# A New Class of Heavy-Tailed Distributions: Modeling and Simulating Actuarial Measures

Authors: Jin Zhao; Zubair Ahmad; Eisa Mahmoudi; E. H. Hafez; Marwa M. Mohie El-Din
Journal: Complexity
 (2021)

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## Abstract

Statistical distributions play a prominent role for modeling data in applied fields, particularly in actuarial, financial sciences, and risk management fields. Among the statistical distributions, the heavy-tailed distributions have proven the best choice to use for modeling heavy-tailed financial data. The actuaries are often in search of such types of distributions to provide the best description of the actuarial and financial data. This study presents a new power transformation to introduce a new family of heavy-tailed distributions useful for modeling heavy-tailed financial data. A submodel, namely, heavy-tailed beta-power transformed Weibull model is considered to demonstrate the adequacy of the proposed method. Some actuarial measures such as value at risk, tail value at risk, tail variance, and tail variance premium are calculated. A brief simulation study based on these measures is provided. Finally, an application to the insurance loss dataset is analyzed, which revealed that the proposed distribution is a superior model among the competitors and could potentially be very adequate in describing and modeling actuarial and financial data.


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## Body

## 1. Introduction

Heavy-tailed distributions have been studied for decades by actuaries to investigate various aspects of financial portfolio theory and risk management problems. As always, most insurance loss datasets share the same properties such as skewed to the right, unimodal shaped, and have a very thick right tail, refer the studies of Kazemi and Noorizadeh [1], Adcock et al. [2], Bhati and Ravi [3], Ahmad et al. [4], Ahmad et al. [5], Ahmad et al. [6], and Ahmad et al. [7]. A distribution possess these characteristics is quite appropriate for modeling insurance loss data and can also be used for estimating the business risk level. These distributions include Pareto (Coorey and Ananada [8]), Lomax (Scollnik [9]), beta (Hogg and Klugman [10], lognormal (Klugman et al. [11]), Burr (Nadarajah and Bakar [12]), and Weibull (Bakar et al. [13]). These are the most appropriate models for modeling of insurance losses. For more details, refer to the studies by Shushi [14], Punzo et al. [15], Punzo et al. [16], and Punzo [17].However, despite these classical distributions having many merits, there are still some deficiencies in these models as they are not flexible enough to provide the best fit to the heavy-tailed datasets. As an example, (i) the Pareto model, which is one of the prominent models used for modeling financial data, sometimes provides poor fitting to many financial applications, (ii) on the other hand, the Weibull model can only cover the behavior of small losses adequately but is not a reasonable candidate model to deal with the behavior of large losses, and (iii) the distribution functions of both the lognormal and the beta distributions have no closed form expressions causing difficulties in the derivation of many mathematical properties and make them less popular to use for analyzing financial datasets (Bhati and Ravi [18]).Due to the importance of the heavy-tailed distributions in the field of actuarial sciences, the actuaries are always in search for introducing new statistical distributions to cover the deficiencies of the traditional distributions. This motivated the researchers to search for new flexible heavy-tailed distributions.In this context, serious efforts have been performed to propose new approaches for developing new flexible distributions such as (i) composition of two or more distributions, (ii) transformation of variables, (iii) finite mixture of distributions, and (iv) compounding of distributions. However, still these methods are suffering from some sort of deficiencies (Ahmad [19] and Ahmad et al. [20]). Thus, it is imperative to develop new statistical models having a closed form of distribution functions, heavy-tailed than the exponential class of distributions, and are capable of adequately modeling the financial risk management problems.In the premises of the above, we further carry this area of distribution theory and propose a new method to introduce new flexible heavy-tailed distributions. The proposed family may be called a new heavy-tailed beta-power transformed (HTBPT) family of distributions. The proposed method may be an interesting alternative to the existing literature for fitting heavy-tailed data. We study a new model called the heavy-tailed beta-power transformed Weibull (HTBPT–Weibull) distribution as a special case of the HTBPT distributions. The proposed model is very flexible and could be chosen as a good candidate model for modeling heavy-tailed data. Some mathematical properties along with the actuarial measures are derived. Based on the actuarial measures, a comparative simulation study of the proposed model and Weibull (also known as superexponential) distributions is conducted. The simulation results show that the HTBPT–Weibull distribution has heavier tail than the superexponential distribution.The study is sectioned as follows. InSection 2, we provided the steps followed in the derivation of the proposed method. In Section 3, we introduced a special case of the proposed family. In Section 4, we introduced some mathematical properties of the proposed family. In Section 5, the maximum likelihood estimation method is used in order to estimate the parameters of the model. A brief Monte Carlo simulation study is provided in Section 6. Actuarial measures of the proposed model are discussed in Section 7. A real-life application of the proposed model to insurance loss data is provided in Section 8. Finally, Section 9 contains the conclusions and the major findings of this research study.


## 2. Steps Followed for Developing the Proposed Family

Suppose that a random variableT∈a1,a2 for −∞≤a1<a2<∞, with density function vt, and let the random variable X has the cumulative distribution function (cdf) WFx;ψ, satisfying(1)
WFx;ψ∈a1,a2(2)
WFx;ψ must be monotonically increasing and can be differentiated(3)
WFx;ψ⟶a1 as x⟶−∞ and WFx;ψ⟶a2 as x⟶∞TheT−X family method was originally proposed by Alzaatreh et al. [21] as follows:(1)Gx=∫a1WFx;ψvtdt,x∈ℝ,where WFx;ψ satisfies the above conditions. Corresponding to (1), the probability density function (pdf) is(2)gx=∂∂xWFx;ψvWFx;ψ,x∈ℝ.More information about theT−X technique for introducing new families of distributions is given by Ahmad et al. [19]. By applying the T−X technique, let us assume that T follows the exponential distribution with the scale parameter, say λ=1; then, its cdf has the form given by(3)Vt=1−e−t,t≥0.By differentiating equation (3) with respect to t, we have(4)vt=e−t,t>0.Assume thatvt is as represented in equation (4) and use WFx;ψ=−log1−β1−Fx;ψ−β1−Fx;ψ in (1). Then, we have(5)Gx;β,ψ=β1−Fx;ψ−β1−Fx;ψ,β>0,β≠1,x∈ℝ,Fx;ψ,β=1,where Fx;ψ is the cdf of the baseline model which depends on the vector parameter ψ∈ℝ. From expression (5), we can see that the proposed cdf reduces to the baseline cdf for β=1. The density function of equation (5) is as follows:(6)gx;β,ψ=fx;ψβ−logββ1−Fx;ψ,x∈ℝ.The expression (6) can also be written as(7)gx;β;ψ=βfx;ψβFx;ψβFx;ψ−logβ,x∈ℝ.The survival function (sf) and hazard rate function (hrf) of the proposed family are given by(8)Sx;β,ψ=1−β1−Fx;ψ−β1−Fx;ψ,x∈ℝ,hx;β,ψ=fx;ψβ−logββ1−Fx;ψ1−β1−Fx;ψ−β1−Fx;ψ,x∈ℝ,respectively.The random variableX with pdf (5) will be denoted by X∼HTBPTx;β,ψ. For the sake of simplicity, we can omit the dependence on the parameters and simply write Gx=Gx;β,ψ.


