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• •   CrossRef (0) A new extended alpha power transformed family of distributions: properties, characterizations and an application to a data set in the insurance sciences  Zubair Ahmada, Eisa Mahmoudi1,a, G.G. Hamedanib

aDepartment of Statistics, Yazd University, Iran;
bDepartment of Mathematical and Statistical Sciences, Marquette University, USA
Correspondence to: 1Department of Statistics, Yazd University, P.O. Box 89175-741, Yazd, Iran.
E-mail: emahmoudi@yazd.ac.ir
Received September 19, 2019; Revised December 15, 2019; Accepted January 10, 2020.
Abstract
Heavy tailed distributions are useful for modeling actuarial and financial risk management problems. Actuaries often search for finding distributions that provide the best fit to heavy tailed data sets. In the present work, we introduce a new class of heavy tailed distributions of a special sub-model of the proposed family, called a new extended alpha power transformedWeibull distribution, useful for modeling heavy tailed data sets. Mathematical properties along with certain characterizations of the proposed distribution are presented. Maximum likelihood estimates of the model parameters are obtained. A simulation study is provided to evaluate the performance of the maximum likelihood estimators. Actuarial measures such as Value at Risk and Tail Value at Risk are also calculated. Further, a simulation study based on the actuarial measures is done. Finally, an application of the proposed model to a heavy tailed data set is presented. The proposed distribution is compared with some well-known (i) two-parameter models, (ii) three-parameter models and (iii) four-parameter models.
Keywords : alpha power transformation, Weibull distribution, family of distributions, actuarial measures, characterizations, maximum likelihood estimation, Monte Carlo simulation
1. Introduction

Statistical distributions play a vital role in modeling data in applied areas such as risk management, banking, economic, financial and actuarial sciences. However, the quality of the procedures primarily depends upon the assumed probability model of the phenomenon under consideration. Among applied fields, insurance data sets are usually positive (Klugman et al., 2012), right skewed (Lane, 2000), unimodal (Cooray and Ananda, 2005) and with heavy tails (Ibragimov and Prokhorov, 2017). Right-skewness may be adequately modeled by the skewed distributions (Bernardi et al., 2012). Therefore, a number of unimodal positively skewed parametric distributions have been deployed to model such data sets (Klugman et al., 2012; Eling, 2012; Bagnato and Punzo, 2013; García et al., 2014; Bakar et al., 2015; Kazemi and Noorizadeh, 2015; Adcock et al., 2015; Landsman et al., 2016; Reynkens et al., 2017; Punzo et al., 2017; Bhati and Ravi, 2018).

Among available literature, the Pareto, Weibull, and gamma are the promising distributions used widely for modeling insurance data sets. Unfortunately, due to the monotonically decreasing shape, the Pareto distribution often does not provide a best fit to many data sets. Weibull distribution is suitable for modeling small losses, but fails to provide best fit to large losses. In addition, gamma distribution is also popular because it does not have a closed form expression of cumulative distribution function (cdf). Consequently, introducing new distributions to address these problems and cater heavy tailed data is an interesting research topic that is quite rich and growing rapidly. Therefore, researchers are often in search of finding more flexible distributions. This has been done through many different approaches such as (i) transformation method, (ii) composition of two or more distributions, (iii) compounding of distributions, and (iv) finite mixture of distributions, for details see Ahmad et al. (2019a).

Among the prominent methods, Mahdavi and Kundu (2017) recently proposed a new method for introducing statistical distributions via the cdf given by

$G (x;α1,ξ)=α1F(x;ξ)-1α1-1, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖα1>0, α1≠1, x∈ŌäØ,$

where F(x; ξ) is the distribution function of the baseline random variable depending on the vector parameter ξ ∈ ŌäØ, and α is the additional parameter.

Ahmad et al. (2019b) proposed another method, called extended alpha power transformed (Ex-APT) family of distributions with cdf given by

$G (x;α1,ξ)=α1F(x;ξ)-eF(x;ξ)α1-e, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖα1>0, α1≠e, x∈ŌäØ.$

The Ex-APT family is an extension of (1.1) by introducing the exponent term, which is a constant quantity. Because of the constant term, the Ex-APT family approach is not a flexible extension of (1.1). In this article, we introduce another new extension of (1.1), called new Ex-APT (NEx-APT) family by introducing an additional parameter β ∈ ŌäØ+. A random variable X follows the NEx-APT distribution, if its cdf is given by

