the proof of proposition 2.3.□

2.7 Order statistics

The expression of i th order statistics of the Bell-G family is obtained here, say f i : n ( x ) , which is based on a random sample of size n taken from the Bell-G and as follows:

f i : n ( x ) = 1 B ( i , n − i + 1 ) ∑ l = 0 n − i ( − 1 ) l n − i l h ( x ) H ( x ) i + l − 1 ,

where B ( . , . ) is a beta function.

Proposition 2.4

The linear expansion of ith order statistics is given by

(22) f i : n ( x ) = ∑ u = 0 ∞ Q i : n ( u ) f u + 1 ( x ) ,

where

Q i : n ( u ) = λ ( 1 + u ) ( − 1 ) u e λ B ( i , n − i + 1 ) ( u + 1 ) u ! [ 1 − exp ( 1 − e λ ) ] i + l × ∑ l = 0 n − i ∑ ψ = 0 i + l − 1 ∑ s , i ∞ e λ s s ! ( − 1 ) ψ + s + i + l i + l − 1 ψ × s i n − i l ( i + 1 ) u ( ψ + 1 ) s .

Proof

Consider I as follows:

I = h ( x ) [ 1 − exp { − e λ [ 1 − e − λ K ( x ) ] } ] i + l − 1 [ 1 − exp [ 1 − e λ ] ] i + l − 1 .

Using Eqs. (17) and (18), after simplifications, yields

I = λ k ( x ) exp { λ [ 1 − K ( x ) ] } { 1 − exp [ 1 − e λ ] } i + l ∑ ψ = 0 ( i + l − 1 ) ∑ s = 0 ∞ ( − 1 ) s ! ψ + s × i + l − 1 ψ [ e λ ( 1 + ψ ) ] s [ 1 − e − λ K ( x ) ] s .

Hence,

I = λ exp ( λ ) { 1 − exp [ 1 − e λ ] } i + l ∑ ψ = 0 ( i + l − 1 ) ∑ s = 0 ∞ ∑ i = 0 ∞ ∑ u = 0 ∞ ( − 1 ) s ! u ! ψ + s + i + u × i + l − 1 ψ s i [ e λ ( 1 + ψ ) ] s × [ λ ( 1 + i ) ] u ( u + 1 ) k ( x ) K ( u + 1 ) − 1 ( x ) .

This completes the proof of Proposition 2.4.□

The expression of p th moment is given by

(23) E ( X i : n p ) = ∑ u = 0 ∞ Q i : n ( u ) μ u + 1 ( p ) ,

where μ u + 1 ( p ) denotes the p th moment of the exp-G distribution with power parameter ( u + 1 ) that is μ u + 1 ( p ) = ( u + 1 ) ∫ 0 ∞ x p k ( x ) K ( u + 1 ) − 1 ( x ) d x .

2.8 Upper record statistics

In many real-world contexts, such as economic data, weather, and sporting events, record value is a crucial measure. Let us consider ( X n ) n ≥ 1 to be a series of distinct independent random variables that have the same distribution. Let X i : n be the previously described i th order statistic and F ( x ) and f ( x ) be the respective CDF and PDF of the BellW distribution. The k th upper record statistic [9] is determined by the following PDF for fixed k ≥ 1 :

(24) f Y n ( k ) ( x ) = k ! ( n − 1 ) ! h ( x ) [ 1 − H ( x ) ] k − 1 [ R ( x ) ] n − 1 ,

where the cumulative HRF associated with H ( x ) represented by R ( x ) = − ln [ 1 − H ( x ) ] . By inserting Eq. (6) into Eq. (24), we have

f Y n ( k ) ( x ) = k ! e λ ( n − 1 ) ( n − 1 ) ! ∑ t = 0 k − 1 ( − 1 ) t k − 1 t [ 1 − e − λ K ( x ) ] n − 1 × h ( x ) H ( x ) t .

Proposition 2.5

The useful expansion of the upper record statistics is given by

f Y n ( k ) ( x ) = ∑ q = 0 ∞ W q f q + 1 ( x ) ,

where

W q = k ! e λ n λ ( q + 1 ) ( n − 1 ) ! ∑ t = 0 k − 1 ∑ s , v , p = 0 ∞ ( − 1 ) s + p + q + v + t v ! q ! × { 1 − exp [ 1 − e λ ] } − ( t + 1 ) t s v + n − 1 p × k − 1 t [ ( s + 1 ) e λ ] v [ ( p + 1 ) λ ] q .

Proof

By setting Eqs. (6) and (11) in Eq. (25) and applying Eq. (17), we obtain

(25) I = h ( x ) [ 1 − e − λ K ( x ) ] n − 1 H ( x ) t ,

I = [ 1 − e − λ K ( x ) ] n − 1 λ k ( x ) exp { λ [ 1 − K ( x ) ] } { 1 − exp [ 1 − e λ ] } t + 1 × ∑ s = 0 ∞ ( − 1 ) s t s exp { − e λ [ 1 − e − λ K ( x ) ] ( s + 1 ) } .

By using Eq. (18), I reduces to as follows:

I = λ e λ ( q + 1 ) { 1 − exp [ 1 − e λ ] } t + 1 ∑ q = 0 ∞ ∑ s , v , p = 0 ∞ ( − 1 ) s + p + v + q v ! q ! × t s v + n − 1 p × [ e λ ( 1 + s ) ] v [ λ ( 1 + p ) ] q ( 1 + q ) k ( x ) K ( x ) ( 1 + q ) − 1 .

The desired expansion of f Y n ( k ) ( x ) is obtained. This completes the proof of Proposition 2.5.□

From Eq. (33), a random sample of 50 is taken by the BellW distribution by setting α = β = λ = 0.5 and k = 3 . Table 1 shows the values for the upper X U ( n ) and lower X L ( n ) records with graphical representation in Figures 1 and 2. This indicates that the BellW model can effectively be used for records value applications. The X U ( n ) and X L ( n ) records value are computed using the R package Records .

Table 1

The upper and lower record values based on the BellW model

k = 3 ; α = β = λ = 0.5 ; n = 50					X U ( n )	X L ( n )
2.35861	3.59107	7.12951	2.56466	1.04267	1.18968	2.35861
1.18968	2.81786	3.23393	0.31025	0.83680	1.83411	1.83411
1.83411	0.38332	5.20245	0.59538	0.04583	2.35861	1.18968
4.05200	3.05768	1.43486	8.28864	4.50876	3.23838	1.03831
0.28231	1.44788	1.71064	1.80866	0.92769	3.59107	1.02814
0.82206	1.84551	4.21741	4.77464	0.56184	4.05200	0.82206
1.03831	0.16857	6.44734	6.09677	0.05419	5.20245	0.38332
1.02814	0.99133	0.88274	11.3902	2.21055	6.39700	0.31025
3.23838	1.67036	5.13104	3.96194	0.34511	6.44734	0.28231
1.85069	6.39700	1.43658	2.30952	5.67188	7.12951	0.16857

Figure 1

Graphical illustration of upper record values based on the BellW distribution.

Figure 2

Graphical illustration of lower record values based on the BellW distribution.

Corollary 2.1

Let δ > 0 . Then the following expression holds:

h ( x ) δ = ∑ b = 0 ∞ Q b ∫ − ∞ + ∞ k ( x ) δ K ( x ) b d x ,

where

(26) Q b = e δ λ λ ( δ + b ) [ 1 − exp ( 1 − e λ ) ] δ b ! ∑ t , s = 0 ∞ 1 t ! ( − 1 ) s + t + b × t s ( δ + s ) b ( δ e λ ) t .

Corollary 2.1 provides mathematical expression of complex measures, such as entropy measures, which are the subject of the next part.

2.9 Entropy measures

Entropy measurements are crucial for highlighting the unpredictability, uncertainty, or diversity of the system. For a complete review, we may refer to ref. [10]. The Rényi (R) entropy is the index of dispersion that is most frequently used in statistics and ecology. Some other well-known entropy measures are the Havrda and Charvat (HC) entropy, the Arimoto (A) entropy and the Tsallis (T) entropy.

The Rényi entropy: It is given by

R δ ( x ) = ( 1 − δ ) − 1 log ∫ − ∞ + ∞ h ( x ) δ d x ,

where δ ≠ 1 and δ > 0 . In the context of the Bell-G family, by using Corollary 2.1, we can expand it as follows:

(27) R δ ( x ) = 1 ( 1 − δ ) log ∑ b = 0 ∞ Q b ∫ − ∞ + ∞ k ( x ) δ K ( x ) b d x ,

where Q b is defined in (26). For a given continuous parent distribution, the remaining integral term is calculable or can be found in many references dealing with exp-G family.