## 3. Description of the Proposed Model

This section deals with the introduction of the new HTBPT–Weibull distribution and its special cases.


### 3.1. Heavy-Tailed Beta-Power Transformed Weibull Distribution

Referring to the cdf of the Weibull distribution with shape parameterα>0 and scale parameter γ>0, Fx;ψ=1−e−γxα,x>0,α,γ>0, and pdf fx;ψ=αγxα−1e−γxα, where ψ=α,γ. A random variable say X has the HTBPT–Weibull distribution, if its cdf is(9)Gx;β,ψ=βe−γxα−βe−γxα,x≥0,α,β,γ>0,β≠1.The pdf, sf, and hrf of the HTBPT–Weibull distributions are given by(10)gx;β,ψ=αγxα−1e−γxαβ−logββe−γxα,x>0,Sx;β,ψ=1−βe−γxα−βe−γxα,x>0,hx;β,ψ=αγxα−1e−γxαβ−logββe−γxα1−βe−γxα−βe−γxα,x>0,respectively.In order to see the effect of addingβ to the proposed distribution, we fixed the distribution’s parameters (α=1.2 and γ=1) and changed the values of β and sketched the plots. From the plots provided in Figure 1, it is clear that as the value of the additional parameter β increases, the proposed model tends to a heavy-tailed distribution.Figure 1
Graphical sketching of the HTBPT–Weibull density function for some parameter values.
(a)(b)Plots for the cdf and sf of the proposed model are provided in Figure2. Whereas, the plots of the hrf are shown in Figure 3.Figure 2
Graphical sketching of the HTBPT–Weibull cdf and sf plots for selected parameter values.
(a)(b)Figure 3
Graphical sketching of the HTBPT–Weibull hrf for some parameter values.


### 3.2. Special Cases of the HTBPT–Weibull Distribution

This subsection deals with the subcases of the proposed distribution. LetX has the HTBPT–Weibull distribution with parameters α,β,γ. Then, we obtainThe Weibull distribution with parametersα and γ, using β=1The one parameter Weibull distribution with parameterα, using β=γ=1The exponential distribution with parameterγ, using β=α=1The Rayleigh distribution with parameterγ, if β=1 and α=2The one parameter HTBPT–Weibull distribution with parametersα and β, if γ=1 (new)The heavy-tailed beta-power transformed exponential distribution with parametersβ and γ, if α=1 (new)The HTBPTR–Rayleigh distribution with parametersβ and γ, if α=2 (new)


## 3.1. Heavy-Tailed Beta-Power Transformed Weibull Distribution

Referring to the cdf of the Weibull distribution with shape parameterα>0 and scale parameter γ>0, Fx;ψ=1−e−γxα,x>0,α,γ>0, and pdf fx;ψ=αγxα−1e−γxα, where ψ=α,γ. A random variable say X has the HTBPT–Weibull distribution, if its cdf is(9)Gx;β,ψ=βe−γxα−βe−γxα,x≥0,α,β,γ>0,β≠1.The pdf, sf, and hrf of the HTBPT–Weibull distributions are given by(10)gx;β,ψ=αγxα−1e−γxαβ−logββe−γxα,x>0,Sx;β,ψ=1−βe−γxα−βe−γxα,x>0,hx;β,ψ=αγxα−1e−γxαβ−logββe−γxα1−βe−γxα−βe−γxα,x>0,respectively.In order to see the effect of addingβ to the proposed distribution, we fixed the distribution’s parameters (α=1.2 and γ=1) and changed the values of β and sketched the plots. From the plots provided in Figure 1, it is clear that as the value of the additional parameter β increases, the proposed model tends to a heavy-tailed distribution.Figure 1
Graphical sketching of the HTBPT–Weibull density function for some parameter values.
(a)(b)Plots for the cdf and sf of the proposed model are provided in Figure2. Whereas, the plots of the hrf are shown in Figure 3.Figure 2
Graphical sketching of the HTBPT–Weibull cdf and sf plots for selected parameter values.
(a)(b)Figure 3
Graphical sketching of the HTBPT–Weibull hrf for some parameter values.


## 3.2. Special Cases of the HTBPT–Weibull Distribution

This subsection deals with the subcases of the proposed distribution. LetX has the HTBPT–Weibull distribution with parameters α,β,γ. Then, we obtainThe Weibull distribution with parametersα and γ, using β=1The one parameter Weibull distribution with parameterα, using β=γ=1The exponential distribution with parameterγ, using β=α=1The Rayleigh distribution with parameterγ, if β=1 and α=2The one parameter HTBPT–Weibull distribution with parametersα and β, if γ=1 (new)The heavy-tailed beta-power transformed exponential distribution with parametersβ and γ, if α=1 (new)The HTBPTR–Rayleigh distribution with parametersβ and γ, if α=2 (new)


## 4. Mathematical Properties

This section is devoted to some basic mathematical properties of the HTBPT family.


### 4.1. Shape of Density and Hazard Functions of the HTBPT Distributions

The shapes of the density function and the hrf of the HTBPT distributions can be expressed by an analytical formula. We can also find the critical points of the density function by differentiating it with respect tox as follows:(11)∂∂xloggx;β,ψ=0.The above equation may have more than one root. Ifx=xo is a root of the above equation, then it corresponds to (i) a local maximum, if ∂2/∂2xloggx;β,ψ<0, (ii) a local minimum, if ∂2/∂2xloggx;β,ψ>0, and (iii) a point of inflection, if ∂2/∂2xloggx;β,ψ=0.The parameter space is the same as in the HTBPT family. In a similar way, the critical points of the hazard function can be obtained by differentiation as follows:(12)∂∂xloghx;β,ψ=0.The above equation may have more than one root. Ifx=xo is a root of the above equation, then it corresponds to (i) a local maximum, if ∂2/∂2xloghx;β,ψ<0, (ii) a local minimum, if ∂2/∂2xloghx;β,ψ>0, and (iii) a point of inflection, if ∂2/∂2xloghx;β,ψ=0.