$G (x;α1,β,ξ)=α1F(x;ξ)-[1-β¯F(x;ξ)]α1-β, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖα1,β>0, α1>β, x∈ŌäØ,$

where, β╠ä = 1 − β. For β = 1, the NEx-APT method captures the characteristics of (1.1) and for α1 = 1, the NEx-APT approach offers the baseline distribution characteristics. The probability density function (pdf) corresponding to (1.2) is given by

$g (x;α1,β,ξ)=f (x,ξ)[log(α1) α1F(x,ξ)+β¯]α1-β, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖ x∈ŌäØ.$

The new pdf is most tractable when F(x, ξ) and f (x, ξ) have simple analytical expressions. A random variable X with pdf (1.3) is represented by X ~ NEx-APT (x; α1, β, ξ). Furthermore, the dependence on the vector of the parameters is omitted and G(x)=G(x; α1, β, ξ) will be used for efficiency. The proposed method is very interesting with a closed form for the cdf and capable of modeling heavy tailed insurance data sets.

The rest of this work is as follows. In Section 2, we introduce a special sub-case of (1.3), called a NEx-APT Weibull (NEx-APTW) distribution by considering Weibull model as a parent distribution. Some mathematical properties are derived in Section 3. Certain characterizations of the proposed distribution are provided in Section 4. Maximum likelihood estimation and Monti Carlo simulation study are provided in Section 5. Section 6 offers the computation of the actuarial measures and simulation study based on these measures. A heavy tailed real-life application from the insurance sciences is discussed in Section 7. Here the proposed model is compared with some prominent distributions. Finally, the article is concluded in Section 8.

2. Sub-model and its special cases

In this section, we introduce a sub-model of the (1.2) and discuss its special cases.

### 2.1. A new extended alpha power transformed Weibull distribution

Let F(x; ξ) and f (x; ξ) be the cdf and pdf of the two parameters Weibull distribution given by F(x; ξ) = 1 − eγxα, x ≥ 0, α, γ > 0, and f (x; ξ) = αγxα−1eγxα, respectively, where ξ = (α, γ). Then, the cdf of the NEx-APTW distribution has the following expression

$G(x)=α1(1-e-γxα)-[1-β¯(1-e-γxα)]α1-β, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖα1,β, α,γ>0, α1>β, x≥0.$

The pdf corresponding (2.1) is given by

$g (x)=αγxα-1e-γxα[log(α1) α1(1-e-γxα)+β¯]α1-β, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖ x>0.$

The survival function (sf) and hazard function (hf) of NEx-APTW distribution are respectively, given by

$S (x)=α1-β-{α1(1-e-γxα)-[1-β¯ (1-e-γxα)]}α1-β, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖ x≥0,h (x)=αγxα-1e-γxα [log (α1) α1(1-e-γxα)+β¯]α1-β-{α1(1-e-γxα)-[1-β¯(1-e-γxα)]}, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖ x≥0.$

Figure 1 sketches different plots for the density function of the NEx-APTW distribution.

### 2.2. Special cases of NEx-APTW distribution

Let X have the NEx-APTW distribution with parameters (α1, β, α, γ). Then, the distribution of X reduces to:

• Alpha power transformed Weibull (APTW) distribution, if β = 1.

• Alpha power transformed exponential (APTE) distribution, if α = 1 and β = 1.

• Weibull distribution, if α1 = 1.

• Rayleigh distribution, if α1 = 1 and α = 2.

• Exponential distribution if α1 = 1 and α = 1.

• One parameter Weibull distribution, if α1 = 1 and γ = 1.

• One parameter New extended alpha power transformed Weibull, if γ = 1. (New)

• New extended alpha power transformed Rayleigh, if α = 2. (New)

• New extended alpha power transformed exponential, if α = 1. (New)

3. Mathematical properties

In this section, we derive some mathematical properties of the NEx-APT distribution, such as quantile function, moments and moment generating function.

### 3.1. Quantile function

The quantile function of the NEx-APT distribution, denoted by Q(u), can be obtained by inverting its cdf (1.2). Therefore, we have

$xq=Q (u)=G-1(u)=F-1(t),$

where t is the solution of the expression $α1t+β¯t-u(α1-β)-1=0$, and u ∈ (0, 1). The nonlinear expression (3.1) can be used to obtain random numbers for the NEx-APT distribution.