The Havrda and Charvat entropy: The Havrda and Charvat entropy of an absolutely continuous distribution having PDF h ( x ) can be expressed as follows:

HC δ ( x ) = 1 2 1 − δ − 1 ∫ − ∞ + ∞ h ( x ) δ d x − 1 .

By using Corollary 2.1, the aforementioned expression further can be expressed as follows:

(28) HC δ ( x ) = 1 2 1 − δ − 1 ∑ b = 0 ∞ Q b ∫ − ∞ + ∞ k ( x ) δ K ( x ) b d x − 1 .

The Arimoto entropy: The Arimoto entropy of an absolutely continuous distribution having PDF h ( x ) can be expressed as follows:

A δ ( x ) = δ 1 − δ ∫ − ∞ + ∞ h ( x ) δ d x 1 δ − 1 .

By using Corollary 2.1, the aforementioned expression further can be expressed as follows:

(29) A δ ( x ) = δ 1 − δ ∑ b = 0 ∞ Q b ∫ − ∞ + ∞ k ( x ) δ K ( x ) b d x 1 δ − 1 .

The Tsallis entropy: It is given by

T δ ( x ) = 1 δ − 1 1 − ∫ − ∞ + ∞ h ( x ) δ d x .

By using Corollary 2.1, the above expression further can be expressed as follows:

(30) T δ ( x ) = 1 δ − 1 1 − ∑ b = 0 ∞ Q b ∫ − ∞ + ∞ k ( x ) δ K ( x ) b d x .

2.10 Stochastic ordering

Another crucial statistical tool for highlighting comparative behavior, particularly in reliability analysis, is stochastic ordering. The readers are referred to ref. [11] for a full demonstration of four stochastic ordering and their well-established relationship. Suppose that the two random variables, say X 1 and X 2 , and under certain circumstances, let say the random variable X 1 is lower than X 2

Proposition 2.6

Let X 1 ∼ Bell-G ( λ 1 ) and X 2 ∼ Bell-G ( λ 2 ) . If λ 1 ≤ λ 2 , then X 1 ≤ l r X 2

Proof

The ratio of pdf is given by

h 1 ( x ) h 2 ( x ) = λ 1 exp ( − e λ 1 [ 1 − e − λ 1 K ( x ) ] ) exp { λ 1 [ 1 − K ( x ) ] } C 1 λ 2 exp ( − e λ 2 [ 1 − e − λ 2 K ( x ) ] ) exp { λ 2 [ 1 − K ( x ) ] } C 2 ,

where C 1 = { 1 − exp [ 1 − e λ 1 ] } − 1 and C 2 = { 1 − exp [ 1 − e λ 2 ] } − 1 . If λ 1 < λ 2 , we obtain

d d x ln h 1 ( x ) h 2 ( x ) = k ( x ) { ( λ 2 − λ 1 ) + λ 2 [ e λ 2 [ 1 − K ( x ) ] ] − λ 1 [ e λ 1 [ 1 − K ( x ) ] ] } < 0 .

Thus, h 1 ( x ) h 2 ( x ) is decreasing in x and hence X 1 ≤ l r X 2 . □

2.11 Estimation

In this section, we utilize the widely used estimation method called maximum likelihood estimation (MLE) to demonstrate parameter estimation. Even when employing a finite sample, the MLEs offer simple approximations that are extremely accessible, and they satisfy the desired properties that can be employed to produce confidence intervals in addition to MLEs. These are just a few advantages that MLEs have over other estimation strategies. The log-likelihood function ℓ ( ⋅ ) for vector parameters ϕ = ( λ , ϖ ⊤ ) ⊤ , where ϖ is a ( p × 1 ) baseline parameter vector is given by

ℓ ( ϕ ) = n ln λ + ∑ i = 1 n ln k ( x i ; ϖ ) + λ ∑ i = 1 n [ 1 − K ( x i ; ϖ ) ] − n e λ + ∑ i = 1 n exp [ λ ( 1 − K ( x i ; ϖ ) ) ] − n ln [ 1 − exp ( 1 − e λ ) ] .

There are several R packages, namely, MaxLik , bbmle , optim , and AdequacyModel that can be employed to maximize the aforementioned equation. These R packages are user friendly and can easily be operated and provide detailed output accuracy measures and goodness-of-fit test summary, and among all others, the Adequac𝗒 Model is the most frequently used package to estimate the model parameters. In R, one can easily install and load the package by using the command install.packages (“ AdequacyModel ”) and then load from library by using command library (“ AdequacyModel ”). The component of the score vector is given by U ( ϕ ) = ∂ ℓ ∂ ϕ = ∂ ℓ ∂ λ , ∂ ℓ ∂ ϖ ⊤

∂ ℓ ∂ λ = n λ − 1 + ∑ i = 1 n [ 1 − K ( x i ; ϖ ) ] − n e λ + ∑ i = 1 n exp [ λ ( 1 − K ( x i ; ϖ ) ) ] ( 1 − K ( x i ; ϖ ) ) − n e λ exp ( 1 − e λ ) 1 − exp ( 1 − e λ ) , ∂ ℓ ∂ ϖ = ∑ i = 1 n k ′ ( x i ; ϖ ) k ( x i ; ϖ ) − λ K ′ ( x i ; ϖ ) − λ ∑ i = 1 n exp [ λ ( 1 − K ( x i ; ϖ ) ) ] K ′ ( x i ; ϖ ) .

By setting ∂ ℓ ∂ λ = 0 and ∂ ℓ ∂ ϖ = 0 , solution of the aforementioned expressions yields the MLEs, where k ′ ( x i ; ϖ ) = ∂ ∂ ϖ k ( x i ; ϖ ) and K ′ ( x i ; ϖ ) = ∂ ∂ ϖ K ( x i ; ϖ ) are the partial derivatives of the same dimension column vectors of ϖ .

3 The BellW distribution

3.1 definition

Here, a special BellW distribution is defined by using Weibull as a baseline model with the following baseline CDF and PDF: K ( x ) = 1 − exp ( − α x β ) and k ( x ) = α β x ( β − 1 ) exp ( − α x β ) , respectively, where x > 0 and α , β > 0 . Then the CDF and PDF of the BellW distribution are given by

(31) H ( x ; λ , α , β ) = 1 − exp [ − e λ ( 1 − e − λ [ 1 − exp ( − α x β ) ] ) ] 1 − exp ( 1 − e λ )

and

(32) h ( x ; λ , α , β ) = λ α β x ( β − 1 ) exp ( − α x β ) × exp ( λ [ exp ( − α x β ) ] ) × exp [ − e λ ( 1 − e − λ [ 1 − exp ( − α x β ) ] ) ] × [ 1 − exp ( 1 − e λ ) ] − 1 ,

respectively. The BellW distribution reduces to the Bell exponential (BellE) if β = 1 in the aforementioned expressions. The graphical illustration of the PDF and HRF of the BellW distribution are shown in Figures 3 and 4 at some parametric values. The BellW distribution accommodates right-skewed, symmetrical, and reversed J-shaped. The shapes of HRF can be increasing, decreasing, constant, bathtub, increasing-decreasing-increasing and unimodal. The shapes of PDF and HRF exhibit enough flexibility to fit a large variety of practical datasets particularly heavily tailed. It stands apart from the BellW distribution from other extended Weibull distributions. The effect of parameter λ and β can be viewed in Figure 5 and shows that by increasing the value of λ and β , the mean, variance, skewness, and kurtosis tend to reduce.

Figure 3

Graphical illustration of PDF based on the BellW distribution.

Figure 4

Graphical illustration of HRF based on the BellW distribution.

Figure 5

A Plot of mean, variance, skewness, and kurtosis based on the BellW model for some parameteric values.

The QF of the BellW distribution is as follows:

(33) Q ( u ) = − 1 α ln 1 λ { ln [ ln ( 1 − u { Q ∗ } ) + e λ ] } 1 / β ,

where Q ∗ = 1 − exp ( 1 − e λ ) and [ 0 ≤ u ≤ 1 ] , for u = 0.5 , Eq. (33) yields the median of the BellW distribution. The SF and HRF of the BellW distribution are given by

SF ( x ) = exp { − e λ [ 1 − e − λ ( 1 − exp ( − α x β ) ) ] } − exp [ 1 − e λ ] ( 1 − exp [ 1 − e λ ] ) − 1 ,

and

HRF ( x ) = λ [ α β x ( β − 1 ) exp ( − α x β ) ] × exp { λ [ 1 − ( 1 − exp ( − α x β ) ) ] } × exp [ − e λ { 1 − e − λ ( 1 − exp ( − α x β ) ) } ] × A ,

respectively, and

A = 1 exp [ − e λ { 1 − e − λ ( 1 − exp ( − α x β ) ) } ] − exp ( 1 − e λ ) .