### 4.2. Asymptotic Behavior of HTBPT Distributions

This subsection is devoted to the asymptotic behavior of the HTBPT distributions.Asx⟶−∞, we have(13)Gx;β,ψ⟶0,gx;β,ψ⟶βfx;ψ1−logβ,Sx;β,ψ⟶1,hx;β,ψ⟶βfx;ψ1−logβ.Asx⟶∞, we have(14)Gx;β,ψ⟶1,gx;β,ψ⟶fx;ψβ−logβ,Sx;β,ψ⟶0,hx;β,ψ⟶∞.


### 4.3. Mixture Representation of the Density of HTBPT Distributions

This subsection deals with the mixture representation of the density of HTBPT distributions. We know thatβv=∑i=0∞logβi/i!vi; therefore, the expression (6) can be formulated as(15)gx;β,ψ=fx;ψβ−∑i=0∞logβi+1i!1−Fx;ψi,gx;β,ψ=fx;ψβ−∑i=0∞∑j=0i−1jlogβi+1i!ijFx;ψj,gx;β,ψ=βfx;ψ−∑i=0∞∑j=0i−1jlogβi+1i!i!i−j!j!fx;ψFx;ψj,gx;β,ψ=βfx;ψ−∑i=0∞∑j=0i−1jlogβi+1i−j!j!fx;ψFx;ψj,gx;β,ψ=βfx;ψ−∑i=0∞∑j=0iΛi,jθfx;ψFx;ψθ−1,where Λi,j=−1jlogβi+1/j+1i−j!j!, and j=θ−1.(16)gx;β,ψ=βfx;ψ−∑i=0∞∑j=0iΛi,jhθx;ψ,where hθx;ψ=θfx;ψFx;ψθ−1 is defined as the density function of exponentiated generated distributions with parameter θ in its power.


### 4.4. Quantile Function

The quantile functions are in extensive use in general statistics and frequently used to generate random samples from probability distributions. LetX denotes the HTBPT random variable; then, the quantile function of X can be obtained as follows:(17)x=Qu=G−1u=F−1t,where t is the solution of the equation β1−t+βt=u+β, and u is the uniform distribution over (0, 1) interval.


### 4.5. Moments

In this part of the study, we derived an important property of the HTBPT distributions called moments, and it has a specific and important role in statistical analysis. It is used in finding the major properties and characteristics of the distribution (e.g., central tendency, dispersion, skewness, and kurtosis). Therth moment of the HTBPT distributions is derived as follows:(18)μr/=∫−∞∞xrgx;β,ψdx=β∫−∞∞xrfx;ψdx−∫−∞∞∑i,j=0∞ηi,jxrfx;ψFx;ψjdx=βυr−∑i,j=0∞ηi,jυr,j,where(19)ηi,j=∑i=0∞logβi+1i!ij,υr=∫−∞∞xrfx;ψdx,υr,j=∫−∞∞xrfx;ψFx;ψjdx.The effects of different values of the parametersα and β on the mean, variance, skewness, and kurtosis of the HTBPT–Weibull distribution with γ=1 are illustrated in Figures 4 and 5.Figure 4
Graphical display of the mean, variance, skewness, and kurtosis of the proposed model.
(a)(b)(c)(d)Figure 5
Graphical display of the mean, variance, skewness, and kurtosis of the proposed model.
(a)(b)(c)(d)The moment generating function (mgf) is another useful tool to calculate the moments of the statistical distributions. The mgf of the HTBPT distributions is given by(20)MXt=∫−∞∞etxgx;ζ,ψdx=∑r=0∞trr!μr/=∑i,r=0∞trr!βυr−∑i,j=0∞ηi,jυr,j.


### 4.6. Incomplete Moments

The shape of many statistical distributions can be conveniently described by the incomplete moments. It plays a very important role in measuring inequalities (e.g., income quantiles as well as Lorenz and Bonferroni curves). Here, we drive therth incomplete moment of HTBPT distributions as(21)Υrx=∫−∞xxrgx;β,ψdx=β∫−∞xxrfx;ψdx−∫−∞x∑i,j=0∞ηi,jxrfx;ψFx;ψjdx=βbr−∑i,j=0∞ηi,jbr,j,where br=∫−∞xxrfx;ψdx, and br,j=∫−∞xxrfx;ψFx;ψjdx. Furthermore, we can find the conditional measures of the HTBPT-distributed random variable X as(22)EXrX>t=1St;ψβ∫−∞∞xrfx;ψdx−1St;ψ∫−∞∞∑i,j=0∞ηi,jxrfx;ψFx;ψjdx=1St;ψβΛr−∑i,j=0∞ηi,jΛr,j,where Λr=∫−∞∞xrfx;ψdx, and Λr,j=∫−∞∞xrfx;ψFx;ψjdx.


## 4.1. Shape of Density and Hazard Functions of the HTBPT Distributions

The shapes of the density function and the hrf of the HTBPT distributions can be expressed by an analytical formula. We can also find the critical points of the density function by differentiating it with respect tox as follows:(11)∂∂xloggx;β,ψ=0.The above equation may have more than one root. Ifx=xo is a root of the above equation, then it corresponds to (i) a local maximum, if ∂2/∂2xloggx;β,ψ<0, (ii) a local minimum, if ∂2/∂2xloggx;β,ψ>0, and (iii) a point of inflection, if ∂2/∂2xloggx;β,ψ=0.The parameter space is the same as in the HTBPT family. In a similar way, the critical points of the hazard function can be obtained by differentiation as follows:(12)∂∂xloghx;β,ψ=0.The above equation may have more than one root. Ifx=xo is a root of the above equation, then it corresponds to (i) a local maximum, if ∂2/∂2xloghx;β,ψ<0, (ii) a local minimum, if ∂2/∂2xloghx;β,ψ>0, and (iii) a point of inflection, if ∂2/∂2xloghx;β,ψ=0.


## 4.2. Asymptotic Behavior of HTBPT Distributions

This subsection is devoted to the asymptotic behavior of the HTBPT distributions.Asx⟶−∞, we have(13)Gx;β,ψ⟶0,gx;β,ψ⟶βfx;ψ1−logβ,Sx;β,ψ⟶1,hx;β,ψ⟶βfx;ψ1−logβ.Asx⟶∞, we have(14)Gx;β,ψ⟶1,gx;β,ψ⟶fx;ψβ−logβ,Sx;β,ψ⟶0,hx;β,ψ⟶∞.