### 3.2. Moments

In this sub-section, we intend to derive the moments and the moment generating function of the NEx-APT distribution. Let X follow (1.3), then, the rth moment of X is derived as

$μr/=∫-∞∞xrg (x;α1,β,ξ) dx,$

and using (1.3) in (3.2), we have

$μr/=∫-∞∞xrf(x,ξ)[log (α1) α1F(x,ξ)+β¯]α1-βdx.$

The function $α1x$ has the Maclaurin series given by

$α1x=∑n=0∞(log (α1))nxnn!.$

Taking x = F (x; ξ) in (3.4), we get

$α1x=∑n=0∞(log (α1))nF (x;ξ)nn!.$

Using (3.5) in (3.3), we have

$μr/=∑n=0∞(log (α1))n+1Λr,n(α1-β) n!+β¯Λr(α1-β),$

where $Λr,n=∫-∞∞xrf (x,ξ) F (x;ξ)n dx$ and $Λr=∫-∞∞xrf (x,ξ) dx$.

Furthermore, the moment generating function of the NEx-APT random variable X, MX (t), is given by

$MX (t)=∑r,n=0∞(log (α1))n+1tr Λr,n(α1-β)n!r!+∑r=0∞trβ¯Λr(α1-β)n!r!.$
4. Characterizations of NEx-APT distribution

In this section, we present certain characterizations of the NEx-APT distribution in the following directions: (i) based on a simple relationship between two truncated moments and (ii) in terms of the reverse hazard function. It should be mentioned that for the characterization (i) the cdf is not required to have a closed form.

We present our characterizations (i)–(ii) in two subsections.

### 4.1. Characterizations based on two truncated moments

In this subsection, we present characterizations of NEx-APT distribution in terms of a simple relationship between two truncated moments. The first characterization result employs a theorem due to Glänzel (1987), see Theorem 4.1 below. Note that the result holds also when the interval H is not closed. Moreover, in it could be also applied when the cdf F does not have a closed form. As shown in Glänzel (1990), this characterization is stable in the sense of weak convergence.

Theorem 1

Let (Ω,Ōä▒, P) be a given probability space and let H = [d, e] be an interval for some d < e (d = −∞, e = ∞ might as well be allowed). Let X : Ω → H be a continuous random variable with the distribution function G and let q1and q2be two real functions defined on H such that

$E [q2(X)ŌłŻX≥x]=E [q1(X)ŌłŻX≥x] η(x), ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx∈H,$

is defined with some real function η. Assume that q1, q2C1(H), ηC2(H) and G is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation ηq1 = q2has no real solution in the interior of H. Then G is uniquely determined by the functions q1, q2, and η, particularly

$G(x)=∫axC|η′(u)η(u)q1(u)-q2(u)|exp(-s(u))du,$

where the function s is a solution of the differential equation s′ = η′ q1/(ηq1q2) and C is the normalization constant, such that ∫H dG = 1.

Proposition 1

Let X : Ω → ŌäØ be a continuous random variable and let$q1(x)=[log(α1)α1F(x;ξ)+β¯]-1$and q2(x) = q1(x)F(x; ξ) for x ∈ ŌäØ. The random variable X has pdf (1.3) only if the function η defined in Theorem 1 has the form

$η(x)=12[1+F (x;ξ)], ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx∈ŌäØ.$
Proof

Let X be a random variable with pdf (1.3), then

$(1-G (x)) E [q1 (X)ŌłŻX≥x]=∫x∞q1 (x)g (x;α1,β,ξ) dx=1α1-β∫x∞f (x;ξ) dx=1α1-β(1-F (x)), ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx∈ŌäØ,$

and

$(1-G (x)) E [q2 (X)ŌłŻX≥x]=∫x∞q2 (x)g (x;α1,β,ξ) dx=1α1-β∫x∞f (x;ξ)F (x;ξ) dx=12(α1-β)(1-F (x)2), ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx∈ŌäØ,$

and finally

$η(x)q1(x)-q2(x)=12(1+F (x))q1 (x)-q1 (x)F (x)=q1 (x)2{1+F (x)-2F (x)}=q1 (x)2{1-F (x)}, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖfor x∈ŌäØ.$

Conversely, if η is given as above, then

$s′(x)=η′(x)q1(x)η(x)q1(x)-q2(x)=f(x;ξ)1-F(x;ξ), ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx∈ŌäØ,$

therefore,

$s(x)=-log [1-F(x;ξ)], ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx∈ŌäØ.$

Now, in view of Theorem 1, X has density (1.3).

### Corollary 1

Let X : Ω → ŌäØ be a continuous random variable and let q1(x) be as in Proposition 1. The pdf of X is (1.3) only if there exist functions q2and η defined in Theorem 1 satisfying the differential equation

$η′(x)q1(x)η(x)q1(x)-q2(x)=f (x;ξ)1-F (x;ξ), ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx∈ŌäØ.$

### Corollary 2

The general solution of the differential equation in Corollary 1 is

$η(x)=[1-F (x;ξ)]-1 [-∫f (x;ξ)(q1 (x))-1 q2(x)+D],$

where D is a constant. Note that a set of functions satisfying the above differential equation is given in Proposition 1 with D = 1/2. However, it should be also noted that there are other triplets (q1, q2, η) satisfying the conditions of Theorem 1.