To gain valuable model properties, we shall first derive a linear representation of the BellW PDF, using Eq. (15)

(34) h ( x ) = ∑ ω = 0 ∞ ( ω + 1 ) [ α β x ( β − 1 ) exp ( − α x β ) ] { 1 − exp ( − α x β ) } ω ,

which is a PDF of the exp-Weibull, and after applying Eq. (17) to Eq. (34), it reduces to

(35) ∑ p = 0 ∞ t p π ( x ; α ( 1 + p ) , β ) ,

where t p = ( − 1 ) p ( p + 1 ) ω p ∑ ω = 0 ∞ v ω ( ω + 1 ) and π ( x ; α ( 1 + p ) , β ) is a Weibull PDF. Since the BellW PDF is a linear combination of Weibull densities, it helps in extracting the numerous properties of the BellW distribution directly from the Weibull distribution. The expression of the r th ordinary moments is as follows:

(36) μ r ′ = Γ r β + 1 ∑ p = 0 ∞ t p [ α ( p + 1 ) ] r / β .

Table 2 displays the ordinary moments ( μ 1 ′ , μ 2 ′ , μ 3 ′ , μ 4 ′ ) , the mean or actual moments ( μ 2 , μ 3 , μ 4 ) , variance ( σ 2 ), coefficient of variation (CV), Pearson’s coefficient of skewness (CS), and Pearson’s coefficient of kurtosis (CK) of the BellW distribution taking different combinations of parametric values, S 1 = [ α = 1.50 , β = 1.0 , λ = 0.20 ] ; S 2 = [ α = 1.50 , β = 1.0 , λ = 1.20 ] ; S 3 = [ α = 1.50 , β = 1.50 , λ = 0.20 ] ; S 4 = [ α = 2.50 , β = 3.85 , λ = 1.20 ] ; S 5 = [ α = 1.50 , β = 3.85 , λ = 1.20 ] and S 6 = [ α = 1.0 , β = 1.70 , λ = 1.0 ] [12]. To acquire the actual moments, the known relationship between actual and raw moments is applied. We obtain the moment-based skewness and kurtosis measures using β 1 = μ 3 2 μ 2 3 and β 2 = μ 4 μ 2 2 , respectively. The square root of β 1 is used to calculate Pearson’s CS, and β 2 − 3 is used to calculate the CK. Based on Table 2, the CK and CS values show that the BellW distribution is platykurtic, leptokurtic, and right-skewed.

Table 2

Dispersion measures (DMs) of the BellW model for some parametric values

DMs	S 1	S 2	S 3	S 4	S 5	S 6
μ 1 ′	0.5985	0.2400	0.6362	0.5209	0.6554	0.5388
μ 2 ′	0.7561	0.1836	0.6088	0.3043	0.4630	0.4585
μ 3 ′	1.4725	0.2847	0.7561	0.1958	0.3490	0.5394
μ 4 ′	3.8753	0.6797	1.1349	0.1372	0.2788	0.8056
σ 2	0.3979	0.1260	0.2041	0.0330	0.0335	0.1682
σ	0.6308	0.3549	0.4518	0.1816	0.1830	0.4101
CV	1.0538	1.4789	0.7101	0.3486	0.2792	0.7612
μ 2	0.3979	0.1260	0.2041	0.0330	0.0335	0.1682
μ 3	0.5436	0.1802	0.1091	0.0030	0.0017	0.1112
μ 4	1.5903	0.4599	0.1978	0.0036	0.0035	0.1888
CK	7.0464	25.9834	1.7491	0.3570	0.1350	3.6744
CS	2.1662	4.0306	1.1837	0.5017	0.2799	1.6115

3.2 Moment generating function

Two representation of the MGF is presented, let X be a random variable with PDF associated to Eq. (32) and M ( t ) = E [ exp ( t x ) ] . Here, we illustrate the MGF by using the Wright generalized hypergeometric function given in Eq. (37) and consider I k ( t ) = ∫ x β − 1 exp [ − ( ζ k x ) β ] d x , whereas π [ ζ k , β ] is a Weibull PDF and ζ k = [ α ( 1 + p ) ] 1 β

(37) Ψ q p ( α 1 , A 1 ) , … … , ( α p , A p ) ( β 1 , B 1 ) , … … , ( β p , B p ) ; x = ∑ n = 0 ∞ Π j = 1 p Γ ( α j + A j n ) Π j = 1 q Γ ( β j + B j n ) x n n ! ,

(38) M ( t ) = ∑ p = 0 ∞ t p ∗ I k ( t ) ,

where t p ∗ = β ( ζ k ) 1 / β ( − 1 ) p ω p ∑ ω = 0 ∞ v ω ( ω + 1 ) .

(39) I k ( t ) = ∑ m = 0 ∞ t m m ! ∫ 0 ∞ x m + β − 1 exp { − ( ζ k x ) β } d x = 1 β ζ k β ∑ m = 0 ∞ [ t / ζ k ] m m ! Γ m β + 1 .

By comparing Eq. (37) to Eq. (39), we obtain

I k ( t ) = 1 β ζ k β Ψ 0 1 1 , 1 / β − ; t ζ k .

Given β > 1 , now Eq. (38) becomes

(40) M ( t ) = ∑ p = 0 ∞ t p ∗ 1 β ζ k β Ψ 0 1 1 , 1 / β − ; t ζ k .

Proposition 3.1

The second representation of MGF is based on Meijer’s G function:

M x ( s ) = ∑ p = 0 ∞ t p α ( 1 − p ) 1 [ − s ] p / q ( 2 π ) 1 − p + q 2 × q ( − 1 / 2 ) p ( p / q − 1 / 2 ) × G p , q q , p − p s p α ( 1 − p ) q q ∣ 1 − i + p / q p , i = 0 , 1 , … , p − 1 j / q , j = 0 , 1 , … , q − 1 .

Proof

The Meijer’s G function is given by

(41) G p , q n , m z ∣ a 1 , … , a p b 1 , … , b q = 1 2 π i ∫ L { ∏ j = 1 m Γ ( b j + s ) } { ∏ j = 1 n Γ ( 1 − a j − s ) } { ∏ j = m + 1 q Γ ( a j + s ) } { ∏ j = n + 1 p Γ ( 1 − b j − s ) } z − s d s ,

where L represents an integration path and i = − 1 is a complex unit. The MGF of X is given as follows:

(42) M x ( s ) = ∫ e s x h ( x ) d x

and

(43) M x ( s ) = ∑ p = 0 ∞ t p α ( 1 − p ) β ∫ 0 ∞ e s x x β − 1 e − α ( 1 − p ) x β d x .

Consider

(44) I = ∫ 0 ∞ e s x x β − 1 e − α ( 1 − p ) x β d x ,

we now display that the integral is proportional to the PDF of the ratio of the random variable X 1 and X 2 , i.e., g 1 ( x 1 ) = c 1 e − x 1 β and g 2 ( x 2 ) = c 2 e s x 2 x 2 β − 2 , where the normalizing constants are c 1 and c 2 . Let u = x 1 x 2 and v = x 2 , so that x 1 = u v and x 2 = v . Equation (44) can be simplified by using inverse Mellin transfer technique and yields

(45) h 1 ( u ) = c 1 c 2 β [ − s ] β 1 2 π i ∫ c − u s − t Γ ( t / β ) Γ ( β − t ) d t ,

when β is a rational number β = p / q in Eq. (45) by setting z = t / p and using Gauss–Legendre multiplication formula

(46) Γ ( q z ) = ( 2 π ) 1 − q 2 q ( q z − 1 / 2 ) ∏ j = 0 q − 1 Γ j q + z

and

(47) Γ ( p / q − p z ) = ( 2 π ) 1 − p 2 p ( p / q − p z − 1 / 2 ) ∏ i = 0 p − 1 Γ i + p / q p − z .

Hence,

I = c 1 c 2 [ − s ] p / q ( 2 π ) 1 − p + q 2 q ( 1 / 2 ) p ( p / q − 1 / 2 − 1 ) × 1 2 π i ∫ c − u p s p q − q − z × ∏ j = 0 q − 1 Γ j q + z ∏ i = 0 p − 1 Γ i + ξ + p / q p − z d z ,

and by using Eq. (41), this completes the proof of proposition 3.1.□

The expression of s th incomplete moment of Bellw model is given by setting s = 1 in Eq. (48), yields the first incomplete moment of Bellw model:

(48) μ s ( x ) = ∑ p = 0 ∞ t p [ α ( 1 + p ) ] − s / β γ s β + 1 , α ( 1 + p ) x β .