## 4.3. Mixture Representation of the Density of HTBPT Distributions

This subsection deals with the mixture representation of the density of HTBPT distributions. We know thatβv=∑i=0∞logβi/i!vi; therefore, the expression (6) can be formulated as(15)gx;β,ψ=fx;ψβ−∑i=0∞logβi+1i!1−Fx;ψi,gx;β,ψ=fx;ψβ−∑i=0∞∑j=0i−1jlogβi+1i!ijFx;ψj,gx;β,ψ=βfx;ψ−∑i=0∞∑j=0i−1jlogβi+1i!i!i−j!j!fx;ψFx;ψj,gx;β,ψ=βfx;ψ−∑i=0∞∑j=0i−1jlogβi+1i−j!j!fx;ψFx;ψj,gx;β,ψ=βfx;ψ−∑i=0∞∑j=0iΛi,jθfx;ψFx;ψθ−1,where Λi,j=−1jlogβi+1/j+1i−j!j!, and j=θ−1.(16)gx;β,ψ=βfx;ψ−∑i=0∞∑j=0iΛi,jhθx;ψ,where hθx;ψ=θfx;ψFx;ψθ−1 is defined as the density function of exponentiated generated distributions with parameter θ in its power.


## 4.4. Quantile Function

The quantile functions are in extensive use in general statistics and frequently used to generate random samples from probability distributions. LetX denotes the HTBPT random variable; then, the quantile function of X can be obtained as follows:(17)x=Qu=G−1u=F−1t,where t is the solution of the equation β1−t+βt=u+β, and u is the uniform distribution over (0, 1) interval.


## 4.5. Moments

In this part of the study, we derived an important property of the HTBPT distributions called moments, and it has a specific and important role in statistical analysis. It is used in finding the major properties and characteristics of the distribution (e.g., central tendency, dispersion, skewness, and kurtosis). Therth moment of the HTBPT distributions is derived as follows:(18)μr/=∫−∞∞xrgx;β,ψdx=β∫−∞∞xrfx;ψdx−∫−∞∞∑i,j=0∞ηi,jxrfx;ψFx;ψjdx=βυr−∑i,j=0∞ηi,jυr,j,where(19)ηi,j=∑i=0∞logβi+1i!ij,υr=∫−∞∞xrfx;ψdx,υr,j=∫−∞∞xrfx;ψFx;ψjdx.The effects of different values of the parametersα and β on the mean, variance, skewness, and kurtosis of the HTBPT–Weibull distribution with γ=1 are illustrated in Figures 4 and 5.Figure 4
Graphical display of the mean, variance, skewness, and kurtosis of the proposed model.
(a)(b)(c)(d)Figure 5
Graphical display of the mean, variance, skewness, and kurtosis of the proposed model.
(a)(b)(c)(d)The moment generating function (mgf) is another useful tool to calculate the moments of the statistical distributions. The mgf of the HTBPT distributions is given by(20)MXt=∫−∞∞etxgx;ζ,ψdx=∑r=0∞trr!μr/=∑i,r=0∞trr!βυr−∑i,j=0∞ηi,jυr,j.


## 4.6. Incomplete Moments

The shape of many statistical distributions can be conveniently described by the incomplete moments. It plays a very important role in measuring inequalities (e.g., income quantiles as well as Lorenz and Bonferroni curves). Here, we drive therth incomplete moment of HTBPT distributions as(21)Υrx=∫−∞xxrgx;β,ψdx=β∫−∞xxrfx;ψdx−∫−∞x∑i,j=0∞ηi,jxrfx;ψFx;ψjdx=βbr−∑i,j=0∞ηi,jbr,j,where br=∫−∞xxrfx;ψdx, and br,j=∫−∞xxrfx;ψFx;ψjdx. Furthermore, we can find the conditional measures of the HTBPT-distributed random variable X as(22)EXrX>t=1St;ψβ∫−∞∞xrfx;ψdx−1St;ψ∫−∞∞∑i,j=0∞ηi,jxrfx;ψFx;ψjdx=1St;ψβΛr−∑i,j=0∞ηi,jΛr,j,where Λr=∫−∞∞xrfx;ψdx, and Λr,j=∫−∞∞xrfx;ψFx;ψjdx.


## 5. Estimation

This part of the study is concerned with the estimation of the unknown parameters of the HTBPT distributions. LetX1,X2,…,Xn be a random sample taken from HTBPT distributions with parameters β,ψ. Then, the corresponding log-likelihood function of this sample is(23)logLxi;β,ψ=∑i=1nlogfxi;ψ+∑i=1nlogβ−logββ1−Fxi;ψ.The function (23) can be maximized either by deploying Ox program (subroutine Max BFGS) or SAS (PROC NLMIXED) or by solving it manually. Obtaining the partial derivatives of (10), we have(24)∂∂βLxi;β,ψ=∑i=1n1−1+logβ1−Fxi;ψ/βFxi;ψβ−logββ1−Fxi;ψ,∂∂ψLxi;β,ψ=∑i=1n∂fxi;ψ/∂ψfxi;ψ+∑i=1nlogβ2β1−Fxi;ψ∂Fxi;ψ/∂ψβ−logββ1−Fxi;ψ.Equating∂/∂βlogLxi;β,ψ and ∂/∂ψlogLxi;β,ψ to zero and simultaneously solving these expressions numerically yields the maximum likelihood estimators (MLEs) of β,ψ.