4.2. Characterization in terms of the reverse hazard function

The reverse hazard function, rG, of a twice differentiable distribution function, G, is defined as

$rG(x)=g(x)G(x), ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx∈support of G.$

In this subsection, we present a characterization of the NEx-APT distribution in terms of the reverse hazard function.

### Proposition 2

Let X : Ω → ŌäØ be a continuous random variable. The random variable X has pdf (1.3) only if its reverse hazard function rG (x) satisfies the following differential equation.

$rG′ (x)-f′ (x;ξ)f (x;ξ)rG (x)=f (x;ξ) ddx {[log (α) αF(x;ξ)+β¯]αF(x;ξ)-[1-β¯F(x;ξ)]}, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx∈ŌäØ.$
Proof

Is straightforward and hence omitted.

5. Estimation and simulation

In this section, we estimate the parameters of the NEx-APT distribution via the method of maximum likelihood and provide Monti Carlo simulation to evaluate the performance of these estimators.

### 5.1. Maximum likelihood estimation

Let x1, x2, . . . , xn be observed values of a random sample from NEx-APT distribution with parameters (α1, β, ξ). The log-likelihood function of this sample is

$Ōäō(Θ)=n log (α1-β)+∑i=1nlog f (xi;ξ)+∑i=1nlog {log (α1) α1F(xi;ξ)+β¯},$

where, Θ = (α1, β, ξ)T be the 3 × 1 parameter vector. The partial derivatives of (5.1) are

$∂Ōäō(Θ)∂α1=n(α1-β)+∑i=1n{log (α1)F (xi;ξ)+1} α1F(xi;ξ)-1{log (α1) α1F(xi;ξ)+β¯},∂Ōäō(Θ)∂β=-n(α1-β)-∑i=1n1{log (α1) α1F(xi;ξ)+β¯},$

and

$∂Ōäō(Θ)∂ξ=∑i=1n∂f (xi;ξ)/∂ξf (xi;ξ)+∑i=1n(log (α1))2 α1F(xi;ξ)∂F (xi;ξ)/∂ξ{log (α1) α1F(xi;ξ)+β¯}.$

Setting (∂/∂α1)Ōäō(Θ), (∂/∂β)Ōäō(Θ), and (∂/∂ξ)Ōäō(Θ) equal to zero and solving numerically these expressions simultaneously, yields the maximum likelihood estimates of (α1, β, ξ).

### 5.2. Asymptotic confidence bounds

In this subsection, we derive the asymptotic confidence intervals of the unknown parameters of the NEx-APT distributions. The simplest large sample approach is to assume that the maximum likelihood estimators (α╠é1, β╠é, ξ╠é) are approximately multivariate normal with mean (α1, β, ξ) and covariance matrix $I0-1$, where $I0-1$ is the inverse of the observed information matrix defined by

$I0-1=-E[∂2Ōäō(Θ)∂α12∂2Ōäō(Θ)∂α1∂β∂2Ōäō(Θ)∂α1∂ξ∂2Ōäō(Θ)∂β∂α1∂2Ōäō(Θ)∂β2∂2Ōäō(Θ)∂β∂ξ∂2Ōäō(Θ)∂ξ∂α1∂2Ōäō(Θ)∂ξ∂β∂2Ōäō(Θ)∂ξ2]-1=[var (α^1)cov(α^1,β^)cov(α^1,ξ^)cov(β^,α^1)var (β^)cov(β,ξ^)cov(ξ^,α^1)cov(ξ^,β)var(ξ^)].$