3.3 The p th moment

The pth moment is given by using Eq. (23)

E ( X i : n p ) = ∑ u = 0 ∞ Q i : n ( u ) ( u + 1 ) × ∫ 0 ∞ x p k ( x , ϖ ) K ( u + 1 ) − 1 ( x , ϖ ) d x .

Hence

(49) E ( X i : n p ) = ∑ v = 0 ∞ t v Γ p β + 1 1 [ α ( v + 1 ) ] p β ,

where t v = ∑ u = 0 ∞ Q i : n ( u ) ( − 1 ) v ( v + 1 ) − 1 u v ( u + 1 ) .

3.4 Entropy measures

We now aim to provide the four expressions of the entropy measures, previously introduced in full generality for the Bell-G family, for the BellW distribution.

The Rényi entropy: Let X ∼ a BellW ( λ , α , β ) distribution, using (27), and the Rényi entropy is expressed as follows:

R δ ( x ) = ∑ i = 0 ∞ Q i E i ∗ ,

where

E i ∗ = α δ β δ − 1 [ α ( i + δ ) ] δ − δ β + 1 β Γ δ − δ β + 1 β ,

and Q i = ∑ b = 0 ∞ Q b ( − 1 ) i b i , and Q b is presented in Corollary 2.1.

The Havrda and Charvat entropy: Based on Eq. (28), it is given as follows:

HC δ ( x ) = 1 2 1 − δ − 1 ∑ i = 0 ∞ Q i E i ∗ − 1 .

The Arimoto entropy: Based on Eq. (29), it is given by

A δ ( x ) = δ 1 − δ ∑ i = 0 ∞ Q i E i ∗ 1 δ − 1 .

The Tsallis entropy: Based on Eq. (30), it is given by

T δ ( x ) = 1 δ − 1 1 − ∑ i = 0 ∞ Q i E i ∗ .

3.5 Reliability

In the fields of engineering, physical science, and economics, reliability has been used in a variety of applications. With the help of reliability, we may estimate the likelihood of failure at a specific period. If X 1 and X 2 are the two random variables that follow the BellW distribution, the component will function satisfactorily if the applied stress does not exceed its strength. The following expression defines reliability. If X 1 and X 2 having independent h ( x ; λ 1 , α , β ) and H ( x ; λ 2 , α , β ) with the identical scale ( α ) and shape ( β ) parameters, the BellW model’s reliability is determined as follows:

R = ∫ 0 ∞ h 1 ( x ) H 2 ( x ) d x .

By using Eqs. (15) and (16), we obtain

h ( x ; λ 1 , α , β ) = ∑ ω = 0 ∞ v ω ( λ 1 ) ( ω + 1 ) × { α β x β − 1 exp ( − α x β ) } × { 1 − exp ( − α x β ) } ω ,

and

H ( x ; λ 2 , α , β ) = ∑ t = 0 ∞ v t ( λ 2 ) { 1 − exp ( − α x β ) } t + 1 .

Hence,

R = ∑ ω = 0 ∞ v ω ( λ 1 ) ( ω + 1 ) ∑ t = 0 ∞ v t ( λ 2 ) Ξ ( α , β , v , t ) ,

where

Ξ ( α , β , ω , t ) = ∫ 0 ∞ { α β x β − 1 exp ( − α x β ) } × { 1 − exp ( − α x β ) } ω + t + 1 d x .

Ξ ( α , β , ω , t ) = ∑ l = 0 ∞ l ζ ( ζ + 1 ) − 1 ,

where l ζ = ( − 1 ) ζ ω + t + 1 ζ .

3.6 Stochastic ordering

Consider the random variable X 1 and X 2 , given that λ 1 < λ 2 , and for X 1 ≤ l r X 2 , if the following results holds, h 1 ( x ) h 2 ( x ) shall be decreasing in x

S = α β x β − 1 exp ( − α x β ) × { ( λ 2 − λ 1 ) + λ 2 [ e λ 2 [ exp ( − α x β ) ] ] − λ 1 [ e λ 1 [ exp ( − α x β ) ] ] } < 0 ,

where S = d d x ln h 1 ( x ) h 2 ( x ) .

3.7 Residual and reversed residual life

The n th moments of the residual life of X is given by

m n ( t ) = 1 1 − H ( t ) ∫ t ∞ ( x − t ) n d H ( x ) .

By using Eq. (35), we obtian

m n ( t ) = 1 1 − H ( t ) ∑ p = 0 ∞ t p ∗ ∫ t ∞ x r π ( x ; α ( p + 1 ) , β ) d x ,

where t p ∗ = t p ∑ r = 0 n n r ( − t ) n − r , and mean residual life of X can be yielded by setting n = 1 in Eq. (50).

(50) m n ( t ) = 1 1 − H ( t ) ∑ p = 0 ∞ t p ∗ [ α ( p + 1 ) ] − n / β × γ n β + 1 , α ( p + 1 ) t β .

The following expression yields the n th moments of reversed residual life:

M n ( t ) = 1 H ( t ) ∫ 0 t ( t − x ) n d F ( x ) .

By using Eq. (35),

M n ( t ) = 1 H ( t ) ∑ p = 0 ∞ t p ∗ ∗ ∫ 0 t x r π ( x ; α ( p + 1 ) , β ) d x ,

where t p ∗ ∗ = t p ∑ r = 0 n n r ( − 1 ) r t n − r , and the mean reverse residual life or mean inactivity time or mean waiting time of X can be yielded by setting n = 1 in Eq. (51):

(51) M n ( t ) = 1 H ( t ) ∑ p = 0 ∞ t p ∗ ∗ [ α ( p + 1 ) ] − n / β × γ n β + 1 , α ( p + 1 ) t β .

3.8 Estimation

The log-likelihood function L for the parameter vector ϕ = ( λ , α , β ) ⊤ for the model provided in Eq. (32) is given as follows:

ℓ ( ϕ ) = n ln [ α β λ ] + ( β − 1 ) ∑ i = 1 n ln x i − α ∑ i = 0 n x i β + λ ∑ i = 1 n [ exp ( − α x i β ) ] − n e λ + ∑ i = 1 n exp [ λ { exp ( − α x i β ) } ] − n ln [ 1 − exp ( 1 − e λ ) ] .

The components of the score vector U ( ϕ ) are as follows:

∂ ℓ ( ϕ ) ∂ λ = n λ − 1 + ∑ i = 1 n [ exp ( − α x i β ) ] − n e λ + ∑ i = 1 n exp [ λ { exp ( − α x i β ) } ] { exp ( − α x i β ) } − n e λ exp [ 1 − e λ ] 1 − exp [ 1 − e λ ] ,

∂ ℓ ( ϕ ) ∂ α = n α − 1 − ∑ i = 0 n x i β − λ ∑ i = 0 n x i β [ exp { − α x i β } ] − λ ∑ i = 0 n x i β exp { λ e − α x i β − α x i β } , ∂ ℓ ( ϕ ) ∂ β = n β − 1 + ∑ i = 1 n ln x i − α ∑ i = 1 n x i β ln [ x i ] − λ α ∑ i = 1 n exp ( − α x i β ) [ x i β ln ( x i ) ] − α λ ∑ i = 1 n exp [ λ { exp ( − α x i β ) } − α x i β ] × { x i β ln ( x i ) } .

The MLEs are given as the vector ϕ ˆ = [ λ ˆ , α ˆ , β ˆ ] ⊤ can be obtained by maximizing the ℓ ( ϕ ) with respect to ϕ and replacing ∂ ℓ ( ϕ ) ∂ λ = 0 , ∂ ℓ ( ϕ ) ∂ α = 0 , and ∂ ℓ ( ϕ ) ∂ β = 0 . We use Adequac𝗒 Model R Package to obtain estimates.

4 Actuarial measures

An increasing trend has been seen using extended models to obtain acturial data. Many authors are referred to Chhetri et al. [13] who proposed new five parameters Kumaraswamy transmuted Pareto distribution and used for Norwegian fire insurance, insurance losses modeled using new beta powered transformed Weibull distribution [14], a class of claim distribution introduced by ref. [15], heavy-tailed exponential distribution for unemployment claim data given by ref. [16], a new class of heavy-tailed distribution proposed by Zhao et al. [17], a new heavy-tailed distribution proposed by Riad et al. [18], truncated Burr X-G family of distributions discussed by Bantan et al. [19] and applied to actuarial data, the generalized exponential family introduced by Khan et al. [11] and applied to premium data and Ahmad et al. [20] proposed exponential T-X family and applied to actuarial data, which exhibits better fits.

