Statistical Analysis and Several Estimation Methods of New Alpha Power-Transformed Pareto Model with Applications in Insurance

Abstract

This article defines a new distribution using a novel alpha power-transformed method extension. The model obtained has three parameters and is quite effective in modeling skewed, complex, symmetric, and asymmetric datasets. The new approach has one additional parameter for the model. Certain distributional and mathematical properties are investigated, notably reliability, quartile, moments, skewness, kurtosis, and order statistics, and several approaches of estimation, notably the maximum likelihood, least square, weighted least square, maximum product spacing, Cramer-Von Mises, and Anderson Darling estimators of the model parameters were obtained. A Monte Carlo simulation study was conducted to evaluate the performance of the proposed techniques of estimation of the model parameters. The actuarial measures are computed for our recommended model. At the end of the paper, two insurance applications are illustrated to check the potential and utility of the suggested distribution. Evaluation using four selection criteria indicates that our recommended model is the most appropriate probability model for modeling insurance datasets.

1. Introduction

Statistical models are commonly applied to describe real-world phenomena and are very much required for data modeling in different disciplines, to analyze and to know the pattern of the data generated by the stochastic nature of the system from the field of knowledge. Further, the baseline distribution is unsuitable for fitting several datasets. To solve this problem, a family of distribution is obtained using numerous techniques. These new techniques extend the classical model to obtain an approach distribution. The suggested tool is more efficient in fitting different real-world phenomena, likely engineering (see Wang et al. [1,2]), biomedical, actuarial science, medicine, insurance, and environmental. Notable examples of these generated distribution families include the beta-normal model, provided by Eugene et al. [3], and the inverse Weibull-G model, considered by Amal et al. [4]. Cordeiro et al. [5] introduced the Kumaraswamy-G family, Marshall, and Olkin [6] and defined a new method for adding a parameter to a family of distributions, Zagrofos and Balakrishnan [7] defined the gamma-G class of distributions. In the same line, Morad Alizadeh et al. [8] investigated the Gompertz-G models and Brito et al. [9] proposed the Topp–Leone odd log-logistic family of distributions. Thomas et al. [10] generated the power generalized DUS family. Alzaatreh et al. [11] have introduced the transformed-transformer(T-X) class of distributions and Almetwaly and Meraou [12] have considered the sine Nadaraja Haghighi model.

Recently, Barry [13] defined Pareto (Par) distribution, which can be used in various applications, particularly in survival, medical science, hydrology, finance, and insurance. Numerous studies and generalized extensions of the Par distribution have been implemented in several fields to bring further flexibility to the Par distribution. One may refer to Abdul Majid et al. [14], Akinsete et al. [15], Bourguignon et al. [16], and Chhetri et al. [17]. The cumulative distribution function (cdf) and density probability function (pdf) of the parameter model can be investigated as

$$\begin{array}{cc}\hfill W(t;\eta ,\delta )& =1-{\left({\displaystyle \frac{\eta }{t}}\right)}^{\delta },t>\eta ,\eta ,\delta >0,\hfill \end{array}$$

(1)

and

$$\begin{array}{cc}\hfill w(t;\eta ,\delta )& ={\displaystyle \frac{\delta {\eta }^{\delta }}{t^{\delta +1}}}.\hfill \end{array}$$

(2)

In the last couple of decades, a useful method for generating a flexible probability model, called a new class of distribution, was suggested by Mahdavi and Kundu [18], listed as the alpha power transformation (APT) family of distributions. The proposed class of probability models can be described as follows. Suppose X is a continuous RV. Then, X is said to have an APT class of distribution if its cdf and pdf are like

$$F\left(x\right)\left\{\begin{array}{cc}{\displaystyle \frac{{\alpha }^{K(x,\psi )}-1}{\alpha -1}},\hfill & \mathrm{if}\alpha >0;\alpha \ne 1,\hfill \\ \\ K(x,\psi ),\hfill & \mathrm{if}\alpha =1,\hfill \end{array}\right.$$

and

$$f\left(x\right)\left\{\begin{array}{cc}{\displaystyle \frac{log\alpha }{\alpha -1}}{\alpha }^{K(x,\psi )}k(x,\psi ),\hfill & \mathrm{if}\alpha >0;\alpha \ne 1,\hfill \\ \\ k(x,\psi ),\hfill & \mathrm{if}\alpha =1,\hfill \end{array}\right.$$

where $\alpha$ is an additional parameter. Many authors applied the APT family frequently in numerous studies. For example, the APT Weibull (APTW) distribution (Nassar et al. [19]), APT generalized exponential (APTGE) model (Dey et al. [20]), APT Gompertz (APTGO) model (Eghwerido et al. [21]), and APT power Lindley (APTPL) distribution (Hassan et al. [22]). In the same context, APT Xgamma (APTXG) distribution (Shivanshi et al. [23]), APT extended exponential (APTEE) model (Hassan et al. [24]), APT Extended power Lindley (APT-EPL) (Fatehi and Chhaya [25]), and APT Dagum (APT-D) model (Reyad et al. [26]). In this work, we introduced an extension of the APT family, which is referred to as a new alpha power-transformed (NAPT) class of distributions. The cdf and pdf of the NAPT model are

$$G(t,\psi ,\alpha )={\alpha }^{-log\left({\displaystyle \frac{1}{K(t,\psi )}}\right)}={\alpha }^{log\left(K(t,\psi )\right)},t\in \mathbb{R},\alpha >1,$$

and

$$g(t,\psi ,\alpha )={\displaystyle \frac{1}{K(t,\psi )}}\left(log\alpha k(t,\psi ){\alpha }^{-log\left({\displaystyle \frac{1}{K(t,\psi )}}\right)}\right)={\displaystyle \frac{1}{K(t,\psi )}}\left(log\alpha k(t,\psi ){\alpha }^{log\left(K(t,\psi )\right)}\right).$$

This article focused on inventing a new model derived from Par distributions to explore more complex datasets and establish various properties. The obtained model was formed using the NAPT technique, and we listed it as the new alpha power-transformed Pareto (NAPT-Par) model. More advantages of the recommended NAPT-Par model can be considered. It can be applied to several sectors, notably insurance, finance, economics, environmental sciences, engineering, survival, and biology. Its applications extend to analyzing income and wealth inequality, firm size data, and Internet traffic datasets, among others.

Now, we can list certain motivations that motivated us to define the new KMIW model:

• The NAPT-Par distribution is adequate for fitting different types data due to its flexibility and utility.
• Another crucial motivation is that the proposed distribution offers mechanisms to introduce or manipulate skewness, allowing for the modeling of datasets where asymmetry is inherent or desired.
• A paramount goal of the NAPT-Par model is to provide several classical estimation procedures to estimate the unknown parameters of the recommended distribution, and a simulation analysis is performed to evaluate the estimator’s performance. Moreover, the final aim is to explore four well-known risk measures of the proposed NAPT-Par model, notably the value at risk (VaR), Tail VaR (TVaR), tail variance (TV), and tail variance premium (TVP).
• the NAPT-Par model can capture various shapes and patterns in data, which often results in better goodness-of-fit compared to other distributions.

The order of the remaining sections is as follows. In Section 2, we provide the useful representations of cdf and pdf and certain important distributional properties of the newly recommended model. In Section 3, several mathematical properties of the NAPT-Par model are investigated. The estimation of model parameters using numerous estimation methods is presented in Section 4. Section 5 offers a brief Monte Carlo (MC) simulation study to evaluate the potential of the proposed estimators. Four risk measures of our distribution are calculated in Section 6, and one real dataset taken from the insurance area is analyzed to select the best-fitting models in Section 7. Finally, the conclusion is given in Section 8.

2. Model Formulation

In this section, the cdf, pdf, and corresponding reliability functions are developed.