The second partial derivatives included in $I0-1$ are given as follows

$∂2Ōäō(Θ)∂ξ∂α1=∑i=1nα1F(x;ξ) (log α1) {α1F(xi;ξ)-1+β¯ (log α1)(∂F (x;ξ)/∂ξ)+2β¯α1-1}((log α1) α1F(x;ξ)+β¯)2,∂2Ōäō(Θ)∂α12=∑i=1nα1F(x;ξ)-1 {(log α1)β¯F (x;ξ)2+β¯+βα1-1α1F(xi;ξ)-1 (F (x;ξ)-1)-α1F(x;ξ)-1}((log α1) α1F(x;ξ)+β¯)2,∂2Ōäō(Θ)∂ξ2=∑i=1nα1F(x;ξ)(∂F (x;ξ)/∂ξ) {(log α1) α1F(x;ξ)+β¯ (log α1)3∂F (x;ξ)/∂ξ)+β¯}((log α1) α1F(x;ξ)+β¯)2+∑i=1nf (x;ξ)(∂2f (x;ξ)/∂ξ2)-(∂F (x;ξ)/∂ξ)2(f (x;ξ))2,∂2Ōäō(Θ)∂α1∂β=n(α1-β)+∑i=1n{(log α1) F (x;ξ)+1} α1F(x;ξ)-1((log α1) α1F(x;ξ)+β¯)2,∂2Ōäō(Θ)∂β2=-n(α1-β)2-∑i=1n1((log α1) α1F(x;ξ)+β¯)2,$

and

$∂2Ōäō(Θ)∂β∂ξ=∑i=1n(log α1) α1F(x;ξ)(∂F (x;ξ)/∂ξ)((log α1) α1F(x;ξ)+β¯)2.$

The above approach is used to derive the (1 − δ)100% confidence intervals for the parameters α1, β, and ξ as in the following forms $α^1±Zδ/2var(α^1),β^±Zδ/2var(β^)$, and $ξ^±Zδ/2var(ξ^)$, where Zδ/2 is the upper (δ/2)th percentile of the standard normal distribution.

### 5.3. Simulation study of the NEx-APTW distribution

In order to evaluate the performances of the maximum likelihood estimators of the (α1, β, ξ), a simulation study is conducted. The process is carried out as:

• We generate N = 750 samples of sizes n = 25, 50, . . . , 750 from the NEx-APTW distribution.

• Initial values for the parameters are selected as given in

• Compute the biases and MSEs given by $Bias(w^)=(1/750) Σi=1750(w^i-w)$ and $MSE(w^)=(1/750) Σi=1750(w^i-w)2$, for w = (α1, β, ξ), respectively.

6. Actuarial measures of the NEx-APTW distribution

One of the most important tasks of actuaries is to evaluate the exposure to market risk in a portfolio of instruments, which arise from changes in underlying variables such as prices of equity, interest rates or exchange rates. In this section, we calculate two well-known and important risk measures Value at Risk (VaR) and Tail VaR (TVaR) for the proposed distribution, which play a crucial role in portfolio optimization under uncertainty. Furthermore, based on these measures, a simulation study is performed and we show that the proposed distribution has heavier tails the Weibull and exponentiated Weibull (EW) distributions.

### 6.1. Value at risk

In the context of actuarial sciences, the measure VaR is widely used by practitioners as a standard financial market risk. It is also known as the quantile risk measure or quantile premium principle. VaR of a random variable X is the qth quantile of its cdf and specified with a given degree of confidence say q (typically 90, 95, or 99%), see Artzner (1999). If X has pdf (1.3), then the VaR of X is given by

$xq=F-1(t),$

where t is the solution of the expression $α1t+β¯t-q (α1-β)-1=0$.

### 6.2. Tail value at risk

Another important measure is TVaR, also known as conditional tail expectation (CTE) used to quantify the expected value of the loss given that an event outside a given probability level has occurred. Let X follow the NEx-APTW distribution, then the TVaR of X is defined as

$TVaR=11-q∫VaRq∞xg (x;α1,β,ξ) dx.$

Using (1.3) in (6.2), we have

$TVaR=αγ(1-q)(α1-β)∫VaRq∞xα+1-1e-γxα [log (α1) α1(1-e-γxα)+β¯] dx.$

Using the series $α1x=Σn=0∞(log (α1))n (xn/n!)$,we have

$TVaR=αγ(1-q)(α1-β)∫VaRq∞xα+1-1e-γxα [∑i=1∞(log α1))i+1n!(1-e-γxα)n+β¯] dx,$

or

$TVaR=Σi,n=0∞(log(α1))i+1(nj)n!Γ (1α+1,γ (j+1)(VaRq)α)(1-q)(α1-β)+β¯Γ (1α+1,γ (VaRq)α)(1-q)(α1-β).$

Based on actuarial measures, a simulation study is performed for the Weibull, exponentiated (EW) and proposed model for the selected parameters values. A model with higher values of the Risk measures (VaR and TVaR) is said to have a heavier tail. The simulated results provided in Tables 2 and 3 show that the proposed model has higher values of the risk measures than the other competitive distributions.