4.1 Value at risk and ES

A common final market risk measure is value at risk, sometimes known as quantile risk or simply VaR. The uncertainty surrounding the international market, commodity prices, and governmental regulations can have a considerable impact on firm earnings and is a factor in many business choices. The value of the loss portfolio is determined by a particular level of confidence that is q . If X follows the BellW, then the VaR = Q ( u ) , where Q ( u ) is given in Eq. (33). Artzner [21] proposed ES and it is given by

(52) ES q ( x ) = 1 q ∫ 0 q VaR x d x , 0 < q < 1 ,

where VaR x = Q ( u ) .

4.2 Tail value at risk

One of the primary issues in risk management is risk quantification. The term tail value at risk refers to the predicted loss value if the loss exceeds the VaR and is given by

(53) TVaR q ( x ) = 1 1 − q ∫ VaR q ∞ x h ( x ) d x .

Using Eq. (35) in Eq. (53) yields the tail value at risk as follows:

(54) TVaR q ( x ) = [ α ( 1 + p ) ] − 1 / β ( 1 − q ) ∑ p = 0 ∞ t p × Γ 1 β + 1 , α ( 1 + p ) [ VaR q ] β .

4.3 Tail variance

TV computes the loss divergence from the mean along the tail of the distribution, and it is defined by ref. [22]

(55) TV q ( x ) = E [ X 2 ∣ X > x q ] − [ TVaR q ] 2 .

Consider I = E [ X 2 ∣ X > x q ]

I = TVaR q ( x ) = 1 1 − q ∫ VaR q ∞ x 2 h ( x ) d x .

Therefore,

(56) I = [ α ( 1 + p ) ] − 2 / β ( 1 − q ) ∑ p = 0 ∞ t p × Γ 2 β + 1 , α ( 1 + p ) [ VaR q ] β .

By substituting Eqs. (54) and (56) in Eq. (55), we obtain the BellW model’s tail variance expression.

4.4 Tail variance premium

Central tendency and dispersion data are combined in the tail variance premium (TVP). It is described by the following relationship:

(57) TV P q ( X ) = TVaR q + δ TV q ,

where 0 < δ < 1 , and by substituting Eqs. (55) and (54) into Eq. (57), for the BellW model, we obtain the TVP.

4.5 Simulation analysis of VaR and ES

Here, we assess the performance of both risk measures namely VaR and ES through simulation analysis at varying parametric values. A simulation analysis is designed by taking a random sample of n = 500 generated using the QF of the BellW distribution. It is replicated r = 1,000 times. The value of VaR and ES along with 95% lower and upper confidence limits (UCLs) are computed at q = seq ( 0.01 , 0.99 , 0.01 ) level of significance (see x -axis of Figure A1). The scale and shape parameters are executed in various combinations, I = [ α = 1 , β = 0.01 , λ = 0.25 ] , II = [ α = 0.50 , β = 0.01 , λ = 0.50 ] , III = [ α = 10.1 , β = 1.00 , λ = 1.00 ] . The graphical and numerical illustrations of Var and ES are shown, respectively, in Figure A1 and Table A1 with 95% lower confidence limit (LCL) and UCL. Based on Table A1, i.e., the values of parameter β and λ increase, the magnitude of both risk measures is reduced. The crucial risk measures such as VaR and ES are provided numerically as well as graphically in Table A2 and Figures A2–A4, respectively, based on MLEs. The VaR and ES of the BellW and the traditional Weibull (W) model are compared. It is important to note that a model is deemed to have a heavier tail if the risk measurements have greater values. The VaR and ES for all datasets are numerically illustrated in Table A2, which shows that the proposed BellW model has larger values of VaR and ES compared to W model. So, the proposed BellW model can be a good choice to use for heavy-tailed data for better estimation. The readers are referred to ref. [23] for numerical computation of ES and VaR using an R package VaRES .

5 Simulation

This section presents the simulation study to highlight the parameter estimates for the proposed BellW model’s performance over predefined replication ( r = 500 ). We generate N = 1,000 samples of sizes ( n = 20 , 30 , 40 , 50 , … , 300 ) , and the parameters α , β , and λ are combined in different ways. For example, take sets I, II, and III: [ α = 2.0 , β = 5.0 , λ = 0.5 ] , [ α = 1.0 , β = 1.85 , λ = 2.5 ] , and [ α = 0.5 , β = 2.5 , λ = 1 ] , respectively. The simulation analysis’s findings indicated that as sample size increased, mean square error (MSE) and bias of the parameters reduced. Therefore, these estimates can be used to build the BellW model’s confidence intervals. The output summary of simulation analysis is presented in Tables A3–A5, whereas the graphical illustrations are presented from Figures A5–A10. The bias and MSE of the estimates are computed using the following expressions:

(58) Bias ( ζ ˆ ) = ∑ i = 1 N ζ i ˆ N − ζ

and

(59) MSE ( ζ ˆ ) = ∑ i = 1 N ( ζ i ˆ − ζ ) 2 N ,

respectively.

6 Applications

6.1 Actuarial data

The three insurance claim datasets are utilized to evaluate the suggested BellW model’s applicability in real-world applications. The first dataset consists of 89 observed premium automobile insurance company complaints ranking premiums per million dollars of insurers in 2016, and data can be assessed from the following link (https:data.world/datasets/insurance/). The data are as follows: 204.173, 84.769, 65.335, 62.505, 46.735, 43.693, 35.072, 32.511, 30.867, 30.397, 29.776, 26.249, 23.911, 23.263, 22.796, 20.956, 19.667, 18.093, 17.419, 16.934, 16.023, 15.572, 15.374, 14.056, 14.044, 14.028, 12.493, 11.733, 11.331, 11.177, 11.008, 10.772, 9.918, 9.881, 9.421, 9.399, 9.336, 9.141, 8.919, 8.253, 8.193, 7.647, 7.589, 7.493, 7.094, 6.635, 6.365, 6.259, 6.217, 5.416, 5.323, 4.405, 4.19, 4.109, 3.764, 3.613, 3.384, 3.106, 3.019, 2.979, 2.89, 2.843, 2.841, 2.77, 2.749, 2.643, 2.616, 2.55, 2.546, 2.433, 2.425, 2.403, 2.392, 2.087, 1.945, 1.833, 1.805, 1.798, 1.741, 1.719, 1.618, 1.584, 1.416, 1.407, 1.364, 1.358, 1.238, 1.199, and 1.048. The second dataset comes from the US Insurance Service Office (ISO) and is based on 1,500 nonlife insurance losses and allocated loss adjustment expenses, which are costs for processing specific insurance claims. In addition, the dataset was examined by Gui et al. [24]. The third data consist of 21 variables, from July 2008 to April 2013, the monthly metrics on claims for unemployment insurance and other parameters were reported by the Department of Labor, Licensing and Regulation, State of Maryland, USA. Recently, Afify et al. [16] used the variable 12 and fitted heavy-tailed exponential distribution, and we used variable 5 in the present study. The data are as follows: 1.2900, 1.0300, 1.2900, 1.2500, 1.0300, 1.1100, 1.4900, 1.1500, 1.3100, 1.0600, 1.0200, 1.3800, 1.4100, 1.4000, 1.5500, 1.4900, 1.0600, 1.3200, 1.3700, 1.1800, 1.3600, 1.5700, 1.2400, 1.7700, 1.7000, 2.0300, 1.8400, 1.7300, 1.5300, 1.5300, 1.6600, 1.4500, 1.4500, 1.4800, 1.4400, 1.6400, 1.6600, 1.7800, 1.7100, 1.7900, 1.6600, 1.2700, 2.0700, 1.6800, 1.9200, 1.8200, 1.9300, 1.9100, 1.9500, 1.9400, 1.5600, 2.6700, 1.8000, 1.4500, 2.0700, 1.5900, 1.4900, and 1.7200.