## 6. Simulation Study

The section offers a brief Monte Carlo simulation study to evaluate the efficiency and consistency properties of the MLEs of the HTBPT–Weibull parameters (a particular case from the proposed family). The random samples are generated from the HTBPT–Weibull distribution using the inverse cdf method. The simulation results are obtained using optim R function with the argument method = “L-BFGS-B.” The steps taken for simulation are given as follows:(1)
N=1000 iterations are made, and different samples of size n=25,50,…,1000 are generated from the HTBPT–Weibull distribution using the inversion cdf method.(2)
The parameter values are estimated using the maximum likelihood estimation method(3)
The biases and mean square errors (MSEs) are calculated using the formulas(25)Biasα^=11000∑i=11000αi^−α,MSEα^=11000∑i=11000αi^−α2,respectively(4)
For calculating the biases as well as the MSEs ofβ,γ, step (3) is repeatedThe simulation results are presented in Tables1 and 2. Whereas, the graphical display of the simulation results is shown in Figures 6–9. It is important to note that, in Figures 6–9, ζ is used for α, and η is used for γ.Table 1
Monte Carlo simulation results of the HTBPT–Weibull distribution.
Set 1:α=0.9, β=1.2, γ=1.5nParametersMLEMSEsBiases25α^0.7350.057−0.164β^2.5883.6111.388γ^2.0490.5520.549100α^0.8110.021−0.088β^1.6840.5710.484γ^1.7020.1040.202200α^0.8440.010−0.055β^1.4690.1870.269γ^1.6220.0430.122300α^0.8580.006−0.041β^1.3970.1120.197γ^1.5900.0280.090400α^0.8660.005−0.033β^1.3480.0730.148γ^1.5690.0190.069500α^0.8740.003−0.025β^1.3130.0520.113γ^1.5530.0150.053600α^0.8760.003−0.023β^1.3050.0460.105γ^1.5520.0130.052700α^0.8800.003−0.019β^1.2890.0400.089γ^1.5390.0110.039800α^0.8820.002−0.017β^1.2720.0320.072γ^1.5350.0090.035900α^0.8840.002−0.015β^1.2710.0290.071γ^1.5350.0080.0351000α^0.8860.002−0.013β^1.2600.0240.060γ^1.5290.0070.029Table 2
Monte Carlo simulation results of the HTBPT–Weibull distribution.
Set 2:α=1.2, β=1.5, γ=0.5nParametersMLEMSEsBiases25α^1.0720.130−0.127β^2.8984.4181.498γ^0.7560.1990.256100α^1.1520.040−0.047β^2.0191.5800.619γ^0.5780.0410.078200α^1.1670.027−0.032β^1.8000.9940.400γ^0.5490.0220.049300α^1.1860.013−0.013β^1.7500.7450.280γ^0.5260.0110.026400α^1.1880.010−0.011β^1.7140.4310.174γ^0.5200.0080.020500α^1.1960.010−0.003β^1.6800.4220.150γ^0.5120.0070.012600α^1.1930.010−0.006β^1.6570.3510.127γ^0.5120.0080.012700α^1.1990.006−0.005β^1.6100.2150.080γ^0.5060.0050.010800α^1.2000.0060.003β^1.5730.2190.073γ^0.5050.0040.009900α^1.2010.0060.001β^1.5300.1790.060γ^0.5050.0040.0071000α^1.1980.004−0.001β^1.5060.0860.030γ^0.5060.0030.006Figure 6
Graphical display of the simulation results provided in Table1. (a) Plot of estimated parameters vs. n. (b) Plot of MSEs vs. n.
(a)(b)Figure 7
Graphical display of the simulation results provided in Table1. (a) Plot of absolute blases vs. n. (b) Plot of blases vs. n.
(a)(b)Figure 8
Graphical display of the simulation results provided in Table2. (a) Plot of estimated parameters vs. n. (b) Plot of MSEs vs. n.
(a)(b)Figure 9
Graphical display of the simulation results provided in Table2. (a) Plot of absolute biases vs. n. (b) Plot of biases vs. n.
(a)(b)From the simulation results provided in Tables1 and 2, it is clear that the estimates of the model parameters are very precise and close to the true value of the parameters as the samples’ size increases. Also, we can see that the values of the estimated biases and absolute biases tend to zero, when the sample size n increases, which shows the accuracy of MLEs. Also, as n increases, MSEs tend to zero. This fact leads to the consistency property of the MLEs of the HTBPT–Weibull distribution.


## 7. Actuarial Measures

One of the primary tasks of actuarial sciences institutions is to predict the market risks in a portfolio of instruments. Henceforth, estimating the risk measures is very important in selling and buying products. In this section of the study, we discuss some important risk measures such as the value at risk (VaR), tail value at risk (TVaR), tail variance (TV), and tail variance premium (TVP) of the proposed family.


### 7.1. Value at Risk

The VaR of a random variableX is the qth quantile of its cdf (Artzner [22]). If X follows the HTBPT distributions with parameters β,ψ, then the VaR of X denoted by VaRqX is given by(26)xq=F−1t,where t is the solution of the equation β1−t+βt=u+β, and u follows the uniform distribution on interval (0, 1).


### 7.2. Tail Value at Risk

LetX follows the HTBPT distributions with parameters β and ψ; then, the TVaR of X is derived as(27)TVaRqX=EX|X>VaRqX,TVaRqX=11−q∫VaRq∞xgx;β,ψdx.Using (6) in (27), we get(28)TVaRqX=11−q∫VaRq∞xfx;ψβ−logββ1−Fx;ψdx,TVaRqX=11−qβ∫VaRq∞xfx;ψdx−11−q∑i,j=0∞−1jlogβi+1i!ij∫VaRq∞xfx;ψFx;ψjdx.Using the pdffx;ψ and cdf Fx;ψ of Weibull distribution in (28), we get(29)TVaRqX=11−qβη1/α∫γVaRqα∞t1/α+1−1e−tdt−11−q∑i=0∞∑j=0i∑t=0j−1j+tlogβi+1i!γ1/αt+11/α+1ijjt×∫γt+1VaRqα∞t1/α+1−1e−tdt.On solving (29), we have(30)TVaRqX=1γ1/α1−qβγ1α+1,γVaRqα−1γ1/α1−q∑i=0∞∑j=0i∑t=0j−1j+tlogβi+1i−j!j−t!t!t+11/α+1×γ1α+1,γt+1VaRqα.


### 7.3. Tail Variance

Among the actuarial measures, TV is the most prominent one and deals with the tail variance beyond VaR. The TV of the HTBPT–Weibull distributed random variable is derived as follows:(31)TVqX=EX2|X>xq−TVaRq2.Consider(32)EX2|X>xq=11−q∫VaRq∞x2fx;ψβ−logββ1−Fx;ψdx.By solving the above equations, we get(33)EX2|X>xq=1γ2/α1−qβγ2α+1,γVaRqα−1γ2/α1−q∑i=0∞∑j=0i∑t=0j−1j+tlogβi+1×γ2/α+1,γt+1VaRqαi−j!j−t!t!t+12/α+1.By referring to equations (30) and (33) and substituting in (31), we get the TV of HTBPT–Weibull distribution.


### 7.4. Tail Variance Premium

In this subsection, we discuss another most important risk measure called TVP. The TVP of the HTBPT–Weibull distribution is derived as follows:(34)TVPqX=TVaRq+δTVq,where 0<δ<1. By substituting (30) and (31) in (34), we can get the TVP of the proposed distribution.