Figure 4 displays the simulation results provided in Table 2; in addition, Figure 5 displays the simulation results provided in

7. An application to heavy tailed insurance data

Actuaries are looking for new distributions to provide an adequate fit to heavy tailed data in actuarial, financial sciences and related areas. In this section, we analyze a real data set from insurance sciences to demonstrate the flexibility of the NEx-APTW distribution. We also calculate actual measures of the Weibull, EW, and NEx-APTW distributions using a real data set.

### 7.1. Hospital cost insurance data

The data set representing hospital costs in the state of Wisconsin is published by the Office of the Health Care Information, Wisconsin’s Department of Health and Human Resources. The data set is available at: https://www.dhs.wisconsin.gov/stats/index.htm. The comparison of the proposed method is made with the other ten (two, three and four parameters) well-known distributions. The cdf’s of the competitive distributions are:

• Weibull distribution

$G (x)=1-e-γxα, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx≥0, α,γ>0.$

• Ex-APTW distribution

$G (x)=α1(1-e-γxα)-e(1-e-γxα)α1-e, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx≥0, α1,α,γ>0, α1≠e.$

• Lomax distribution

$G (x)=1-(1+γx)-α, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx≥0, α,γ>0.$

• Burr-XII (B-XII) distribution

$G (x)=1-(1+xγ)-α, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx≥0, α,γ>0.$

• Pareto distribution

$G (x)=1-(γx)α, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx≥0, α,γ>0.$

• The alpha power transformed Weibull (APTW) distribution

$G (x)=α1(1-e-γxα)-1α1-1, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx≥0, α1,α,γ>0, α1≠1.$

• The Marshall-Olkin (MOW) distribution

$G (x)=(1-e-γxα)σ+(1-σ)(1-e-γxα), ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx≥0, α,γ,σ>0.$

• EW distribution

$G (x)=(1-e-γxα)a, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx≥0, α,γ,a>0.$

• Kumaraswamy Weibull (Ku-W) distribution

$G (x)=1-{1-(1-e-γxα)a}b, ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx≥0, α,γ,a,b>0.$

• The beta Weibull (BW) distribution

$G (x)=I(1-e-γxα)(a,b), ŌĆŖŌĆŖ ŌĆŖŌĆŖ ŌĆŖŌĆŖx≥0, α,γ,a,b>0.$

To determine the goodness-of-fit among the applied distributions, we consider certain goodness-of-fit measures such as Cramer-Von-Messes (CM) test statistic, Anderson Darling (AD) test statistic and Kolmogorov-Simonrove (KS) test statistic with corresponding p-value. The measure are given by:

• The AD test statistic

$AD=-n-1n∑i=1n(2i-1) [log G(xi)+log{1-G (xn-i+1)}],$

where n = the sample size, xi = the ith sample, calculated when the data is sorted in ascending order.

• The CM test statistic

$CM=112n+∑i=1n[2i-12n-G (xi)]2.$

• The KS test statistic is given by

$KS=supx [Gn (x)-G (x)],$

where Gn (x) is the empirical cdf and supx is the supremum of the set of distances.

A distribution with lower values of these analytical measures is considered as a good candidate model among the applied distributions for the underlying data sets. Based on the considered measures, the NEx-APTW distribution has the lowest values among all fitted models for the hospital cost insurance data. Table 4 reports parameter values with standard errors in parenthesis. In support of the numerical measures provided in Table 5, the empirical cdf and sf of the NEx-APTW are plotted in Figure 6. From Figure 6, we can see that the proposed model fit empirical cdf and sf very closely. In addition, the PP plot of the NEx-APTW distribution for the respective data set is plotted in Figure 7 and shows that the proposed provide best fit to the considered data. The box plot of the data set is also sketched in Figure 7 showing that the hospital cost insurance data is skewed to the right.

### 7.2. Computation of actuarial measures using real data set

In this sub-section, we compute VaR and TVaR measures of Weibull, EW and the NEx-APTW distributions using estimated parameters values analyzed in Subsection 7.1. Table 6 reports the numerical results. A model with higher values of the risk measures possesses the heavier tails. The numerical results for the actuarial measures of the proposed and the other distributions show that the proposed distribution has a heavier tail than Weibull and EW distributions. In addition, it can be used as a good candidate model for modeling heavy tailed insurance data sets.

8. Concluding remarks

In this article, a new family of distributions called a new extended alpha power transformed family has been proposed. The proposed method examines a four-parameter special model of a new extended alpha power transformed Weibull distribution. Actuarial measures of the proposed model are also calculated and a simulation study is conducted to show the usefulness of the proposed method in actuarial sciences. A practical application to the heavy tailed insurance data is analyzed and the comparison of the proposed model with the other nine well-known competitors are presented. Actuarial measures based on a real data set is also calculated which shows that the proposed model may be a good candidate model to analyze actuarial data sets. We hope that the proposed method will attract a wider applications in actuarial sciences and related fields.