6.2 COVID-19 data

Arif et al. [25] proposed a novel extended exponentiated family of distributions and fitted a new extended exponentiated Weibull (NEE-W) model to Mexican COVID-19 mortality rates from 4 March to 20 July 2020. The goodness-of-fit measures based on the proposed NEE-W model are as follows: AIC(569.64), CAIC(539.37), BIC(547.69), CAIC(542.90), AD(0.419), CM(0.070), and KS (0.074). The same data (data-4) are fitted to our proposed BellW model and yield considerable improvement in the goodness-of-fit tests shown in Table 3. The descriptive summary of datasets is presented in Table A6 and revealed that datasets are right-skewed as CS > 0 and can be fitted effectively through BellW distribution. We compared the newly proposed three-parameter BellW model, with some well-established Weibull-based extended models such as three-parameter Poisson–Weibull (PW), the three-parameter [26] Weibull-Claim (W-Claim) distribution, a class of claim distribution is recently proposed by Ahmad et al. [15], four-parameter Weibull–Weibull (WW) [7], three-parameter alpha power-Weibull (APW) [27], three-parameter Gull alpha power-Weibull (GAPW) [28], three-parameter transmuted-Weibull (TW) [29], and two-parameter Weibull (W) distributions. Table A6 shows the descriptive information such as the total number of observations in the data ( n ), the minimum ( x min ) , maximum ( x max ), mean of the data ( x ¯ ), the median ( x ˜ ) , lower ( Q 1 ) and upper ( Q 3 ) quartile, standard deviation ( σ ) , and CS and CK of the datasets. The total time on a test (TTT) plot is an effective but rough way to extract some information about the HRF. From Figure 6, it is obvious that the first data showed decreasing (increasing) hazard rate, the second data confirmed the decreasing hazard rate, and the third data showed an increasing hazard rate. This suggests that the proposed model can be powerful as a long way as HRF shape is concerned. All statistical analysis is carried out by using the R - Statistica𝗅 Computin𝗀 Environment , following R packages, namely, AdequacyModel , Survival , and VaRES , which are used to perform parameter estimation via MLE and actuarial numerical analysis (VaR and ES). Table 4 shows all fitted models estimated (Est) parameters along with standard errors (SEs) of the estimates for all four datasets. The parameter values based on ML estimates are chosen in a way that maximizes the possibility that the model’s process genuinely created the data that were actually seen or observed.

Table 3

The detailed summary of model selection measures

Dist.	− 2 ℓ ˆ	AIC	CAIC	BIC	HQIC	AD	CM	KS	P -Value
Data 1
BellW	313.99	633.98	634.26	641.45	636.99	1.2732	0.1850	0.0996	0.3193
PW	315.11	636.22	636.50	643.68	639.23	1.4201	0.2058	0.1004	0.3096
W-Claim	315.01	636.01	636.29	643.48	639.02	1.3485	0.1902	0.1027	0.2850
APW	315.11	636.22	636.50	643.68	639.23	1.4215	0.2060	0.1011	0.3024
GAPW	316.97	639.93	640.21	647.40	642.94	1.5594	0.2243	0.1089	0.2253
TW	317.88	641.75	642.03	649.22	644.76	1.6526	0.2387	0.1140	0.1833
WW	321.34	650.68	651.15	660.63	654.69	1.9431	0.2820	0.1316	0.0835
W	320.41	644.82	644.96	649.80	646.83	1.8608	0.2689	0.1257	0.1099
Data 2
BellW	5054.17	10114.34	10114.36	10130.28	10120.28	1.2389	0.2190	0.0255	0.2825
PW	5065.69	10137.38	10137.40	10153.32	10143.32	2.7418	0.4737	0.0331	0.0746
W-Claim	5076.95	10159.90	10159.92	10175.84	10165.84	3.7561	0.6277	0.0398	0.0173
APW	5065.70	10137.40	10137.42	10153.34	10143.34	2.7225	0.4705	0.0329	0.0774
GAPW	5088.92	10183.85	10183.86	10199.79	10189.79	5.1164	0.8659	0.0448	0.0048
TW	5098.73	10203.46	10203.48	10219.40	10209.40	6.3642	1.0783	0.0484	0.0018
WW	5145.74	10299.48	10299.51	10320.74	10307.40	10.8692	1.8211	0.0638	9.8 × 1 0 − 6
W	5133.53	10271.06	10271.06	10281.68	10275.01	9.8825	1.6547	0.0605	3.4 × 1 0 − 5
Data 3
BellW	14.660	35.319	35.764	41.500	37.727	0.2160	0.0231	0.0516	0.9978
PW	15.118	36.237	36.681	42.418	38.644	0.2527	0.0282	0.0604	0.9840
W-Claim	15.762	37.524	37.969	43.705	39.932	0.2628	0.0241	0.0561	0.9931
APW	15.126	36.253	36.697	42.434	38.661	0.2502	0.0277	0.0585	0.9888
GAPW	16.297	38.595	39.039	44.776	41.002	0.3249	0.0357	0.0649	0.9677
TW	16.731	39.462	39.907	45.644	41.870	0.3686	0.0424	0.0694	0.9429
WW	18.820	45.639	46.394	53.881	48.849	0.5132	0.0589	0.0839	0.8084
W	18.583	41.166	41.384	45.287	42.771	0.4932	0.0561	0.0816	0.8349
Data 4
BellW	262.49	530.98	531.22	538.97	534.22	0.4019	0.0674	0.0649	0.7640
PW	262.80	531.60	531.83	539.59	534.84	0.4500	0.0737	0.0696	0.6833
W-Claim	265.53	537.06	537.30	545.05	540.30	0.7268	0.1173	0.0707	0.6643
APW	262.78	531.56	531.79	539.55	534.80	0.4444	0.0734	0.0690	0.6943
GAPW	262.62	531.25	531.48	539.24	534.48	0.4102	0.0680	0.0744	0.5998
TW	263.46	532.92	533.15	540.91	536.15	0.5430	0.0867	0.0709	0.6602
WW	264.47	536.94	537.33	547.59	541.26	0.6788	0.1054	0.0685	0.7030
W	264.29	532.59	532.70	537.91	534.75	0.6562	0.1021	0.0687	0.6984

Figure 6

Graphical illustration of TTT of data 1–3.

Table 4

Summary of estimated parameters with S.Es of fitted models

Dist.	Param	Data-1		Data-2		Data-3		Data-4
		Est	SE	Est	SE	Est	SE	Est	SE
BellW	α ˆ	0.014	0.007	0.015	0.002	0.005	0.002	0.006	0.003
	β ˆ	1.084	0.081	0.969	0.020	6.449	0.618	2.304	0.141
	λ ˆ	1.458	0.274	1.689	0.081	1.707	0.278	0.923	0.256
PW	α ˆ	0.021	0.009	0.029	0.003	0.009	0.004	0.008	0.004
	β ˆ	1.019	0.078	0.907	0.018	6.137	0.568	2.216	0.151
	λ ˆ	4.605	1.616	5.097	0.511	5.440	1.998	1.960	1.042
W-Claim	α ˆ	0.275	0.123	0.549	0.094	0.236	0.212	1.650	0.874
	β ˆ	0.663	0.092	0.522	0.029	3.709	0.884	0.616	0.153
	σ ˆ	1.763	1.768	0.685	0.274	1.068	1.942	0.010	0.013
APW	α ˆ	0.038	0.025	0.030	0.003	0.010	0.004	0.008	0.004
	β ˆ	1.179	0.145	0.907	0.018	6.097	0.565	2.220	0.144
	γ ˆ	0.118	0.171	0.006	0.002	0.005	0.007	0.114	0.122
GAPW	α ˆ	0.047	0.013	0.069	0.004	0.025	0.009	0.501	0.128
	β ˆ	0.939	0.070	0.841	0.016	5.637	0.517	0.909	0.099
	γ ˆ	2.343	0.322	2.391	0.080	2.464	0.275	0.004	0.005
TW	α ˆ	0.064	0.016	0.093	0.005	0.035	0.012	0.015	0.006
	β ˆ	0.896	0.067	0.804	0.015	5.380	0.498	2.058	0.145
	λ ˆ	0.773	0.193	0.801	0.047	0.857	0.157	0.539	0.346
WW	α ˆ	0.083	0.319	66.02	18.00	15.93	166.01	14.35	85.65
	β ˆ	13.37	12.43	1.249	0.077	27.70	29.23	33.65	38.24
	a ˆ	0.711	0.145	0.009	0.001	0.602	0.144	0.605	0.192
	b ˆ	0.042	0.038	0.574	0.034	0.131	0.135	0.042	0.051
W	α ˆ	0.127	0.028	0.181	0.009	0.077	0.024	0.027	0.008
	β ˆ	0.823	0.061	0.842	0.014	4.925	0.451	1.918	0.139

The well-known accuracy measures, namely, AIC, CAIC, BIC, and HQIC, are used to select the best-fitted model for the insurance claim datasets. This criterion emphasizes a fitted model with the least information-criterion value considered the best model among all competitive models. It is obvious from Table 3, that the proposed BellW model is outperforming compared to PW, W-Claim, APW, GAPW, TW, WW, and W as all information-criterion for the datasets are the least. Table 3 also showed the goodness-of-fit measures, Anderson–Darling (AD) test, Cramer-Von-Mises (CM) test, Kolmogorov-Smirnov (KS) test, and P -value. The proposed model outperforms compared to the competitive models with the least goodness-of-fit measures and higher P -value for all datasets. The graphical illustration of estimated PDFs, CDFs, probability–probability (P–P) plot, Kaplan–Maier (K–M), and hazard rates are presented in Figures 7, 8, 9, 10, 11. The graphical presentation shows good agreement between actual and predict under the proposed BellW model for all datasets (Figure 12).