A continuous random variable Z is said to have the recommended NAPT-Par distribution with parameters $\eta ,\delta$, and $\alpha$ if its cdf and pdf have the below forms

$$\begin{array}{cc}\hfill M(z,\eta ,\delta ,\alpha )& ={\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)},z>\eta ,\mathrm{and}\alpha >1,\eta ,\delta >0,\hfill \end{array}$$

(3)

and

$$\begin{array}{cc}\hfill m(z,\eta ,\delta ,\alpha )& ={\displaystyle \frac{1}{1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }}}\left\{log\alpha {\displaystyle \frac{\delta {\eta }^{\delta }}{z^{\delta +1}}}{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right\}.\hfill \end{array}$$

(4)

For different values of the parameters $\eta ,\delta$, and $\alpha$, several representations of the pdf of the NAPT-Par distribution are displayed in Figure 1. The pdf of the random variable Z confirms a variety of shapes. The proposed NAPT-Par model is decreasing, positively skewed, and unimodal.

• Furthermore, the survival function (sf) and hazard rate function (hrf) of the random variable Z are obtained:

$$S(z,\eta ,\delta ,\alpha )=1-{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)},$$

and

$$h(z,\eta ,\delta ,\alpha )={\displaystyle \frac{log\alpha {\displaystyle \frac{\delta {\eta }^{\delta }}{z^{\delta +1}}}{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}}{\left[1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right]\left[1-{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right]}}.$$

The plot of hrf for the NAPT-Par is summarized in Figure 2. The plots considering the hrf of the suggested model are obtained by applying numerous selected parameter values. The hrf of Z decreases, and the unimodal and reverse J-shaped functions are also decreasing. Consequently, the correct behavior of the hrf confirms that the NAPT-Par is a valid and reliable model for analyzing various datasets.

• Henceforth, the cumulative hrf with the reversed hrf of the random variable Z are written as follows:

$$\begin{array}{cc}\hfill H(z,\eta ,\delta ,\alpha )=& -log\left\{1-{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right\},\hfill \end{array}$$

and

$$R(z,\eta ,\delta ,\alpha )={\displaystyle \frac{log\alpha {\displaystyle \frac{\delta {\eta }^{\delta }}{z^{\delta +1}}}}{1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }}}.$$

3. Some Statistical Properties of NAPT-Par Model

We established certain mathematical properties of the NAPT-Par distribution. These properties covered the quantile function, $k^{th}$ moment, mean, variance, moment generating function (MGF), and the distribution of the order statistics.

3.1. Quantile Function

Let Z follows the NAPT-Par distribution. The corresponding quantile function (q) is obtained by the inverse of Equation (3). That is,

$$q\left(u\right)=\eta {\left(1-e^{\frac{logu}{log\alpha }}\right)}^{-1/\delta },0<u<1,$$

(5)

So, a random sample from the random variable Z may be generated using the above equation with u having a uniform random number (0,1).

• The median of Z is given by

$$z_{0.5}=-\eta {\left(1-e^{\frac{log0.5}{log\alpha }}\right)}^{-1/\delta }.$$

Next, both coefficients of skewness ($SK$) and the kurtosis ($KR$) of the recommended NAPT-Par model can be derived as

$$SK={\displaystyle \frac{q\left(0.25\right)+q\left(0.75\right)-2q\left(0.5\right)}{q\left(0.75\right)-q\left(0.25\right)}},$$

and

$$KR={\displaystyle \frac{q\left(0.875\right)-q\left(0.625\right)+q\left(0.375\right)-q\left(0.125\right)}{q\left(0.75\right)-q\left(0.25\right)}}.$$

3.2. Useful Expansion

First, the generalized binomial linear representation can be written as

$$\begin{array}{c}\hfill {\displaystyle {\alpha }^t=\sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^i}{i!}}{t}^i.}\end{array}$$

As a result, we can obtain the cdf and pdf of NAPT-Par distribution, and they are given by

$$\begin{array}{ccc}\hfill M(z,\eta ,\delta ,\alpha )& =& {\displaystyle \sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^{i+1}}{i!}}{\left[log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)\right]}^i}\hfill \\ & =& {\displaystyle \sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^{i+1}}{i!}}{\left[logK(z,\eta ,\delta )\right]}^i,}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill m(z,\eta ,\delta ,\alpha )& =& {\displaystyle {\displaystyle \frac{\delta {\eta }^{\delta }}{z^{\delta +1}\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}}\sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^{i+1}}{i!}}{\left[log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)\right]}^i}\hfill \\ & =& {\displaystyle {\displaystyle \frac{k(z,\eta ,\delta )}{K(z,\eta ,\delta )}}\sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^{i+1}}{i!}}{\left[logK(z,\eta ,\delta )\right]}^i,}\hfill \end{array}$$

where, $K(z,\eta ,\delta )$ and $k(z,\eta ,\delta )$ are given in (1) and (2).

3.3. The $k^{th}$ Moment of the NAPT-Par

Now, by applying the new expression pdf of the RV Z, which follows the NAPT-Par model, we can express the $k^{th}$-moment of Z, and it is written as

$${\displaystyle u_k^{\prime }=\sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^{i+1}}{i!}}{{\rm Y}}_{i,k}(z,\eta ,\delta ),}$$

(6)

with ${\displaystyle {{\rm Y}}_{i,k}(z,\eta ,\delta )={\int }_0^{\infty }{z}^k{\displaystyle \frac{k(z,\eta ,\delta )}{K(z,\eta ,\delta )}}{\left[logK(z,\eta ,\delta )\right]}^{i}dz}$.

• Consequently, the expected value (mean) and variance of Z are

$${\displaystyle u_1^{\prime }=\sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^{i+1}}{i!}}{{\rm Y}}_{i,1}(z,\eta ,\delta ),}$$

and

$${\displaystyle V=\sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^{i+1}}{i!}}\left\{{\displaystyle {{\rm Y}}_{i,2}(z,\eta ,\delta )-\sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^i}{i!}}{{\rm Y}}_{i,1}^2(z,\eta ,\delta )}\right\}.}$$

The index of dispersion of the RV Z can thus be obtained using the formula

$$ID={\displaystyle \frac{V}{u_1^{\prime }}}.$$

Finally, the MGR of Z is

$${\displaystyle M\left(t\right)=\sum _{k=0}^{\infty }\sum _{i=0}^{\infty }{\displaystyle \frac{t^k{(log\alpha )}^{i+1}}{k!i!}}{{\rm Y}}_{i,k}(z,\eta ,\delta ).}$$

(7)

Table 1. Several characteristic properties for the NAPT-Par model at $\alpha =1.75$.

$\mathit{\delta }$	${\mathit{\mu }}_1^{\prime }$	V	ID	$\mathit{SK}$	$\mathit{KR}$
$\eta$ = 2   3	2.6433	1.6734	0.6331	9.8944	307.896
6	2.2631	0.165	0.0729	4.0128	30.177
9	2.1655	0.0574	0.0265	3.3245	18.362
12	2.1207	0.0288	0.0136	3.0614	14.827
$\eta$ = 4   3	5.2866	6.6935	1.2661	9.8944	307.896
6	4.5262	0.6602	0.1459	4.0128	30.177
9	4.3309	0.2298	0.0531	3.3245	18.362
12	4.2414	0.1154	0.0272	3.0614	14.827
$\eta$ = 6   3	7.9299	15.0603	1.8992	9.8944	307.896
6	6.7892	1.4854	0.2188	4.0128	30.177
9	6.4964	0.5170	0.0796	3.3245	18.362
12	6.3621	0.2596	0.0408	3.0614	14.827

and

Table 2. Several characteristic properties for the NAPT-Par model at $\alpha =3$.