Acknowledgments
The authors are grateful to the Editor-in-Chief, the Associate Editor and anonymous referees for their valuable comments and suggestions that helped improve the manuscript. The first two authors also acknowledge the support of Yazd University, Iran.
Data availability statement
This work is a methodological development and has been applied on secondary data related to the hospital cost insurance data if required, data can be provided to interested readers.
Dedication
This article is drafted from the Ph.D. work of the first author who would like to dedicate this article to the memory of his late parents.
Figures Fig. 1. Different plots for the density function of the NEx-APTW distribution. NEx-APTW = new extended alpha power transformed Weibull Fig. 2. Plots of the estimated parameters and MSEs of the new extended alpha power transformed Weibull distribution for . MSE = mean squared error. Fig. 3. Plot of the biases of the parameters of the new extended alpha power transformed Weibull distribution for . Fig. 4. Graphical sketching of the results provided in . VaR = Value at Risk; TVaR = Tail VaR; EW = exponentiated Weibull; NEx-APTW = new extended alpha power transformed Weibull; W = Weibull. Fig. 5. Graphical sketching of the results provided in . VaR = Value at Risk; TVaR = Tail VaR; EW = exponentiated Weibull; NEx-APTW = new extended alpha power transformed Weibull. Fig. 6. Estimated cdf and Kaplan-Meier survival plots of the new extended alpha power transformed Weibull distribution for the hospital cost insurance data. Fig. 7. PP plot of the new extended alpha power transformedWeibull distribution and box plot for the hospital cost insurance data.
TABLES

### Table 1

Simulation results for different combination of parameters values of new extended alpha power transformed Weibull distribution

α1 = 1.2, α = 0.5, γ = 1, β = 0.9

nParametersMLEBiasMSE
25$α1^$2.9051.7057.485
α̂2.1971.6976.860
γ̂4.0983.09812.38
β̂1.7180.8183.805

50$α1^$2.5501.3506.123
α̂1.9141.4145.932
γ̂3.5692.56910.25
β̂1.6820.7823.525

100$α1^$1.9450.7453.829
α̂1.5651.0654.843
γ̂2.8251.8257.272
β̂1.6030.7033.020

150$α1^$1.7920.5922.957
α̂1.2250.7253.381
γ̂2.4001.4005.543
β̂1.3820.4822.127

250$α1^$1.3980.1981.127
α̂0.8440.3441.718
γ̂1.6950.6952.767
β̂1.0840.1840.951

350$α1^$1.2700.0700.545
α̂0.7390.2391.192
γ̂1.4520.4521.790
β̂1.0350.8650.681

450$α1^$1.2450.0450.309
α̂0.5900.0900.495
γ̂1.2660.2661.044
β̂0.9480.0480.268

550$α1^$1.2160.0160.157
α̂0.5780.0780.410
γ̂1.1690.1690.671
β̂0.9290.0290.196

650$α1^$1.2260.0260.143
α̂0.5490.0490.244
γ̂1.1830.0830.331
β̂0.9340.0140.101

750$α1^$1.1940.0050.031
α̂0.5200.0400.189
γ̂1.0520.0520.209
β̂0.9200.0100.096

MLE = maximum likelihood estimators; MSE = mean squared error.

### Table 2

Simulation results for VaR abd TVaR of the NEx-APTW and other fitted distributions

DistributionParametersLevel of significanceVaRTVaR
Weibull0.7003.69511.208
0.7504.67412.617
0.8005.99514.447
α̂ = 0.50.8507.88516.970
γ̂ = 0.70.90010.88920.828
0.95016.88528.199
0.97523.88636.494
0.99967.95585.868

EW0.7001.4143.076
0.7501.7003.381
â = 0.50.8002.0603.758
α̂ = 0.50.8502.5344.250
γ̂ = 0.70.9003.2174.949
0.9504.4086.155
0.9755.6147.369
0.99911.27213.033

NEx-APTW0.70036.02946.156
0.75037.73348.016
$α1^=0.9$0.80039.85950.332
α̂ = 0.50.85042.66553.380
γ̂ = 0.70.90046.74357.792
β̂ = 0.30.95054.03865.646
0.97561.73173.881
0.999102.339116.836

VaR = Value at Risk; TVaR = Tail VaR; EW = exponentiated Weibull; NEx-APTW = new extended alpha power transformed Weibull.