Figure 7

Estimated plots of PDF of data 1–3.

Figure 8

Estimated plots of CDF of data 1–3.

Figure 9

Graphical illustration of estimated K-M of data 1–3.

Figure 10

Plots of the estimated HRF of data 1–3.

Figure 11

Estimated P–P plots of data 1–3.

Figure 12

Graphical illustration of estimated PDF, CDF and P–P of data 4.

7 Concluding remarks

As an alternative to the compounded Poisson generalized family of distributions, we proposed the Bell generalized class of distributions. We also derived some broad characteristics of the proposed family including QF, analytical shapes of the PDF and HRF, linear representation of PDF, CDF, PWMs, order statistics, upper record values, entropy measures, and stochastic ordering. A special model of the Bell-G class called the BellW model is presented with several properties including the two representations of the MGF. The ES and VaR are also defined for the BellW model and are shown numerically and graphically. To assess how well the BellW model estimates perform, a simulation study is conducted. The usefulness of the proposed BellW model is examined by employing insurance claims and COVID-19 datasets and compared with some well-known models such as PW, W-Claim, APW, GAPW, TW, WW, and W. The results of the data analysis showed that the new BellW model is superior to its competitors. We thought the family of distributions that has been proposed is a valuable addition to the literature. First, it is developed utilizing discrete Bell distribution as its genesis, retaining the intriguing characteristics of Bell distribution. Second, the Poisson-G family of distributions is widely employed in many applied domains. Recently, Maurya and Nadarajah [4] established the significance of the Poisson-G family by conducting a thorough survey related to it and discovering 77 distributions, 12 power series distributions, and 23 transformed distributions based on the Poisson-G family. Therefore, the suggested Bell-G family of distributions is a good alternative and has the potential to surpass the well-known Poisson-G.

Acknowledgments

Researchers Supporting Project number (RSPD2023R548), King Saud University, Riyadh, Saudi Arabia.

Funding information: Researchers Supporting Project number (RSPD2023R548), King Saud University, Riyadh, Saudi Arabia.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Since no datasets were created or examined during the current investigation, sharing of data does not apply to this article.

Appendix

Table A1

Simulated values of VaR and ES based on the BellW model

q	VaR	LCL	UCL	ES	LCL	UCL
Set-I
0.6500	88.3877	77.4478	99.3275	3637.535	3196.601	4078.47
0.7000	102.2639	90.0796	114.4483	4071.224	3598.614	4543.834
0.7500	118.9576	104.8381	133.0771	4550.967	4043.548	5168.239
0.8000	139.4857	122.9911	155.9802	5089.35	4541.657	5890.627
0.8500	166.6608	146.5801	186.7416	5707.891	5114.725	6301.057
0.9000	205.1772	179.3730	230.9814	6444.77	5799.244	7090.297
0.9500	271.4791	234.0882	308.8701	7384.715	6671.112	8098.319
0.9900	424.1444	339.0910	509.1978	8502.145	7701.081	9303.209
Set-II
0.6500	142.5281	123.2526	161.8036	5713.835	4975.68	6451.99
0.7000	166.6386	144.8402	188.4371	6429.584	5626.54	7232.629
0.7500	196.0917	170.2490	221.9344	7230.563	6355.281	8105.844
0.8000	233.5295	202.6949	264.3640	8143.633	7188.217	9099.048
0.8500	282.6087	244.5700	320.6475	9208.097	8162.929	10253.27
0.9000	355.5750	306.2569	404.8932	10497.75	9341.484	11654.01
0.9500	484.6419	410.6784	558.6054	12190.5	10892.03	13488.96
0.9900	786.6001	623.1763	950.0238	14284.34	12826.74	15741.95
Set-III
0.6500	0.0396	0.0340	0.0451	1.531126	1.330328	1.731924
0.7000	0.0470	0.0404	0.0535	1.734639	1.512121	1.957156
0.7500	0.0563	0.0483	0.0644	1.967022	1.719835	2.214209
0.8000	0.0685	0.0581	0.0789	2.2381	1.962918	2.513283
0.8500	0.0856	0.0723	0.0989	2.564945	2.255092	2.874798
0.9000	0.1123	0.0938	0.1309	2.979524	2.624393	3.334655
0.9500	0.1648	0.1354	0.1943	3.559803	3.136991	3.982615
0.9900	0.3029	0.2247	0.3811	4.351739	3.82473	4.878748

Table A2

Summary of ES and VaR based on the MLEs

Data-1			Data-2			Data-3
q	BellW	W	q	BellW	W	q	BellW	W
ES
0.5000	3.7647	3.4897	0.5000	2.7076	2.2745	0.5000	1.2295	1.2855
0.6000	4.2804	4.0223	0.6000	3.1434	2.6127	0.6000	1.3943	1.3142
0.6500	4.8512	4.6102	0.6500	3.5529	2.9847	0.6500	1.5756	1.3420
0.7000	5.4909	5.2642	0.7000	4.0091	3.3971	0.7000	1.7775	1.3692
0.7500	6.2201	5.9996	0.7500	4.5254	3.8591	0.7500	2.0058	1.3960
0.8000	7.0709	6.8390	0.8000	5.1226	4.3845	0.8000	2.2697	1.4228
0.8500	8.0982	7.8187	0.8500	5.8364	4.9953	0.8500	2.5848	1.4499
0.9000	9.4109	9.0045	0.9000	6.7369	5.7311	0.9000	2.9818	1.4778
0.9500	11.291	10.546	0.9500	8.0075	6.6822	0.9500	3.5409	1.5075
0.9900	14.059	12.404	0.9900	9.8850	7.8199	0.9900	4.365	1.5347
VaR
0.5000	9.1753	9.0757	0.5000	6.6849	5.8285	0.5000	2.9633	1.6070
0.6000	10.7762	10.7272	0.6000	7.8179	6.8632	0.6000	3.4644	1.6525
0.6500	12.6833	12.6554	0.6500	9.1566	8.0667	0.6500	4.0559	1.6988
0.7000	15.0154	14.9476	0.7000	10.7785	9.4921	0.7000	4.7721	1.7467
0.7500	17.9697	17.7412	0.7500	12.8115	11.2225	0.7500	5.6688	1.7974
0.8000	21.9033	21.2687	0.8000	15.4862	13.3990	0.8000	6.8473	1.8527
0.8500	27.5529	25.9730	0.8500	19.2766	16.2891	0.8500	8.5148	1.9156
0.9000	36.7877	32.8651	0.9000	25.3883	20.5023	0.9000	11.1984	1.9925
0.9500	56.6867	45.2482	0.9500	38.4960	28.0243	0.9500	16.9405	2.1018
0.9900	125.762	76.2967	0.9900	88.3114	46.7003	0.9900	38.808	2.2936

Table A3

Simulation analysis output, set I

	MSE			Bias
n	α	β	λ	α	β	λ
20	3.77413	16.38253	16.64894	0.5940432	− 1.2661358	1.964895
30	2.29277	10.28492	12.65996	0.5085802	− 1.4535885	2.005294
40	2.40445	9.46550	10.13697	0.5918333	− 1.4731461	1.976829
50	1.64813	7.40166	7.86629	0.6830247	− 1.6863477	2.086634
60	1.66703	7.19316	7.96233	0.6985617	− 1.7524321	2.219253
70	1.44537	6.53853	6.86792	0.6990432	− 1.5347099	1.992568
80	1.27638	6.38979	5.95072	0.5760309	− 1.7050123	1.974228
90	1.16537	6.01545	5.90977	0.5127654	− 1.5902675	1.933788
100	1.04046	5.31250	5.58657	0.5834074	− 1.8214218	2.196819
110	0.23275	2.76304	5.12636	0.2290679	− 1.4073621	2.107471
120	0.22381	2.70340	4.86893	0.2196481	− 1.3248519	1.999093
130	0.18746	2.64205	4.82490	0.2106543	− 1.4080062	2.056049
140	0.15937	2.50893	4.76296	0.2212037	− 1.4237284	2.086827
150	0.16862	2.53059	4.79671	0.2146111	− 1.4442284	2.117698
160	0.16677	2.54126	4.73405	0.2165309	− 1.4278333	2.093944
170	0.14167	2.53499	4.78943	0.2270679	− 1.444321	2.118401
180	0.14384	2.31951	4.73884	0.1973704	− 1.3757593	2.10687
190	0.13444	2.28179	4.56824	0.2146296	− 1.3648519	2.089463
200	0.12603	2.22517	4.57590	0.2177037	− 1.3492716	2.074228
210	0.13425	2.23014	4.58987	0.2106296	− 1.3479383	2.093228
220	0.11850	2.13895	4.51197	0.2286667	− 1.3261481	2.091852
230	0.12587	1.90859	4.18211	0.2081481	− 1.2400864	2.007414
240	0.10610	1.39624	3.51171	0.2022593	− 1.0248148	1.839852
250	0.10223	1.31406	3.39938	0.2075926	− 0.9633457	1.796654
260	0.09097	1.20292	3.39565	0.1867407	− 0.9414815	1.805685
270	0.09605	1.18366	3.34358	0.2043889	− 0.9326358	1.804198
280	0.08277	1.08549	3.30803	0.1935556	− 0.8812469	1.796086
290	0.07632	1.09373	3.30110	0.1976667	− 0.8752222	1.799778
300	0.06328	0.93660	3.23674	0.1745556	− 0.7740247	1.781475