### 7.5. Numerical Study of the Risk Measures

The importance of this subsection is due to the numerical study of VaR and TVaR for the Weibull and HTBPT–Weibull distributions for different sets of parameters. The followed steps are given by(1)
A random sample of sizen=100 is generated from the Weibull and HTBPT–Weibull distributions, and the maximum likelihood method is adopted to estimate the model parameters.(2)
Repetitions of 1000 iterations are made to find the values of VaR and TVaR for these distributionsTables3–5 contain the simulation results of the risk measures. The simulation results are also displayed graphically corresponding to each table in Figures 10–12, respectively.Table 3
Forn=100, the simulation study of the risk measures for Weibull and HTBPT–Weibull distributions.
DistributionsParametersSignificance levelVaRTVaRWeibullα = 0.80.7001.66882.23560.7501.79532.33650.8001.93972.45420.8502.11222.5980γ = 10.9002.33522.78800.9502.67633.08630.9752.98103.35890.9994.12604.4174HTBPT–Weibullα = 0.80.7002.30735.03870.7502.60345.55650.8002.99036.2489β = 20.8503.53767.2499γ = 10.9004.42488.90540.9506.356912.58020.9759.014717.69780.99943.667785.0932Table 4
Forn=100, the simulation study of the risk measures for Weibull and HTBPT–Weibull distributions.
DistributionsParametersSignificance levelVaRTVaRWeibullα = 0.50.7002.56284.94480.7502.98245.38060.8003.50155.91780.8504.17886.6160γ = 10.9005.14647.60950.9506.82969.33020.9758.542611.07470.99916.770319.4062HTBPT–Weibullα = 0.50.7004.18399.41910.7504.736610.41310.8005.461111.7463β = 0.50.8506.490413.6812γ = 10.9008.168116.89770.95011.851724.09370.97516.968734.21110.99985.8277171.7096Table 5
Forn=100, the simulation study of the risk measures for Weibull and HTBPT–Weibull distributions.
DistributionsParametersSignificance levelVaRTVaRWeibullα = 1.40.7003.52236.53980.7504.06747.09050.8004.73657.76580.8505.60198.6383γ = 10.9006.82619.87120.9508.928811.98650.97511.041514.10960.99920.942624.0443HTBPT–Weibullα = 1.40.7005.274011.87330.7505.970713.12630.8006.883914.8067β = 0.90.8508.181517.2459γ = 10.90010.296221.30040.95014.939730.37140.97521.389943.12480.999108.1901216.4486Figure 10
Graphical sketching of the numerical results provided in Table3.
(a)(b)Figure 11
Graphical sketching of the numerical results provided in Table4.
(a)(b)Figure 12
Graphical sketching of the numerical results provided in Table5.
(a)(b)After performing the simulation study based on VaR and TVaR, we observed that the proposed model is better than the other compared models because it has higher values of the risk measures (VaR and TVaR).


## 7.1. Value at Risk

The VaR of a random variableX is the qth quantile of its cdf (Artzner [22]). If X follows the HTBPT distributions with parameters β,ψ, then the VaR of X denoted by VaRqX is given by(26)xq=F−1t,where t is the solution of the equation β1−t+βt=u+β, and u follows the uniform distribution on interval (0, 1).


## 7.2. Tail Value at Risk

LetX follows the HTBPT distributions with parameters β and ψ; then, the TVaR of X is derived as(27)TVaRqX=EX|X>VaRqX,TVaRqX=11−q∫VaRq∞xgx;β,ψdx.Using (6) in (27), we get(28)TVaRqX=11−q∫VaRq∞xfx;ψβ−logββ1−Fx;ψdx,TVaRqX=11−qβ∫VaRq∞xfx;ψdx−11−q∑i,j=0∞−1jlogβi+1i!ij∫VaRq∞xfx;ψFx;ψjdx.Using the pdffx;ψ and cdf Fx;ψ of Weibull distribution in (28), we get(29)TVaRqX=11−qβη1/α∫γVaRqα∞t1/α+1−1e−tdt−11−q∑i=0∞∑j=0i∑t=0j−1j+tlogβi+1i!γ1/αt+11/α+1ijjt×∫γt+1VaRqα∞t1/α+1−1e−tdt.On solving (29), we have(30)TVaRqX=1γ1/α1−qβγ1α+1,γVaRqα−1γ1/α1−q∑i=0∞∑j=0i∑t=0j−1j+tlogβi+1i−j!j−t!t!t+11/α+1×γ1α+1,γt+1VaRqα.


## 7.3. Tail Variance

Among the actuarial measures, TV is the most prominent one and deals with the tail variance beyond VaR. The TV of the HTBPT–Weibull distributed random variable is derived as follows:(31)TVqX=EX2|X>xq−TVaRq2.Consider(32)EX2|X>xq=11−q∫VaRq∞x2fx;ψβ−logββ1−Fx;ψdx.By solving the above equations, we get(33)EX2|X>xq=1γ2/α1−qβγ2α+1,γVaRqα−1γ2/α1−q∑i=0∞∑j=0i∑t=0j−1j+tlogβi+1×γ2/α+1,γt+1VaRqαi−j!j−t!t!t+12/α+1.By referring to equations (30) and (33) and substituting in (31), we get the TV of HTBPT–Weibull distribution.


## 7.4. Tail Variance Premium

In this subsection, we discuss another most important risk measure called TVP. The TVP of the HTBPT–Weibull distribution is derived as follows:(34)TVPqX=TVaRq+δTVq,where 0<δ<1. By substituting (30) and (31) in (34), we can get the TVP of the proposed distribution.


