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

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.*e actuaries are often in search of such types of distributions to provide the best description of the actuarial and financial data. *is 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 proposedmethod. 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.


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. ese 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]). ese 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. is 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]). us, 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. e proposed family may be called a new heavy-tailed beta-power transformed (HTBPT) family of distributions. e proposed method may be an interesting alternative to the existing literature for fitting heavytailed data. We study a new model called the heavy-tailed betapower transformed Weibull (HTBPT-Weibull) distribution as a special case of the HTBPT distributions. e 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. e simulation results show that the HTBPT-Weibull distribution has heavier tail than the superexponential distribution. e study is sectioned as follows. In Section 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.

Steps Followed for Developing the Proposed Family
Suppose that a random variable T ∈ [a 1 , a 2 ] for − ∞ ≤ a 1 < a 2 < ∞, with density function v(t), and let the random variable X has the cumulative distribution function (2) W[F(x; ψ)] must be monotonically increasing and can be differentiated e T − X family method was originally proposed by Alzaatreh et al. [21] as follows: where W[F(x; ψ)] satisfies the above conditions. Corresponding to (1), the probability density function (pdf ) is More information about the T − 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 By differentiating equation (3) with respect to t, we have Assume that v(t) is as represented in equation (4) and where F(x; 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. e density function of equation (5) is as follows: e expression (6) can also be written as e survival function (sf ) and hazard rate function (hrf ) of the proposed family are given by respectively. e random variable X with pdf (5) will be denoted by X ∼ HTBPT(x; β, ψ). For the sake of simplicity, we can omit the dependence on the parameters and simply write G(x) � G(x; β, ψ).

Description of the Proposed Model
is section deals with the introduction of the new HTBPT-Weibull distribution and its special cases.

Heavy-Tailed Beta-Power Transformed Weibull
Distribution. Referring to the cdf of the Weibull distribution with shape parameter α > 0 and scale parameter c > 0, c). A random variable say X has the HTBPT-Weibull distribution, if its cdf is e pdf, sf, and hrf of the HTBPT-Weibull distributions are given by respectively. In order to see the effect of adding β to the proposed distribution, we fixed the distribution's parameters (α � 1.2 and c � 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.
Plots for the cdf and sf of the proposed model are provided in Figure 2. Whereas, the plots of the hrf are shown in Figure 3.

Special Cases of the HTBPT-Weibull Distribution.
is subsection deals with the subcases of the proposed distribution. Let X has the HTBPT-Weibull distribution with parameters (α, β, c). en, we obtain e Weibull distribution with parameters α and c, using β � 1 e one parameter Weibull distribution with parameter α, using β � c � 1 e exponential distribution with parameter c, using β � α � 1 e Rayleigh distribution with parameter c, if β � 1 and α � 2 e one parameter HTBPT-Weibull distribution with parameters α and β, if c � 1 (new) e heavy-tailed beta-power transformed exponential distribution with parameters β and c, if α � 1 (new) e HTBPTR-Rayleigh distribution with parameters β and c, if α � 2 (new)

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

Shape of Density and Hazard Functions of the HTBPT Distributions.
e 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 to x as follows: e above equation may have more than one root. If x � x o is a root of the above equation, then it corresponds to (i) a local maximum, if (z 2 /z 2 x)log g(x; β, ψ) < 0, (ii) a local minimum, if (z 2 /z 2 x)log g(x; β, ψ) > 0, and (iii) a point of inflection, if (z 2 /z 2 x)log g(x; β, ψ) � 0. e 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: e above equation may have more than one root. If

Asymptotic Behavior of HTBPT Distributions.
is subsection is devoted to the asymptotic behavior of the HTBPT distributions.

Mixture Representation of the Density of HTBPT Distributions.
is subsection deals with the mixture representation of the density of HTBPT distributions. We where where h θ (x; ψ) � θf(x; ψ)F(x; ψ) θ− 1 is defined as the density function of exponentiated generated distributions with parameter θ in its power.