### Table 3

Simulation results for VaR abd TVaR of the NEx-APTW and other fitted distributions.

DistributionParametersLevel of significanceVaRTVaR
Weibull0.7003.60115.695
0.7504.83117.998
0.8006.59321.084
α̂ = 0.50.8509.28825.500
γ̂ = 0.70.90013.90532.569
0.95024.05946.976
0.97537.11964.354
0.999137.146185.066

EW0.7001.0522.439
0.7501.2862.694
â = 0.50.8001.5823.010
α̂ = 0.50.8501.9763.424
γ̂ = 0.70.9002.5484.015
0.9503.5545.039
0.9754.5786.072
0.9999.39810.902

NEx-APTW0.70056.13178.680
0.75059.72482.844
$α1^=0.9$0.80064.26588.080
α̂ = 0.50.85070.35295.060
γ̂ = 0.70.90079.371105.338
β̂ = 0.30.95095.977124.064
0.975114.069144.267
0.999217.212257.047

VaR = Value at Risk; TVaR = Tail VaR; EW = exponentiated Weibull; NEx-APTW = new extended alpha power transformed Weibull.

### Table 4

Estimated values with standard error (in parenthesis) of the proposed and other competitive models for the hospital cost insurance data

Distribution$α1^$α̂γ̂σ̂β̂â
NEx-APTW3.419 (2.751)0.696 (0.028)1.469 (0.231)0.521 (1.613)
Ex-APTW3.893 (2.751)0.865 (0.183)1.665 (0.069)
Weibull0.674 (0.022)1.918 (0.084)
Lomax1.569 (0.207)0.365 (0.074)
B-XII2.885 (0.129)0.813 (0.026)
Pareto2.970 (0.420)1.909 (2.895)
APTW0.185 (2.910)1.229 (0.403)0.762 (0.945)
MOW0.808 (0.955)1.186 (0.863)0.387 (0.145)
EW1.944 (0.459)2.564 (0.261)0.487 (0.063)
Ku-W0.847 (0.805)1.086 (1.290)2.567 (2.906)4.583 (1.095)
BW0.441 (0.089)0.828 (0.928)2.097 (0.761)3.804 (1.092)

NEx-APTW = new Ex-APTW; Ex-APTW = extended APTW; B-XII = Burr-XII; APTW = alpha power transformed Weibull; MOW = Marshall-Olkin; EW = exponentiated Weibull; Ku-W = Kumaraswamy Weibull; BW = beta Weibull.

### Table 5

Goodness-of-fit measures of the proposed and other competitive models for the hospital cost insurance data

DistributionCMADKSp-value
NEx-APTW0.0930.8900.0310.695
Ex-APTW0.1090.9320.0340.585
Weibull0.1120.9360.0350.035
Lomax0.4642.6230.0780.002
B-XII0.2501.4260.0450.229
Pareto0.5601.7080.0580.247
APTW0.1520.9570.0380.414
MOW0.1550.9610.0370.447
EW0.1680.9970.0490.150
Ku-W0.1590.9440.0390.396
BW1.9481.9120.1970.206

CM = Cramer-Von-Messes; AD = Anderson Darling; KS = Kolmogorov-Simonrove; NEx-APTW = new Ex-APTW; Ex-APTW = extended APTW; B-XII = Burr-XII; APTW = alpha power transformed Weibull; MOW = Marshall-Olkin; EW = exponentiated Weibull; Ku-W = Kumaraswamy Weibull; BW = beta Weibull.

### Table 6

Results for the actuarial measures using health care insurance data

DistributionParametersLevel of significanceVaRTVaR
Weibull0.7000.7100.914
0.7501.0471.275
0.8001.3231.709
α̂ = 0.6740.8501.7931.907
γ̂ = 1.9180.9002.1342.086
0.9502.6002.395
0.9752.9062.706
0.9993.1452.875

EW0.7000.0230.021
0.7500.1080.067
â = 0.4870.8000.2900.153
α̂ = 1.9440.8500.7870.490
γ̂ = 2.5640.9000.9730.630
0.9501.1220.959
0.9751.6451.141
0.9991.7601.397

NEx-APTW0.7000.9251.563
0.7501.7542.901
$α1^=3.419$0.8001.8953.396
α̂ = 0.6960.8502.3655.303
γ̂ = 1.4690.9003.9335.925
β̂ = 0.5210.9504.8846.964
0.9755.5107.313
0.9996.3397.890

VaR = Value at Risk; TVaR = Tail VaR; EW = exponentiated Weibull; NEx-APTW = new extended alpha power transformed Weibull.

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