## 7.5. Numerical Study of the Risk Measures

The importance of this subsection is due to the numerical study of VaR and TVaR for the Weibull and HTBPT–Weibull distributions for different sets of parameters. The followed steps are given by(1)
A random sample of sizen=100 is generated from the Weibull and HTBPT–Weibull distributions, and the maximum likelihood method is adopted to estimate the model parameters.(2)
Repetitions of 1000 iterations are made to find the values of VaR and TVaR for these distributionsTables3–5 contain the simulation results of the risk measures. The simulation results are also displayed graphically corresponding to each table in Figures 10–12, respectively.Table 3
Forn=100, the simulation study of the risk measures for Weibull and HTBPT–Weibull distributions.
DistributionsParametersSignificance levelVaRTVaRWeibullα = 0.80.7001.66882.23560.7501.79532.33650.8001.93972.45420.8502.11222.5980γ = 10.9002.33522.78800.9502.67633.08630.9752.98103.35890.9994.12604.4174HTBPT–Weibullα = 0.80.7002.30735.03870.7502.60345.55650.8002.99036.2489β = 20.8503.53767.2499γ = 10.9004.42488.90540.9506.356912.58020.9759.014717.69780.99943.667785.0932Table 4
Forn=100, the simulation study of the risk measures for Weibull and HTBPT–Weibull distributions.
DistributionsParametersSignificance levelVaRTVaRWeibullα = 0.50.7002.56284.94480.7502.98245.38060.8003.50155.91780.8504.17886.6160γ = 10.9005.14647.60950.9506.82969.33020.9758.542611.07470.99916.770319.4062HTBPT–Weibullα = 0.50.7004.18399.41910.7504.736610.41310.8005.461111.7463β = 0.50.8506.490413.6812γ = 10.9008.168116.89770.95011.851724.09370.97516.968734.21110.99985.8277171.7096Table 5
Forn=100, the simulation study of the risk measures for Weibull and HTBPT–Weibull distributions.
DistributionsParametersSignificance levelVaRTVaRWeibullα = 1.40.7003.52236.53980.7504.06747.09050.8004.73657.76580.8505.60198.6383γ = 10.9006.82619.87120.9508.928811.98650.97511.041514.10960.99920.942624.0443HTBPT–Weibullα = 1.40.7005.274011.87330.7505.970713.12630.8006.883914.8067β = 0.90.8508.181517.2459γ = 10.90010.296221.30040.95014.939730.37140.97521.389943.12480.999108.1901216.4486Figure 10
Graphical sketching of the numerical results provided in Table3.
(a)(b)Figure 11
Graphical sketching of the numerical results provided in Table4.
(a)(b)Figure 12
Graphical sketching of the numerical results provided in Table5.
(a)(b)After performing the simulation study based on VaR and TVaR, we observed that the proposed model is better than the other compared models because it has higher values of the risk measures (VaR and TVaR).


## 8. Practical Illustration of the Proposed Model

This section deals with the practical illustration of the proposed model using a real-life insurance dataset. Additionally, we use the same data to calculate the actuarial measures of the Weibull and HTBPT–Weibull distributions.


### 8.1. An Application to Vehicle Insurance Loss Data

In this part of the study, we consider a dataset from insurance sciences to show the practicability of the proposed distribution. These data have also been studied by Zhao et al. [23] and can be found at http://www.businessandeconomics.mq.edu.au. The comparison of the HTBPT–Weibull distribution is made with the other well-known distributions including the two-parameter Weibull, Lomax, Burr-XII (B-XII), three-parameter new beta-power transformed Weibull (NBPTW), four-parameter Kumaraswamy–Weibull (Ku–W), and new Weibull Burr-XII (NWB-XI) distributions. The cdfs of the competing distributions are(1)
Weibull distribution(35)Gx;α,γ=1−e−γxα,x>0,α,γ>0.(2)
W-Loss distribution(36)Gx;α,γ,σ=1−σe−γxασ+γxα,x>0,α,γ,σ>0.(3)
NBPTW distribution(37)Gx;α,β,γ=β1−e−γxα−e−γxαβ,x>0,α,β,γ>0,β≠1.(4)
Kumaraswamy–Weibull(38)Gx;α,γ,c,t=1−1−1−e−γxαct,x>0,α,γ,c,t>0.(5)
Lomax(39)Gx;α,γ=1−1+xγ−α,x>0,α,γ>0.(6)
NWB-XII(40)Gx;α,γ,c,t=1−exp−αtlog1+xcγ,x>0,c,t,α,γ>0.(7)
B-XII(41)Gx;c,t=1−1+xc−t,x>0,c,t>0.The comparison is made using some certain analytical measures called the Akaike information criterion (AIC) (Akaike [24]), Bayesian information criterion (BIC) (Schwarz [25]), Hannan–Quinn information criterion (HQIC) (Hannan and Quinn [26]), and consistent Akaike information criterion (CAIC) (Bozdogan [27]). These measures are as follows:The AIC is given by(42)AIC=2k−2ℓ.The CAIC is(43)CAIC=2nkn−k−1−2ℓ.The BIC is calculated as follows:(44)BIC=klogn−2ℓ.The HQIC is(45)HQIC=2kloglogn−2ℓ,where k is the number of model parameters, n is the sample size, and ℓ refers to the log-likelihood function evaluated at MLEs.For the calculation of the analytical measures, we use the optim R function with the argument method = “SANN.” Smaller values of these analytical measures indicate the best fit. The MLEs of the model parameters along with the analytical measures are reported in Tables6 and 7, respectively. Referring to the results provided in Table 7, we found that the HTBPT–Weibull distribution is a superior model and provides the best fit among the other considered competitors. In support of Table 7, the estimated pdf and cdf of the proposed distribution are shown in Figure 13, which closely fit the proposed model. The probability-probability (PP) plot and Kaplan–Meier survival plot of the HTBPT–Weibull model are shown in Figure 14. From the PP and Kaplan–Meier survival plots, it is clear that the proposed is a good candidate model for modeling heavy-tailed insurance datasets. Furthermore, the quantile-quantile (QQ) plot and box plot of the data are shown in Figure 15.Table 6
The MLEs of the competing models for the vehicle insurance loss data.
Distributionsα^β^γ^c^t^σ^HTBPT–Weibull0.991.060.004Weibull1.010.003W-Loss0.941.291.63Ku–W0.870.0081.501.12NBPTW0.901.180.86Lomax0.4930.00NWB-XII0.0825.550.061.22B-XII0.044.42Table 7
Analytical measures of the competing models for the vehicle insurance loss data.
DistributionsAICBICCIACHQICHTBPT–Weibull430.6435.0431.4432.0Weibull432.3439.2434.0435.7W-Loss432.1437.5433.7434.9Ku–W432.2438.1433.7434.1NBPTW434.8440.0435.5437.2Lomax460.1463.0460.5461.1NWB-XII431.7437.6433.2433.6B-XII503.4506.3503.8504.4Figure 13
The estimated pdf and cdf of the HTBPT–Weibull distribution for the vehicle insurance loss data.
(a)(b)Figure 14
The PP and Kaplan–Meier survival survival plots of the HTBPT–Weibull distribution for the vehicle insurance loss data.
(a)(b)Figure 15
The QQ and box plots of the vehicle insurance loss data.
(a)(b)