Quantile Function.
e quantile functions are in extensive use in general statistics and frequently used to generate random samples from probability distributions. Let X denotes the HTBPT random variable; then, the quantile function of X can be obtained as follows: where t is the solution of the equation β 1− t + βt � u + β, and u is the uniform distribution over (0, 1) interval.

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, 4 Complexity dispersion, skewness, and kurtosis). e r th moment of the HTBPT distributions is derived as follows: where e effects of different values of the parameters α and β on the mean, variance, skewness, and kurtosis of the HTBPT-Weibull distribution with c � 1 are illustrated in Figures 4 and 5. e moment generating function (mgf ) is another useful tool to calculate the moments of the statistical distributions. e mgf of the HTBPT distributions is given by 4.6. Incomplete Moments. e 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 the r th incomplete moment of HTBPT distributions as A lp h a

Estimation
is part of the study is concerned with the estimation of the unknown parameters of the HTBPT distributions. Let X 1 , X 2 , . . . , X n be a random sample taken from HTBPT distributions with parameters (β, ψ).
en, the corresponding log-likelihood function of this sample is e 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 Equating (z/zβ)log L(x i ; β, ψ) and (z/zψ)log L(x i ; β, ψ) to zero and simultaneously solving these expressions numerically yields the maximum likelihood estimators (MLEs) of (β, ψ).

Simulation Study
e 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). e random samples are generated from the HTBPT-Weibull distribution using the inverse cdf method. e simulation results are obtained using optim R function with the argument method � "L-BFGS-B." e 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) e parameter values are estimated using the maximum likelihood estimation method (3) e biases and mean square errors (MSEs) are calculated using the formulas respectively (4) For calculating the biases as well as the MSEs of (β, c), step (3) is repeated e simulation results are presented in Tables 1 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 c.
From the simulation results provided in Tables 1 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. is fact leads to the consistency property of the MLEs of the HTBPT-Weibull distribution.

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.

Value at Risk.
e VaR of a random variable X is the q th quantile of its cdf (Artzner [22]). If X follows the HTBPT distributions with parameters (β, ψ), then the VaR of X denoted by VaR q (X) is given by where t is the solution of the equation β 1− t + βt � u + β, and u follows the uniform distribution on interval (0, 1).

Tail Value at Risk.
Let X follows the HTBPT distributions with parameters β and ψ; then, the TVaR of X is derived as Using (6) in (27), we get Using the pdf f(x; ψ) and cdf F(x; ψ) of Weibull distribution in (28), we get On solving (29), we have

Tail Variance. Among the actuarial measures, TV is the most prominent one and deals with the tail variance beyond
VaR. e TV of the HTBPT-Weibull distributed random variable is derived as follows: (32) By solving the above equations, we get By referring to equations (30) and (33) and substituting in (31), we get the TV of HTBPT-Weibull distribution.

Tail Variance Premium.
In this subsection, we discuss another most important risk measure called TVP. e TVP of the HTBPT-Weibull distribution is derived as follows: TVP q (X) � TVaR q + δTV q , (34) where 0 < δ < 1. By substituting (30) and (31) in (34), we can get the TVP of the proposed distribution.

Numerical Study of the Risk Measures.
e 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. e followed steps are given by (1) A random sample of size n � 100 is generated from the Weibull and HTBPT-Weibull distributions, and the maximum likelihood method is adopted to estimate the model parameters.  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).

Practical Illustration of the Proposed Model
is 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.

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. ese data have also been studied by Zhao et al. [23] and can be found at http://www.businessandeconomics.mq. edu.au. e comparison of the HTBPT-Weibull distribution (2) W-Loss distribution (38) (6) NWB-XII (40) e 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]). ese measures are as follows: e AIC is given by  Table 4.  Table 3.
Complexity 13 e BIC is calculated as follows: e HQIC is 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. e MLEs of the model parameters along with the analytical measures are reported in Tables 6  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. e 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.

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.
e numerical results are reported in Table 8.

Conflicts of Interest
e authors declare that they have no conflicts of interest.