$\mathit{\delta }$	${\mathit{\mu }}_1^{\prime }$	V	ID	$\mathit{SK}$	$\mathit{KR}$
$\eta$ = 2   3	3.0702	2.889	0.9410	8.8719	258.264
6	2.4267	0.2512	0.1035	3.3636	22.196
9	2.2664	0.0843	0.0372	2.7186	12.867
12	2.1936	0.0416	0.019	2.4707	10.110
$\eta$ = 4   3	6.1403	11.556	1.8820	8.8719	258.264
6	4.8535	1.0049	0.2070	3.3636	22.196
9	4.5328	0.3371	0.0744	2.7186	12.867
12	4.3872	0.1663	0.0379	2.4707	10.110
$\eta$ = 6   3	9.2105	26.001	2.8230	8.8719	258.264
6	7.2802	2.2610	0.3106	3.3636	22.196
9	6.7991	0.7584	0.1115	2.7186	12.867
12	6.5808	0.3742	0.0569	2.4707	10.110

record numerous recommended mathematical properties as discussed previously, and Figure 3 and Figure 4 demonstrate the 3d plots of these statistical measures. All these obtained values and representations show that our NAPT-Par distribution is a more suitable model for modeling several types of datasets.

3.4. Order Statistics

Let $Y_{(1:n)},Y_{(2:n)},\dots ,Y_{(n:n)}$ be an order random sample (RS) drawn from our suggested model. The pdf of the $j^{th}$ order statistic of the RV Z is

$$\begin{array}{ccc}\hfill f_{(j:n)}\left(z\right)& =& {\displaystyle \frac{n!}{(j-1)!(n-j)!}}m(z,\eta ,\delta ,\alpha ){\left[M(z,\eta ,\delta ,\alpha )\right]}^{j-1}{[1-M(z,\eta ,\delta ,\alpha )]}^{n-j}\hfill \\ & =& {\displaystyle \frac{n!{\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}^{-1}}{(j-1)!(n-j)!}}\left\{log\alpha {\displaystyle \frac{\delta {\eta }^{\delta }}{z^{\delta +1}}}{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right\}{\left({\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right)}^{j-1}\hfill \\ & \times & {\left(1-{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right)}^{n-j}.\hfill \end{array}$$

This gives the pdf of $Z_{(n:n)}=max\{z_1,z_2,\dots ,z_n\}$ as

$$\begin{array}{ccc}\hfill f_{(n:n)}\left(z\right)& =& n{\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}^{-1}\left\{log\alpha {\displaystyle \frac{\delta {\eta }^{\delta }}{z^{\delta +1}}}{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right\}{\left({\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right)}^{n-1},\hfill \end{array}$$

Also, we have the pdf of $Z_{\left(1\right)}=min\{z_1,z_2,\dots ,z_n\}$ as

$$\begin{array}{ccc}\hfill f_{(1:n)}\left(z\right)& =& n{\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}^{-1}\left\{log\alpha {\displaystyle \frac{\delta {\eta }^{\delta }}{z^{\delta +1}}}{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right\}{\left(1-{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right)}^{n-1}.\hfill \end{array}$$

Now, the cdf of the $j^{th}$ order statistic of Z is obtained to be

$$\begin{array}{ccc}\hfill F_{(j:n)}\left(z\right)& =& {\displaystyle \sum _{l=j}^n{M}^l(z,\eta ,\delta ,\alpha ){[1-M(z,\eta ,\delta ,\alpha )]}^{n-l}}\hfill \\ & =& {\displaystyle \sum _{l=j}^n{\left({\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right)}^l{\left(1-{\alpha }^{log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)}\right)}^{n-l}.}\hfill \end{array}$$

4. Parameter Estimation Procedures

In this part of the study, we discuss six parameter estimation processes for estimating the unknown parameters of the proposed model.

4.1. Maximum Likelihood Estimation (MLE)

Suppose $(z_1,z_2\dots ,z_n)$ is a set of samples drawn from the suggested NAPT-Par model, and $\Theta =(\eta ,\delta ,\alpha )$ represents the parameter vector. Corresponding to Equation (4), the log-likelihood function, say $\mathcal{LL}(z;\Theta )$, is

$$\begin{array}{ccc}\hfill \mathcal{LL}(z;\Theta )& =& {\displaystyle \sum _{i=1}^{n}logm(z;\Theta )}\hfill \\ & =& {\displaystyle nlog(log\left(\alpha \right))+nlog\left(\delta \right)+n\delta log\eta -(\delta +1)\sum _{i=1}^{n}log{z}_i+(log\alpha -1)\sum _{i=1}^{n}log\left(1-{\left({\displaystyle \frac{\eta }{z_i}}\right)}^{\delta }\right).}\hfill \end{array}$$

(8)

Corresponding to Equation (8), the partial derivatives with respect parameters are obtained to be

$${\displaystyle {\displaystyle \frac{\partial \mathcal{LL}(z;\Theta )}{\partial \eta }}=\frac{n\delta }{\eta }+(1-log\alpha )\delta {\eta }^{\delta -1}\sum _{i=1}^n{\left[z_i^{\delta }\left(1-{\left({\displaystyle \frac{\eta }{z_i}}\right)}^{\delta }\right)\right]}^{-1},}$$

(9)

$${\displaystyle {\displaystyle \frac{\partial \mathcal{LL}(z;\Theta )}{\partial \delta }}=\frac{n}{\delta }+nlog\eta -\sum _{i=1}^{n}log{z}_i+(1-log\alpha )\sum _{i=1}^n{\displaystyle \frac{log\left({\displaystyle \frac{\eta }{z_i}}\right){\left({\displaystyle \frac{\eta }{z_i}}\right)}^{\delta }}{1-{\left({\displaystyle \frac{\eta }{z_i}}\right)}^{\delta }}},}$$

(10)

and

$${\displaystyle {\displaystyle \frac{\partial \mathcal{LL}(z;\Theta )}{\partial \alpha }}={\displaystyle \frac{n}{\alpha log\alpha }}+{\displaystyle \frac{1}{\alpha }}\sum _{i=1}^{n}log\left(1-{\left({\displaystyle \frac{\eta }{z_i}}\right)}^{\delta }\right).}$$

(11)

Based on Equations (9)–(11), it is difficult to compute them directly. To overcome this issue, we can use several iterative non-linear procedures, likely Newton–Raphson, secant, bisection, and fixed point techniques, to find the MLEs, say ${\varpi }_1$ of $\Theta =(\eta ,\delta ,\alpha )$.

4.2. Least Squares Estimator (LSE)

Swain et al. [27] introduced an extensive tool to determine the estimation of unknown parameters for any distribution. Assume that an RS $z_1,\dots z_n$ of size n is drawn from a distribution function $M(z;\eta \delta ,\alpha )$, and $z_{(1:n)},\dots z_{(n:n)}$ represents its order statistics. The proposed LSEs technique, say ${\varpi }_2$ of the $\Theta =(\eta ,\delta ,\alpha )$, is computed by minimizing

$${\varpi }_2=\sum _{i=1}^n{\left[M(z_{(i:n)};\Theta )-A_1\right]}^2,$$

where, $A_1={\displaystyle \frac{i}{n+1}}$. Consequently, we obtain the final estimates of $\Theta$, and they are expressed as follows:

$${\displaystyle \sum _{i=1}^n\left[M(z_{(i:n)};\Theta )-{\displaystyle \frac{i}{n+1}}\right]{\Omega }_1(z_{(i:n)};\Theta )=0,}$$

$${\displaystyle \sum _{i=1}^n\left[M(z_{(i:n)};\Theta )-{\displaystyle \frac{i}{n+1}}\right]{\Omega }_2(z_{(i:n)};\Theta )=0,}$$

and

$${\displaystyle \sum _{i=1}^n\left[M(z_{(i:n)};\Theta )-{\displaystyle \frac{i}{n+1}}\right]{\Omega }_3(z_{(i:n)};\Theta )=0,}$$

with

$${\Omega }_1(z_{(i:n)};\Theta )={\displaystyle \frac{\partial }{\partial \eta }}M(z_{(i:n)};\Theta ),$$