Table A4

Simulation analysis output, set II

	MSE			Bias
n	α	β	λ	α	β	λ
20	1.785247	6.491527	6.908129	0.405643	0.220548	− 0.08778
30	1.789787	5.367643	4.993281	0.56644	0.221259	0.297545
40	1.234879	4.227957	3.427934	0.391434	0.174074	0.291071
50	0.922273	3.250408	2.671558	0.355262	0.084359	0.326031
60	0.544576	2.557375	1.848972	0.225693	− 0.08025	0.31404
70	0.450849	1.769083	1.169157	0.115781	− 0.05262	0.234032
80	0.30907	0.984029	0.606557	0.068217	− 0.04196	0.197843
90	0.223356	0.53912	0.366776	− 0.05923	0.059919	0.061315
100	0.119661	0.232743	0.171505	− 0.06556	0.082887	0.058056
110	0.087098	0.117095	0.088301	− 0.07656	0.113933	0.067518
120	0.030873	0.050678	0.057753	− 0.02421	0.082592	0.091184
130	0.012247	0.022292	0.021934	− 0.00876	0.075357	0.106807
140	0.003042	0.014953	0.020056	0.000987	0.075727	0.10545
150	0.000821	0.013074	0.021562	0.00148	0.06919	0.11137
160	0.001848	0.013485	0.020398	0.00333	0.07067	0.10804
170	0.000411	0.013485	0.020261	0.00222	0.07289	0.10952
180	0.000958	0.0128	0.020467	0.00518	0.06919	0.11063
190	0.000479	0.011637	0.021972	0.00259	0.0629	0.11877
200	0.000616	0.011842	0.021767	0.00333	0.06401	0.11766
210	0.000205	0.013416	0.020535	0.00111	0.07252	0.111
220	0.000411	0.013142	0.020672	0.00222	0.07104	0.11174
230	0.000548	0.011842	0.021836	0.00296	0.06401	0.11803
240	0.000342	0.013279	0.020603	0.00185	0.07178	0.11137
250	0.000274	0.011979	0.021972	0.00148	0.06475	0.11877
260	0.000342	0.012253	0.021562	0.00185	0.06623	0.11655
270	0.000205	0.012253	0.021767	0.00111	0.06623	0.11766
280	0.000548	0.011842	0.021836	0.00296	0.06401	0.11803
290	0.000411	0.010884	0.022931	0.00222	0.05883	0.12395
300	0.000137	0.012047	0.022041	0.00074	0.06512	0.11914

Table A5

Simulation analysis output, set III

	MSE			Bias
n	α	β	λ	α	β	λ
20	3.341773	4.263579	5.957978	1.302571	− 0.830980	1.351857
30	1.864175	2.279767	2.894353	0.989586	− 0.630181	0.975984
40	1.404340	1.729717	1.877772	0.911046	− 0.613673	0.849590
50	1.019405	1.635210	1.586530	0.765364	− 0.443914	0.653543
60	0.726377	1.180563	0.843355	0.653352	− 0.511759	0.662568
70	0.417494	0.760631	0.620780	0.473568	− 0.309568	0.359062
80	0.245108	0.553505	0.341244	0.332407	− 0.223167	0.179722
90	0.121816	0.225846	0.114502	0.260185	− 0.098481	0.082815
100	0.072120	0.068205	0.026039	0.225593	− 0.009574	0.025611
110	0.055250	0.008625	0.002375	0.214000	0.024500	0.006500
120	0.055500	0.006750	0.002000	0.219000	0.024000	0.005000
130	0.055500	0.005625	0.001250	0.222000	0.022500	0.005000
140	0.057625	0.003750	0.000750	0.230500	0.015000	0.003000
150	0.057000	0.004375	0.000875	0.228000	0.017500	0.003500
160	0.057625	0.003875	0.001000	0.230500	0.015500	0.004000
170	0.057875	0.004375	0.000125	0.231500	0.017500	0.000500
180	0.058500	0.003250	0.000500	0.234000	0.013000	0.002000
190	0.059125	0.002625	0.000750	0.236500	0.010500	0.003000
200	0.059625	0.002625	0.000125	0.238500	0.010500	0.000500
210	0.059750	0.002625	0.000125	0.239000	0.010500	0.000500
220	0.059250	0.003000	0.000250	0.237000	0.012000	0.001000
230	0.060000	0.002000	0.000500	0.240000	0.008000	0.002000
240	0.060375	0.001750	0.000250	0.241500	0.007000	0.001000
250	0.059750	0.002125	0.000500	0.239000	0.008500	0.002000
260	0.060625	0.001875	0.000000	0.242500	0.007500	0.000000
270	0.061375	0.000875	0.000250	0.245500	0.003500	0.001000
280	0.061125	0.001250	0.000125	0.244500	0.005000	0.000500
290	0.061125	0.001375	0.000000	0.244500	0.005500	0.000000
300	0.061250	0.001250	0.000000	0.245000	0.005000	0.000000

Table A6

Descriptive statistics of the datasets

	n	x min	x max	x ¯	x ˜	Q 1	Q 3	σ	CS	CK
Data-1	89.0000	1.0500	204.170	14.080	7.090	2.620	15.370	25.270	5.312	37.97
Data-2	1500.00	0.0150	501.863	12.588	5.471	2.333	12.572	28.146	9.250	127.7
Data-3	58.0000	1.0200	2.67000	1.5530	1.530	1.330	1.7600	0.3189	0.608	4.069
Data-4	106.000	1.0410	16.4980	5.8220	5.279	3.289	7.5940	3.2498	0.973	3.666

Figure A1

Plot of VaR (left) and ES (right) of the BellW model for some parametric values.

Figure A2

Graphical illustration of ES and VaR of Data-1.

Figure A3

Graphical illustration of ES and VaR of Data-2.

Figure A4

Graphical illustration of ES and VaR of Data-3.

$Figure A5 The biases plots of parameters ( α , β , λ ) \left(\alpha ,\beta ,\lambda ) at varying sample sizes set-I.$

Figure A5

The biases plots of parameters ( α , β , λ ) at varying sample sizes set-I.

$Figure A6 The MSEs plots of parameters ( α , β , λ ) \left(\alpha ,\beta ,\lambda ) at varying sample sizes set-I.$

Figure A6

The MSEs plots of parameters ( α , β , λ ) at varying sample sizes set-I.

$Figure A7 The biases plots of parameters ( α , β , λ ) \left(\alpha ,\beta ,\lambda ) at varying sample sizes set-II.$

Figure A7

The biases plots of parameters ( α , β , λ ) at varying sample sizes set-II.

$Figure A8 The MSEs plots of parameters ( α , β , λ ) \left(\alpha ,\beta ,\lambda ) at varying sample sizes set-II.$

Figure A8

The MSEs plots of parameters ( α , β , λ ) at varying sample sizes set-II.

$Figure A9 The biases plots of parameters ( α , β , λ ) \left(\alpha ,\beta ,\lambda ) at varying sample sizes set-III.$

Figure A9

The biases plots of parameters ( α , β , λ ) at varying sample sizes set-III.

$Figure A10 The MSEs plots of parameters ( α , β , λ ) \left(\alpha ,\beta ,\lambda ) at varying sample sizes set-III.$

Figure A10

The MSEs plots of parameters ( α , β , λ ) at varying sample sizes set-III.

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Received: 2022-12-06

Revised: 2023-03-31

Accepted: 2023-04-01

Published Online: 2023-04-27

This work is licensed under the Creative Commons Attribution 4.0 International License.

Artikel in diesem Heft

https://doi.org/10.1515/phys-2022-0242

Schlagwörter für diesen Artikel

actuarial measures; Bell numbers; compounded family; entropy measures; Poisson generalized family

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