### 8.2. Computation of the Actuarial Measures Using Real Data

In this subsection, we calculate and compare the values of the VaR and TVaR measures of the Weibull and HTBPT–Weibull distributions using the data discussed in subsection 8.1. The numerical results are reported in Table 8.Table 8
Results for the actuarial measures using the vehicle insurance loss data.
DistributionsParametersSignificance levelVaRTVaRWeibullα^ = 1.0120.700326.949593.7770.750375.850642.4000.800435.601701.8440.850512.495778.385γ^ = 0.0030.900620.646886.1050.950805.0421069.8860.975988.9441253.2820.9991838.5902101.309HTBPT–Weibullα^ = 0.9910.700335.147691.1380.750386.355748.6440.800449.103819.083β^ = 1.0610.850530.105909.978γ^ = 0.0040.900644.4431038.2380.950840.2901257.8590.9751036.5311477.8690.9991951.4232503.161


## 8.1. An Application to Vehicle Insurance Loss Data

In this part of the study, we consider a dataset from insurance sciences to show the practicability of the proposed distribution. These data have also been studied by Zhao et al. [23] and can be found at http://www.businessandeconomics.mq.edu.au. The comparison of the HTBPT–Weibull distribution is made with the other well-known distributions including the two-parameter Weibull, Lomax, Burr-XII (B-XII), three-parameter new beta-power transformed Weibull (NBPTW), four-parameter Kumaraswamy–Weibull (Ku–W), and new Weibull Burr-XII (NWB-XI) distributions. The cdfs of the competing distributions are(1)
Weibull distribution(35)Gx;α,γ=1−e−γxα,x>0,α,γ>0.(2)
W-Loss distribution(36)Gx;α,γ,σ=1−σe−γxασ+γxα,x>0,α,γ,σ>0.(3)
NBPTW distribution(37)Gx;α,β,γ=β1−e−γxα−e−γxαβ,x>0,α,β,γ>0,β≠1.(4)
Kumaraswamy–Weibull(38)Gx;α,γ,c,t=1−1−1−e−γxαct,x>0,α,γ,c,t>0.(5)
Lomax(39)Gx;α,γ=1−1+xγ−α,x>0,α,γ>0.(6)
NWB-XII(40)Gx;α,γ,c,t=1−exp−αtlog1+xcγ,x>0,c,t,α,γ>0.(7)
B-XII(41)Gx;c,t=1−1+xc−t,x>0,c,t>0.The comparison is made using some certain analytical measures called the Akaike information criterion (AIC) (Akaike [24]), Bayesian information criterion (BIC) (Schwarz [25]), Hannan–Quinn information criterion (HQIC) (Hannan and Quinn [26]), and consistent Akaike information criterion (CAIC) (Bozdogan [27]). These measures are as follows:The AIC is given by(42)AIC=2k−2ℓ.The CAIC is(43)CAIC=2nkn−k−1−2ℓ.The BIC is calculated as follows:(44)BIC=klogn−2ℓ.The HQIC is(45)HQIC=2kloglogn−2ℓ,where k is the number of model parameters, n is the sample size, and ℓ refers to the log-likelihood function evaluated at MLEs.For the calculation of the analytical measures, we use the optim R function with the argument method = “SANN.” Smaller values of these analytical measures indicate the best fit. The MLEs of the model parameters along with the analytical measures are reported in Tables6 and 7, respectively. Referring to the results provided in Table 7, we found that the HTBPT–Weibull distribution is a superior model and provides the best fit among the other considered competitors. In support of Table 7, the estimated pdf and cdf of the proposed distribution are shown in Figure 13, which closely fit the proposed model. The probability-probability (PP) plot and Kaplan–Meier survival plot of the HTBPT–Weibull model are shown in Figure 14. From the PP and Kaplan–Meier survival plots, it is clear that the proposed is a good candidate model for modeling heavy-tailed insurance datasets. Furthermore, the quantile-quantile (QQ) plot and box plot of the data are shown in Figure 15.Table 6
The MLEs of the competing models for the vehicle insurance loss data.
Distributionsα^β^γ^c^t^σ^HTBPT–Weibull0.991.060.004Weibull1.010.003W-Loss0.941.291.63Ku–W0.870.0081.501.12NBPTW0.901.180.86Lomax0.4930.00NWB-XII0.0825.550.061.22B-XII0.044.42Table 7
Analytical measures of the competing models for the vehicle insurance loss data.
DistributionsAICBICCIACHQICHTBPT–Weibull430.6435.0431.4432.0Weibull432.3439.2434.0435.7W-Loss432.1437.5433.7434.9Ku–W432.2438.1433.7434.1NBPTW434.8440.0435.5437.2Lomax460.1463.0460.5461.1NWB-XII431.7437.6433.2433.6B-XII503.4506.3503.8504.4Figure 13
The estimated pdf and cdf of the HTBPT–Weibull distribution for the vehicle insurance loss data.
(a)(b)Figure 14
The PP and Kaplan–Meier survival survival plots of the HTBPT–Weibull distribution for the vehicle insurance loss data.
(a)(b)Figure 15
The QQ and box plots of the vehicle insurance loss data.
(a)(b)


## 8.2. Computation of the Actuarial Measures Using Real Data

In this subsection, we calculate and compare the values of the VaR and TVaR measures of the Weibull and HTBPT–Weibull distributions using the data discussed in subsection 8.1. The numerical results are reported in Table 8.Table 8
Results for the actuarial measures using the vehicle insurance loss data.
DistributionsParametersSignificance levelVaRTVaRWeibullα^ = 1.0120.700326.949593.7770.750375.850642.4000.800435.601701.8440.850512.495778.385γ^ = 0.0030.900620.646886.1050.950805.0421069.8860.975988.9441253.2820.9991838.5902101.309HTBPT–Weibullα^ = 0.9910.700335.147691.1380.750386.355748.6440.800449.103819.083β^ = 1.0610.850530.105909.978γ^ = 0.0040.900644.4431038.2380.950840.2901257.8590.9751036.5311477.8690.9991951.4232503.161


## 9. Concluded Remarks

The statistical distributions are very useful for modeling financial and actuarial datasets. Among them, the heavy-tailed distributions are very prominent for modeling financial and other related datasets. A number of methods to define heavy-tailed distributions have been studied in the literature. However, unfortunately, these methods still have certain limitations. To address the issues associated with the former methods, we introduced a new heavy-tailed family of distributions. By using the proposed method, a new three-parameter special model, called the heavy-tailed beta power transformed Weibull distribution is studied in detail. Actuarial measures of the proposed model are calculated, and a simulation study is conducted to show the usefulness of the proposed method in actuarial sciences. The applicability of the family of distributions has been illustrated using a real-life heavy-tailed dataset from insurance and shown that the proposed model performs reasonably well compared to other well-known heavy-tailed distributions.



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*Source: 5580228-2021-04-02.xml*