(12)

$${\Omega }_2(z_{(i:n)};\Theta )={\displaystyle \frac{\partial }{\partial \delta }}M(z_{(i:n)};\Theta ),$$

(13)

and

$${\Omega }_2(z_{(i:n)};\Theta )={\displaystyle \frac{\partial }{\partial \alpha }}M(z_{(i:n)};\Theta ),$$

(14)

4.3. Weighted Least Squares Estimator (WLSE)

Similarly, by minimizing the below equation, we obtained the WLSE estimators, say ${\varpi }_3$ of $\Theta$. For more details, see

$${\displaystyle {\varpi }_3=\sum _{i=1}^n{A}_2{\left[M(z_{(i:n)};\Theta ))-{\displaystyle \frac{i}{n+1}}\right]}^2,}$$

with $A_2={\displaystyle \frac{{(n+1)}^2(n+2)}{i(n-i+1)}}$. Consequently, we obtain the final estimates of $\Theta$, and they can be written as

$${\displaystyle \sum _{i=1}^n{A}_2\left[M(z_{(i:n)};\Theta )-{\displaystyle \frac{i}{n+1}}\right]{\Omega }_1(z_{(i:n)};\Theta )=0,}$$

$${\displaystyle \sum _{i=1}^n{A}_2\left[M(z_{(i:n)};\Theta )-{\displaystyle \frac{i}{n+1}}\right]{\Omega }_2(z_{(i:n)};\Theta )=0,}$$

and

$${\displaystyle \sum _{i=1}^n{A}_2\left[M(z_{(i:n)};\Theta )-{\displaystyle \frac{i}{n+1}}\right]{\Omega }_3(z_{(i:n)};\Theta )=0.}$$

4.4. Maximum Product of Spacings (MPS)

To obtain the estimates of $\Theta$ of the proposed NAPT-Par distribution using the MPS technique, we described this tool with the following steps: First, let

$$B_i=M(z_{(i:n)};\Theta )-M(z_{(i-1:n)};\Theta );i=1,\dots ,n+1,$$

where

$$M(z_{\left(0\right)};\Theta )=0,\mathrm{and}M(z_{(n+1)};\Theta )=1.$$

Evidently ${\displaystyle \sum _{i=1}^{n+1}{B}_i=1}$. Secondly, by minimizing the below equation, the MPS estimators, say ${\varpi }_4$ of $\Theta$, are obtained to be

$${\displaystyle {\varpi }_4={\displaystyle \frac{1}{n+1}}\sum _{i=1}^{n+1}log\mathcal{MPS}.}$$

By taking the derivative of the preceding equation, we obtain the final estimates of $\Theta$, and they are

$${\displaystyle {\displaystyle \frac{\partial {\varpi }_4}{\partial \eta }}={\displaystyle \frac{1}{n+1}}\sum _{i=1}^{n+1}{\displaystyle \frac{1}{{\varpi }_4}}\left\{{\Omega }_1(z_{(i:n)};\Theta )-{\Omega }_1(z_{(i:n)};\Theta )\right\}=0,}$$

$${\displaystyle {\displaystyle \frac{\partial {\varpi }_4}{\partial \delta }}={\displaystyle \frac{1}{n+1}}\sum _{i=1}^{n+1}{\displaystyle \frac{1}{{\varpi }_4}}\left\{{\Omega }_2(z_{(i:n)};\Theta )-{\Omega }_2(z_{(i:n)};\Theta )\right\}=0,}$$

and

$${\displaystyle {\displaystyle \frac{\partial {\varpi }_4}{\partial \alpha }}={\displaystyle \frac{1}{n+1}}\sum _{i=1}^{n+1}{\displaystyle \frac{1}{{\varpi }_4}}\left\{{\Omega }_3(z_{(i:n)};\Theta )-{\Omega }_3(z_{(i:n)};\Theta )\right\}=0,}$$

4.5. Cramer-Von Mises Method of Estimation (CVE)

Macdonald [28] defined empirical evidence that the bias of the estimator is smaller than the other minimum distance estimators. This proposed estimator is based on the Cramer-von-Mises statistics, say ${\varpi }_5$, provided by minimizing

$${\varpi }_5={\displaystyle \frac{1}{12n}}+\sum _{i=1}^n{\left[M(z_{(i:n)};\Theta )-{\displaystyle \frac{2i-1}{2n}}\right]}^2.$$

Now, the final estimates of $\Theta$ can be obtained to be

$${\displaystyle \sum _{i=1}^n\left[M(z_{(i:n)};\Theta )-{\displaystyle \frac{2i-1}{2n}}\right]{\Omega }_1(z_{(i:n)};\Theta )=0,}$$

$${\displaystyle \sum _{i=1}^n\left[M(z_{(i:n)};\Theta )-{\displaystyle \frac{2i-1}{2n}}\right]{\Omega }_2(z_{(i:n)};\Theta )=0,}$$

and

$${\displaystyle \sum _{i=1}^n\left[M(z_{(i:n)};\Theta )-{\displaystyle \frac{2i-1}{2n}}\right]{\Omega }_3(z_{(i:n)};\Theta )=0.}$$

4.6. Anderson–Darling Method of Estimation (ADE)

This proposed estimator is based on Anderson–Darling statistics, say ${\varpi }_6$, obtained by minimizing

$${\displaystyle {\varpi }_6=-n-{\displaystyle \frac{1}{n}}\sum _{i=1}^n(2i-1)\left\{logM(z_{(i:n)};\Theta )+log(1-M(z_{(i:n)};\Theta ))\right\}.}$$

By taking the derivative of the preceding equation, we obtain the final estimates of $\Theta$, and they are formulated as

$${\displaystyle \sum _{i=1}^n(2i-1)\left[{\displaystyle \frac{{\Omega }_1(z_{(i:n)};\Theta }{M(z_{(i:n)};\Theta )}}-{\displaystyle \frac{{\Omega }_1(z_{(i:n)};\Theta }{1-M(z_{(i:n)};\Theta )}}\right]=0,}$$

$${\displaystyle \sum _{i=1}^n(2i-1)\left[{\displaystyle \frac{{\Omega }_2(z_{(i:n)};\Theta }{M(z_{(i:n)};\Theta )}}-{\displaystyle \frac{{\Omega }_2(z_{(i:n)};\Theta }{1-M(z_{(i:n)};\Theta )}}\right]=0,}$$

and

$${\displaystyle \sum _{i=1}^n(2i-1)\left[{\displaystyle \frac{{\Omega }_3(z_{(i:n)};\Theta }{M(z_{(i:n)};\Theta )}}-{\displaystyle \frac{{\Omega }_3(z_{(i:n)};\Theta }{1-M(z_{(i:n)};\Theta )}}\right]=0.}$$

5. Simulation Experiment Study

A Monte Carlo simulation analysis is conducted over time using the R programming language to assess the performance of methods discussed in this study.

