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BY 4.0 license Open Access Published by De Gruyter Open Access April 27, 2023

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 and Mohammed Elgarhy EMAIL logo
From the journal Open Physics

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.

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 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.
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 ( ) for vector parameters ϕ = ( λ , ϖ ) , where ϖ is a ( p × 1 ) baseline parameter vector is given by

( 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 ) = ϕ = λ , ϖ

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 ) , 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 λ = 0 and 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 3

Graphical illustration of PDF based on the BellW distribution.

Figure 4 
                  Graphical illustration of HRF 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.
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