• The samples are obtained through the application of the quantile function, given by

$$y_u=-\eta {\left(1-e^{\frac{logu}{log\alpha }}\right)}^{-1/\delta },0<u<1,$$

(15)

The simulation study is undertaken for an arbitrarily chosen set of parameter values—Set 1: $\Theta =(0.85,0.5,2.0)$ or Set 2: $\Theta =(1.0,0.75,2.3)$—and different selecting random samples, such as $n=25,50,70,75,100$. Several criteria are applied to assess the simulation results, such as mean estimates (AE), mean bias (ABs), and mean square error (MSE). These criteria are detailed below

$${\displaystyle \mathrm{Mean}=\frac{1}{1000}\sum _{i=1}^{1000}{\widehat{\Theta }}_i,}$$

$${\displaystyle \mathrm{MSE}=\frac{1}{1000}\sum _{i=1}^{1000}\mid {\widehat{\Theta }}_i-\Theta \mid ,}$$

$${\displaystyle \mathrm{MSE}=\frac{1}{1000}\sum _{i=1}^{1000}{({\widehat{\Theta }}_i-\Theta )}^2.}$$

The comparison is made using several metrics of deviations, notably the mean estimates (AE), mean bias (AB), and mean square errors (MSE) based on various sample sizes, $n=\{25,50,75,100\}$. By taking N = 1000 replications of the process, A random sample from the basic NAPT-Par model is generated using Equation (15).

Table 3. The numerical representation of the simulation study of the NAPT-Par distribution is provided for Set 1 based on classical methods.

n	Method		$\widehat{\mathit{\eta }}$						$\widehat{\mathit{\delta }}$				$\widehat{\mathit{\alpha }}$
			AE	AB		MSE		AE	AB		MSE		AE	AB		MSE
25	${\varpi }_1$		0.8722	0.0222		0.0007		0.5475	0.0475		0.0157		1.9871	0.0128		0.0807
	${\varpi }_2$		0.8244	0.0255		0.0044		0.5389	0.0389		0.0311		2.1824	0.1824		0.2619
	${\varpi }_3$		0.8319	0.0180		0.0059		0.4876	0.0123		0.0475		2.0850	0.0850		0.2802
	${\varpi }_4$		0.2603	0.0103		0.0020		0.4957	0.0042		0.0281		2.0863	0.0863		0.2018
	${\varpi }_5$		0.8655	0.0155		0.0023		0.4347	0.0652		0.0199		1.7637	0.2362		0.1360
	${\varpi }_6$		0.2457	0.6042		0.4925		0.4061	0.0938		0.0753		1.8016	0.1983		0.3972
50	${\varpi }_1$		0.8614	0.0114		0.0005		0.5192	0.0192		0.0052		1.9940	0.0059		0.0317
	${\varpi }_2$		0.8381	0.0118		0.0039		0.4483	0.0516		0.0223		2.0253	0.0253		0.2328
	${\varpi }_3$		0.8423	0.0076		0.0056		0.4844	0.0155		0.0324		2.035	0.035		0.2445
	${\varpi }_4$		0.8683	0.0183		0.0016		0.4938	0.0061		0.0086		1.8139	0.1860		0.0590
	${\varpi }_5$		0.8747	0.0247		0.0015		0.5050	0.0050		0.0159		1.9911	0.0088		0.0539
	${\varpi }_6$		0.3910	0.2589		0.2504		0.3195	0.1804		0.0355		1.8938	0.0061		0.3516
75	${\varpi }_1$		0.8619	0.0119		0.0003		0.5478	0.0478		0.0042		2.0187	0.0187		0.0176
	${\varpi }_2$		0.8482	0.0017		0.0032		0.4762	0.02378		0.0069		2.0539	0.0539		0.1208
	${\varpi }_3$		0.8461	0.0038		0.0042		0.4603	0.0396		0.0079		1.9661	0.0338		0.1500
	${\varpi }_4$		0.8544	0.0044		0.0008		0.4842	0.0157		0.0048		1.9414	0.0585		0.0196
	${\varpi }_5$		0.8651	0.0151		0.0006		0.4683	0.0316		0.0157		1.8970	0.1029		0.0111
	${\varpi }_6$		0.7289	0.1211		0.1529		0.4342	0.0618		0.0321		1.9277	0.0723		0.2356
100	${\varpi }_1$		0.8506	0.0006		0.0001		0.5180	0.0180		0.0007		2.0282	0.0282		0.0091
	${\varpi }_2$		0.8382	0.0117		0.0007		0.5191	0.0191		0.0037		2.1477	0.1477		0.0674
	${\varpi }_3$		0.8691	0.0191		0.0032		0.5123	0.0123		0.0062		0.5212	0.0212		0.0041
	${\varpi }_4$		0.8571	0.0071		0.0005		0.4784	0.0215		0.0013		1.9583	0.0416		0.0173
	${\varpi }_5$		0.8545	0.0045		0.0004		0.5330	0.0330		0.0011		2.0888	0.0888		0.0106
	${\varpi }_6$		0.7723	0.0773		0.0925		0.4841	0.0159		0.0294		1.9729	0.0271		0.1241

and

Table 4. The numerical representation of the simulation study of the NAPT-Par distribution is provided for Set 2 based on classical methods.

n	Method		$\widehat{\mathit{\eta }}$						$\widehat{\mathit{\delta }}$				$\widehat{\mathit{\alpha }}$
			AE	AB		MSE		AE	AB		MSE		AE	AB		MSE
25	${\varpi }_1$		1.0311	0.0311		0.0073		0.9074	0.1574		0.0711		1.9452	0.3547		0.2026
	${\varpi }_2$		1.0066	0.0066		0.0128		0.8026	0.0526		0.1216		2.7011	0.4011		1.7191
	${\varpi }_3$		0.9701	0.0298		0.0145		0.9141	0.1641		0.1482		2.8453	0.5453		2.4655
	${\varpi }_4$		1.0830	0.0830		0.0119		0.6733	0.0766		0.0879		2.0292	0.2707		0.2537
	${\varpi }_5$		0.9979	0.0020		0.0123		0.8638	0.1138		0.0882		2.8908	0.5908		0.2950
	${\varpi }_6$		0.2062	0.7937		0.7833		0.4988	0.2511		0.2767		2.9385	0.6385		3.4141
50	${\varpi }_1$		1.0185	0.0185		0.0014		0.7173	0.0326		0.0154		2.0668	0.2331		0.1474
	${\varpi }_2$		0.9716	0.0283		0.0070		0.7329	0.0170		0.0559		2.6513	0.3513		0.8264
	${\varpi }_3$		0.9785	0.0214		0.0059		0.8049	0.0549		0.0440		2.7088	0.4088		0.9355
	${\varpi }_4$		1.0302	0.0302		0.0023		0.6311	0.1188		0.0206		1.9985	0.3014		0.1986
	${\varpi }_5$		0.9876	0.0123		0.0033		0.7889	0.03897		0.0429		2.5659	0.2659		0.2838
	${\varpi }_6$		0.1741	0.8258		0.6884		0.5328	0.2171		0.1588		1.9327	0.3672		1.4878
75	${\varpi }_1$		1.0113	0.0113		0.0002		0.7419	0.0080		0.0061		2.0891	0.2108		0.0895
	${\varpi }_2$		0.9790	0.020		0.0059		0.7274	0.0225		0.0327		2.3480	0.0480		0.4795
	${\varpi }_3$		0.9899	0.010		0.0017		0.7705	0.0205		0.0310		2.6016	0.3016		0.4383
	${\varpi }_4$		1.0152	0.0152		0.0006		0.6836	0.0663		0.0238		2.0259	0.2740		0.1118
	${\varpi }_5$		0.9851	0.0148		0.0008		0.7930	0.0430		0.0300		2.4752	0.1752		0.2152
	${\varpi }_6$		0.3222	0.6777		0.6590		0.6799	0.0401		0.1425		2.8221	0.5221		1.1142
100	${\varpi }_1$		1.0076	0.0076		0.0001		0.7367	0.0132		0.0034		2.3036	0.0036		0.0397
	${\varpi }_2$		1.0135	0.0135		0.0025		0.7358	0.0141		0.0132		2.2810	0.0189		0.3020
	${\varpi }_3$		0.9998	0.0001		0.0011		0.7593	0.0093		0.0234		2.2676	0.0323		0.0717
	${\varpi }_4$		1.0085	0.0085		0.0003		0.7421	0.0078		0.0045		2.2052	0.0947		0.0548
	${\varpi }_5$		0.9901	0.0098		0.0007		0.7785	0.0285		0.0063		2.3779	0.07796		0.1223
	${\varpi }_6$		0.6203	0.3797		0.3995		0.7091	0.0408		0.1204		2.8369	0.5369		0.4652

display the obtained results. The findings from the simulation study of the NAPT-Par distribution, as detailed in Table 3 and Table 4, demonstrate that:

• As n continues to increase, the AE is observed to approach the initial value of parameters for all estimates procedures, which shows that all techniques are asymptotically unbiased.
• As n continues to increase, the MSE values decline, showing that all techniques are consistent.
• Since the MSE values of the MLE method are smaller, the MLE technique outperforms other proposed tools.

6. Actuarial Metrics

In this part of the work, we evaluated several metric measures for the NAPT-Par model, including VaR, TVaR, TV, and TVP. In the literature, many authors applied these actuarial measures, for example, the studies of Meraou et al. [29,30], Affify et al. [31], and Teamah et al. [32].

6.1. VaR Metric

The VaR, listed as (${\mathcal{X}}_1$), of our NAPT-Par model, is obtained by finding the inverse of the Equation (3). The final expression of ${\mathcal{X}}_1$ can be expressed as

$${\mathcal{X}}_1=\eta {\left(1-e^{\frac{logq}{log\alpha }}\right)}^{-1/\delta },0<q<1.$$

6.2. TVaR Metric

Another crucial metric measure is named the TVaR metric (${\mathcal{X}}_2$). It represents the amount utilized to perform the financial impact of a loss. For each random variable, Z follows the NAPT-Par model, and its ${\mathcal{X}}_2$ is denoted by

$$\begin{array}{ccc}\hfill {\mathcal{X}}_2& =& {\displaystyle \frac{1}{(1-q)}}{\int }_{{\mathcal{V}}_1}^{\infty }zm(z,\eta ,\delta ,\alpha )dz\hfill \\ \\ & =& {\displaystyle \frac{\delta {\eta }^{\delta }}{(1-q)}\sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^{i+1}}{i!}}{\Phi }_i(z,\eta ,\delta ),}\hfill \end{array}$$

where ${\displaystyle {\Phi }_i(z,\eta ,\delta )={\int }_{{\mathcal{X}}_1}^{\infty }{\left[z^{\delta }\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)\right]}^{-1}{\left[log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)\right]}^{i}dz}$.

6.3. TV Mteric

The TV metric (${\mathcal{X}}_3$) for our recommended NAPT-Par model can be defined by the formula below

$$\begin{array}{ccc}\hfill {\mathcal{X}}_3& =& {\displaystyle \frac{1}{(1-p)}}{\int }_{{\mathcal{X}}_1}^{\infty }{z}^{2}m(z,\eta ,\delta ,\alpha )dy-{\left({\mathcal{X}}_2\right)}^2\hfill \\ \\ & =& {\displaystyle \frac{\delta {\eta }^{\delta }}{(1-q)}\sum _{i=0}^{\infty }{\displaystyle \frac{{(log\alpha )}^{i+1}}{i!}}{\Phi }_i^{\prime }(z,\eta ,\delta )-{\left({\mathcal{X}}_2\right)}^2,}\hfill \end{array}$$

where ${\displaystyle {\Phi }_i^{\prime }(z,\eta ,\delta )={\int }_{{\mathcal{X}}_1}^{\infty }{\left[z^{\delta +1}\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)\right]}^{-1}{\left[log\left(1-{\left({\displaystyle \frac{\eta }{z}}\right)}^{\delta }\right)\right]}^{i}dz}$

6.4. TVP Metric

Another type of metric measure is referred to as the TVP metric, which is denoted by (${\mathcal{X}}_4$). It is a criterion used by the insurance industry. The expression of the TVP metric is defined as follows.

$$\begin{array}{c}\hfill {\mathcal{X}}_4={\mathcal{X}}_2+q{\mathcal{X}}_3,\end{array}$$

6.5. Simulation Results of the Risk Measures for the NAPT-Par Model

In this part, the results for ${\mathcal{X}}_1$, ${\mathcal{X}}_2$, ${\mathcal{X}}_3$, and ${\mathcal{X}}_3$ using our proposed Par models based on two cases of the numerical values of the unknown parameters are performed.

Table 5. The numerical representation of the simulation study of ${\mathcal{X}}_1$, ${\mathcal{X}}_2$, ${\mathcal{X}}_3$, and ${\mathcal{X}}_4$ for the NAPT-Par and Par models.

Model	Parameters	q	${\mathcal{X}}_1$	${\mathcal{X}}_2$	${\mathcal{X}}_3$	${\mathcal{X}}_4$
NAPT-Par	$\eta =2.25,,\delta =2.25,\alpha =3.0$	0.70	3.979	7.063	29.579	27.769
		0.75	4.3205	7.6479	33.445	32.730
		0.80	4.7766	8.4258	38.777	39.447
		0.85	5.4343	9.5401	46.724	49.256
		0.90	6.5145	11.349	60.220	65.547
		0.95	8.8742	15.198	90.356	101.037
Par	$\eta =2.25,,\delta =2.25$	0.70	3.8421	6.7967	27.693	26.182
		0.75	4.1664	7.3566	31.350	30.869
		0.80	4.6008	8.1027	36.399	37.222
		0.85	5.2283	9.1727	43.942	46.524
		0.90	6.260	10.911	56.795	62.028
		0.95	8.5195	14.620	85.677	96.013

and

Table 6. The numerical representation of the simulation study of ${\mathcal{X}}_1$, ${\mathcal{X}}_2$, ${\mathcal{X}}_3$, and ${\mathcal{X}}_4$ for the NAPT-Par and Par models.

Model	Parameters	q	${\mathcal{X}}_1$	${\mathcal{X}}_2$	${\mathcal{X}}_3$	${\mathcal{X}}_4$
NAPT-Par	$\eta =3.0,,\delta =1.5,\alpha =4.0$	0.70	8.0653	17.548	195.701	154.539
		0.75	9.1609	19.341	215.542	180.997
		0.80	10.689	21.706	241.398	214.825
		0.85	13.018	25.021	277.758	261.116
		0.90	17.146	30.106	338.372	334.642
		0.95	27.352	38.932	512.671	525.971
Par	$\eta =3.0,,\delta =1.5$	0.70	6.6943	14.860	161.729	128.071
		0.75	7.5595	16.411	179.626	151.131
		0.80	8.7720	18.482	203.060	180.930
		0.85	10.626	21.435	235.784	221.852
		0.90	13.924	26.107	287.742	285.075
		0.95	22.104	34.978	412.776	427.116

reported the obtained results. Furthermore, Figure 5 and Figure 6 demonstrated the visual comparisons between the NAPT-Par and Par distributions. It is clear from the numerical experiments and Figure 5 and Figure 6 that the NAPT-Par model is better than the other distribution since our recommended model has higher values of the actuarial measures than the Par distribution. This ensures that the NAPT-Par model is more efficient in heavy-tailed datasets.

7. Statistical Modeling

7.1. Dataset 1

A real-life application taken from the insurance sector was analyzed in this section to demonstrate the utility and performance of our NAPT-Par model. The considered dataset reported the monthly unemployment insurance metrics from 8 July 2008 to 13 April 2013, and it was drawn from https://opendata.maryland.gov/w/3x6e-7i3k/gz96-f9ea?cur=aQL1UsyAvhe&from=oyc8IB21VJ5, accessed on 15 September 2021. Henceforth, it was applied by Riad et al. [33]. The considered dataset can be tabulated in

Table 7. The monthly unemployment insurance values.

48.2	48.2	55.6	49.4	54.7	84.4	85.6	92.1	115.7	92.7	87.4	104.1	86.6	96.7
74.1	70.4	84	80.3	86.5	86.9	104.7	74.3	66.7	79.1	65.7	78.6	59.9	59
72.6	66.6	89.9	75.2	72.1	63.2	70.3	58.9	59.7	72.6	55.3	62.6	55.5	60.4
84.2	68.9	65.2	70	56.6	56.3	71.9	60.3	54, 65.2	56.3	69.6	68.8	66	63.5
64.9

.

7.2. Dataset 2

These data define the values of the stream health cost (% of GDP) for Saudi Arabia (KSA). The dataset values are taken from the World Bank (accessed on 15 September 2021) as well from Emam [34] and Muqrin [35]. The records are tabulated in

Table 8. The current health expenditure values for KSA.

	42.1159410	44.6173573	42.4937630	39.7909737	35.8400607
	34.1867280	36.1922503	35.6228733	29.7100425	42.9041958
	36.4785600	37.1177721	40.1962376	44.6568298	52.2795486
	59.9834490	58.3562946	62.6256323	57.4845695	56.8828773

.

7.3. Dataset 3

These data represent 43 observations of long patients with head and neck cancer surviving. The dataset is provided by Ceren et al. [36], and its records are are tabulated in

Table 9. 43 observations of long patients with head and neck cancer survive.

12.20	23.56	23.74	25.87	31.98	37	41.35	47.38	55.46	58.36
63.47	68.46	78.26	74.47	81.43	84	92	94	110	112
119	127	130	133	140	146	155	159	173	179
194	195	209	249	281	319	339	432	469	519
633	725	817

.

• A summary of statistics and some non-parametric plots of the proposed three datasets, notably the scaled total time on the test (TTT), Q-Q, and box plots, were depicted, respectively, in Table 10 and Figure 7. It is well known that the scaled TTT transform plot indicates the shape of the hazard function. For example, if the plot is a concave (convex) function, then it indicates that the hrf function is an increasing (decreasing) function. The TTT plot demonstrates that the hrfs for the three datasets are monotonically increasing. It also demonstrates that the NAPT-Par has a asymmetric bimodal density with a right tail.

Table 10. Summary statistics for the three proposed datasets.

Dataset	${\mathit{Q}}_1$	Median	Mean	${\mathit{Q}}_3$	ID	$\mathcal{K}$	$\mathcal{R}$
1	60.0	71.52	69.25	83.08	3.1232	0.7244	0.0749
2	36.104	41.345	42.477	53.430	4.4251	0.8688	1.4129
3	65.97	127.00	187.37	202.00	1.0235	1.7327	2.3258

Table 10. Summary statistics for the three proposed datasets.

Dataset	${\mathit{Q}}_1$	Median	Mean	${\mathit{Q}}_3$	ID	$\mathcal{K}$	$\mathcal{R}$
1	60.0	71.52	69.25	83.08	3.1232	0.7244	0.0749
2	36.104	41.345	42.477	53.430	4.4251	0.8688	1.4129
3	65.97	127.00	187.37	202.00	1.0235	1.7327	2.3258

Figure 7. Theoretical-TTT, QQ, and box plots using the three proposed real datasets.

Figure 7. Theoretical-TTT, QQ, and box plots using the three proposed real datasets.

• Henceforth, we consider certain fitted distributions to select the more suitable model for analyzing the three real datasets. For this purpose, we consider power inverse Nadarajah Haghighi (PINH, [37]), Burr X (BurXD, [38]), Truncated Poisson Lindley (TPLind, [39]), Truncated Poisson log-normal (TPLN, [40]), and Truncated Poisson exponential (TPExp, [41]) models. The fitted cdfs of the comparison models can be formulated by

• BurXD:

  $$F\left(z\right)=1-e^{-{\left(\eta z^{\delta }\right)}^2},z>0,\eta ,\delta >0.$$
• TPLN:

  $$F\left(z\right)={\displaystyle \frac{e^{\alpha \Phi \left({\displaystyle \frac{lnz-\eta }{\delta }}\right)}-1}{e^{\alpha }-1}},z>0,\eta \in \mathbb{R},\delta ,\alpha >0.$$
• TPExp:

  $$F\left(z\right)={\displaystyle \frac{e^{\eta (1-e^{-\delta z})}-1}{e^{\eta }-1}},z,\delta ,\eta >0.$$
• TPLind:

  $$F\left(z\right)={\displaystyle \frac{e^{\eta \left[1-\left({\displaystyle \frac{\delta z}{\delta +1}}+1\right)e^{-\delta z}\right]}-1}{e^{\eta }-1}},z>0,\eta ,\delta >0,$$
• PINH:

$$F\left(y\right)=e^{1-{(1+\eta /y^{\delta })}^{\alpha }},z,\eta ;\delta ,\alpha >0.$$

Next, for validation purposes, we employ several statistical criteria: Kolmogorov–Smirnov ($\mathcal{KS}$) statistics with their associated $\mathcal{P}$-values, Akaike Information Criterion (${\mathcal{C}}_1$), and Bayesian Information Criterion (${\mathcal{C}}_2$). The results are summarized in

Table 11. Estimation and values of the selection criterion of the fitted distributions.

Data	Distribution	$\widehat{\mathit{\eta }}$	$\widehat{\mathit{\delta }}$	$\widehat{\mathit{\alpha }}$	$\mathcal{KS}$	$\mathcal{P}$-Value	${\mathcal{C}}_1$	${\mathcal{C}}_2$
	NAPT-Par	44.544	4.6620	76.957	0.0934	0.6914	481.028	487.209
	PINH	6061.75	933.85	3.8538	0.1049	0.5452	484.328	490.509
	TPLN	5.2787	0.4761	41.589	0.1026	0.5744	482.184	488.366
1	TPExp	0.0517	26.273		0.1485	0.1548	493.665	497.786
	Par	5.7369	0.3861		0.5603	3.33 $\times {10}^{-16}$	719.425	723.546
	BurXD	0.0021	1.4294		0.2824	0.0001	510.927	515.047
	TPLind	0.0673	14.255		0.1393	0.2104	489.537	493.658
	NAPT-Par	27.678	4.5090	49.720	0.1209	0.8981	152.155	155.142
	PINH	135.703	120.570	2.7258	0.2054	0.3221	155.896	158.883
	TPLN	3.6550	0.2314	1.8068	0.1580	0.6438	152.623	155.610
2	TPExp	0.0669	12.255		0.2388	0.1731	160.912	162.903
	Par	29.475	2.4721		0.2739	0.0811	157.124	159.115
	BurXD	0.0328	0.89592		0.4039	0.0018	174.493	176.485
	TPLind	0.0937	8.2562		0.2177	0.2593	157.876	159.868
	NAPT-Par	0.8572	7.9343	334.21	0.1186	0.5405	537.577	542.861
	PINH	52.544	0.2261	0.6795	0.1357	0.3730	552.007	557.291
	TPLN	0.3492	5.8578	10.247	0.1324	0.4023	542.679	552.861
3	TPExp	0.0086	2.3503		0.149	0.2678	544.912	548.435
	Par	0.4162	11.214		0.2998	0.0006	575.730	579.252
	BurXD	0.0356	0.6259		0.1374	0.3577	541.156	544.678
	TPLind	0.0127	1.2503		0.2048	0.0464	554.6856	558.208

. From the numerical values of Table 11, our recommended NAPT-Par model is a suitable distribution for the three datasets. Additionally, the fitted pdf, cdf, and sf plots of the proposed model and other competitor distributions presented in Figure 8, Figure 9, Figure 10 and Figure 11 ensure that our suggested model fits the three datasets well.

• Further, the suggested datasets were used to estimate the unknown parameters using the proposed estimation method. Table 12 reported the obtained results.

Table 12. Several proposed estimation techniques for the NAPT-Par model using the three proposed datasets.

Data	Parameters	${\mathit{\varpi }}_2$	${\mathit{\varpi }}_3$	${\mathit{\varpi }}_4$	${\mathit{\varpi }}_5$	${\mathit{\varpi }}_6$
	$\eta$	47.497	45.301	44.297	48.568	41.038
1	$\delta$	4.7525	5.1417	4.2351	4.8083	3.9132
	$\alpha$	33.8735	34.892	37.463	34.210	30.302
	$\eta$	24.743	22.389	21.694	26.138	25.851
2	$\delta$	4.5329	4.5587	4.5438	4.5197	4.5637
	$\alpha$	49.684	49.781	49.718	49.704	49.735
	$\eta$	0.8327	0.8719	0.8368	0.8411	0.8479
3	$\delta$	7.9237	7.9008	7.9164	7.9185	7.9209
	$\alpha$	332.97	331.52	330.87	333.49	333.51

Table 12. Several proposed estimation techniques for the NAPT-Par model using the three proposed datasets.

Data	Parameters	${\mathit{\varpi }}_2$	${\mathit{\varpi }}_3$	${\mathit{\varpi }}_4$	${\mathit{\varpi }}_5$	${\mathit{\varpi }}_6$
	$\eta$	47.497	45.301	44.297	48.568	41.038
1	$\delta$	4.7525	5.1417	4.2351	4.8083	3.9132
	$\alpha$	33.8735	34.892	37.463	34.210	30.302
	$\eta$	24.743	22.389	21.694	26.138	25.851
2	$\delta$	4.5329	4.5587	4.5438	4.5197	4.5637
	$\alpha$	49.684	49.781	49.718	49.704	49.735
	$\eta$	0.8327	0.8719	0.8368	0.8411	0.8479
3	$\delta$	7.9237	7.9008	7.9164	7.9185	7.9209
	$\alpha$	332.97	331.52	330.87	333.49	333.51

• At the end, we performed the values of risk measures of the NAPT-Par and Par model by applying the three datasets.

  Table 13. Results of ${\mathcal{X}}_1$, ${\mathcal{X}}_2$, ${\mathcal{X}}_3$, and ${\mathcal{X}}_4$ using the insurance dataset.

  Data	Model	Parameters	q	${\mathcal{X}}_1$	${\mathcal{X}}_2$	${\mathcal{X}}_3$	${\mathcal{X}}_4$
  	Empirical		0.55	73.398	84.666	75.254	126.056
  			0.65	77.279	87.337	64.291	129.127
  			0.75	81.601	90.518	53.976	131.001
  			0.85	87.011	94.752	43.534	131.756
  1	NAPT-Par	$\widehat{\eta }=$ 44.544	0.55	69.161	90.523	721.410	487.298
  		$\widehat{\delta }=$ 4.6620	0.65	73.898	95.977	793.115	611.502
  		$\widehat{\alpha }=$ 76.957	0.75	80.303	103.605	905.375	782.636
  			0.85	90.489	116.070	1114.7	1063.645
  	Par	$\widehat{\eta }=$ 5.7369	0.55	45.378	160.839	44954.2	24885.6
  		$\widehat{\delta }=0.3861$	0.65	87.002	188.769	54247.5	35449.6
  			0.75	207.968	210.083	73886.7	55625.1
  			0.85	780.872	77.016	62139.1	52895.2
  	Empirical		0.55	42.678	53.310	302.493	218.327
  			0.65	44.631	56.038	329.461	270.329
  			0.75	53.430	59.067	378.129	356.113
  			0.85	57.615	60.322	479.517	475.953
  2	NAPT-Par	$\widehat{\eta }=$ 27.678	0.55	42.678	56.384	305.956	224.660
  		$\widehat{\delta }=$ 4.5090	0.65	45.682	59.889	337.883	279.513
  		$\widehat{\alpha }=$ 49.720	0.75	49.762	64.805	387.897	355.728
  			0.85	56.283	72.871	481.522	482.165
  	Par	$\widehat{\eta }=$ 29.475	0.55	40.713	67.7463	2166.00	1259.04
  		$\widehat{\delta }=2.4721$	0.65	45.069	74.884	2555.13	1735.72
  			0.75	51.641	85.599	3173.93	2466.04
  			0.85	63.494	104.756	4364.70	3814.75
  	Empirical		0.55	133.70	333.315	187.371	436.370
  			0.65	163.2	382.200	191.542	506.702
  			0.75	202.0	453.818	191.542	597.475
  			0.85	333.0	562.00	191.542	724.811
  3	NAPT-Par	$\widehat{\eta }=$ 0.8572	0.55	131.10	311.89	185.269	434.219
  		$\widehat{\delta }=$ 7.9343	0.65	161.47	379.579	188.374	503.981
  		$\widehat{\alpha }=$ 334.21	0.75	197.268	449.197	189.067	592.741
  			0.85	329.197	558.341	189.643	719.358
  	Par	$\widehat{\eta }=$ 0.4162	0.55	123.478	297.367	167.394	411.378
  		$\widehat{\delta }=$ 11.214	0.65	154.297	334.297	176.218	456.297
  			0.75	172.974	384.379	177.397	529.648
  			0.85	281.397	453.671	178.397	642.167

  reported the obtained results. From Table 13, the risk measure values of the NAPT-Par distribution are approaches to their associated empirical values for the three datasets. Thus, the proposed model is the best distribution for modeling the suggested datasets.

8. Conclusions

In light of the relevance of the Par distribution as well as its extraordinary flexibility in applications in the fields of dependability, engineering, and actuarial sciences, amongst others, a novel alpha power-transformed Pareto (NAPT-Par) model with three parameters has been investigated. We provided distributional and mathematical properties of this new distribution, notably density function, cumulative probability function, survival function, hazard function, quantile, mean, moment generating function, and distribution of order statistics. Furthermore, several estimation methods are considered to estimate the model parameters, and it is shown that the MLE procedure is the most suitable estimator by several simulation experiments for estimating the unknown parameters. Additionally, we established several indicator actuarial sciences of NAPT-Par distribution. Finally, two real insurance datasets have been considered for applications. By applying different statistical criteria, it is observed that the proposed NAPT-Par distribution was the best candidate model for analyzing insurance datasets.

• The following are some future research directions:

• We plan to explore and apply various goodness-of-fit (GOF) statistical tests for right-censored distributional validation.
• Investigate the extension of the proposed novel probabilistic Pareto model to the multivariate setting. Explore the mathematical properties and implications of incorporating different copulas to model dependencies among multiple variables.
• Researchers may compare the proposed NAPT-Par model with other asymmetric well-known distributions to determine its advantages and limitations in capturing the features of complex data.
• Future research might explore Bayesian methods to estimate the model parameters, especially when dealing with limited data or incorporating prior information into the estimation process.

9. Limitations of Study

It is well documented that limitations accompany our findings. Though advantageous for engineering and insurance loss, the proposed mode might not universally apply across all types of datasets. This specificity raises questions about the model’s versatility and adaptability to other complex datasets. Further, the effectiveness of the NAPT-Par distribution is deeply intertwined with the quality and comprehensiveness of the data it analyzes. In scenarios where data are sparse, incomplete, or biased, the model’s performance could be significantly compromised, potentially affecting the reliability of survival time predictions.