# A New Exponential-X Family: Modeling Extreme Value Data in the Finance Sector

Authors: Zubair Ahmad; Eisa Mahmoudi; Rasool Roozegar; Morad Alizadeh; Ahmed Z. Afify
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a family of statistical models, namely, a new exponential-X family is proposed. A subcase of the introduced family, called the new exponential-Weibull (NE-Weibull) model, is studied. The NE-Weibull model is very competent and possesses heavy-tailed properties. The maximum likelihood estimators of its parameters are derived. The consistency and efficiency of these estimators are assessed in a brief simulation study. Finally, the effectiveness of the NE-Weibull distribution is illustrated by modeling real insurance claims data. The practical analysis shows that the NE-Weibull distribution outclassed other distributions and it can be a better choice for modeling data in the finance sector.


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## Body

## 1. Introduction

The extreme value phenomena such as financial returns and other related events can be modeled effectively by extreme value methods. The heavy-tailed (HT) distributions have been proven to be substantial in modeling HT and extreme value data. Researchers have shown a deep concern in the financial sector to study new HT distributions. Among the applicability of the statistical distributions in the applied area, the HT distributions have received much attention for modeling financial phenomena. Most of the data sets in the financial sector possesses HT behavior with long right tail (see Lane [1]; Cooray and Ananda [2]; Wang et al. [3]; Ahn et al. [4]; Jelenković and Tan [5]; Forbes and Wraith [6]; Guo [7]; Punzo et al. [8]; Bhati and Ravi [9]; Punzo [10]; Ke et al. [11]; Dos Reis et al. [12]; and Niu et al. [13]).Recently, different copulas and new approaches of introducing HT distribution have been studied due to the importance of HT distributions in the financial sector (for example, Bladt et al. [14]; Tikhomirov [15]; Lugosi and Mendelson [16]; and Yousri et al. [17]). For more information about the usefulness of statistical distributions, one can refer to studies by Ramos et al. [18, 19]; He et al. [20]; Alfaer et al. [21]; Afify et al. [22]; and Al Mutairi et al. [23].A statistical distribution with SF (survival function), sayM¯x;Δ=1−Mx;Δ, is said to be a HT model, if its SF verifies(1)limx⟶∞expηx1−Mx;Δ=∞,for all η>0. For more details, see the study by Resnick [24].An interesting characteristic of the HT distributions is the regularly varying behavior. A statistical distribution is said to possess the regularly varying behavior, if it satisfies(2)limx⟶∞1−Mqx;Δ1−Mx;Δ=qb,where b∈0,∞, and it is also known as an index of regular variation.The statistical models possessing such property are very prominent models for modeling HT phenomena in the financial sector [25]. Furthermore, actuaries are very interested in looking for new flexible HT models (see Nadarajah and Bakar [26]; and Ahmad et al. [27]).Alzaatreh et al. [28] proposed a prominent approach called the T-X family method. Let ut represent the PDF (probability density function) of T, where T is a RV (random variable) belonging to δ1,δ2 and −∞≤δ1<δ2<∞. Suppose that OHx;Δ is a function of Hx;Δ of a RV-X and it has the following conditions:(i)
OHx;Δ∈δ1,δ2(ii)
OHx;Δ is a differentiable function as well as monotonically increasing(iii)
OHx;Δ⟶δ1 for x⟶−∞ and OHx;Δ⟶δ2 for x⟶∞The CDF (cumulative distribution function) of the T-X distributions is(3)Mx;Δ=∫δ1OHx;Δutdt,x∈ℝ,where OHx;Δ fulfills the aforementioned conditions. The corresponding PDF of equation (3), say mx;Δ, reduces to(4)mx;Δ=∂∂xOHx;ΔuOHx;Δ.More details about the T-X approach can be found in the work of Ahmad et al. [29]. By implementing the T-X method, the survival distribution family [30] can be obtained via the CDF:(5)Mx;Δ=1−∫δ1OH¯x;Δutdt,x∈ℝ,where H¯x;Δ=1−Hx;Δ, representing the SF of X.In this study, a new exponential-X (NE-X) family is proposed based on equation (3). Suppose T∼exp1; then, its respective CDF is(6)Ut=1−exp−t,t≥0.Linking this to equation (6), the PDF is(7)ut=exp−t.By using ut and OHx;β,Δ=−log1−Hβx;Δ/eHβx;Δ in equation (3), the CDF of the NE-X family is as follows:(8)Mx;β,Δ=1−1−Hβx;ΔeHβx;Δ,β>0,x,Δ∈ℝ,where Hx;Δ is a baseline CDF with parametric space Δ∈ℝ. Next, in Propositions 1 and 2, we are going to prove that the expression provided in equation (8) is a CDF.Proposition 1.
For theMx;β,Δ defined in equation (8), limx⟶−∞Mx;β,Δ=0 and limx⟶∞Mx;β,Δ=1.Proof.
(9)limx⟶∞Mx;β,Δ=limx⟶−∞1−1−Hβx;ΔeHβx;Δ=1−1−Hβ−∞;ΔeHβ−∞;Δ=1−1−0e0=0,limx⟶∞Mx;β,Δ=limx⟶∞1−1−Hβx;ΔeHβx;Δ=1−1−Hβ∞;ΔeHβ∞;Δ=1−1−1e=1.Proposition 2.
The CDFMx;β,Δ is right continuous and differentiable.Proof.
(10)ddxMx;β,Δ=mx;β,Δ.
From the results proved in Propositions1 and 2, we arrive at the conclusion that the function Mx;β,Δ defined in equation (8) is differentiable, right continuous, and a compact CDF. Moreover, taking into account the differentiability property of Mx;β,Δ for all x∈ℝ, we have the following theorem.Theorem 1.
Letκ1x=eHβx;Δ+Hβx;Δ−1 and κ2x=eHβx;Δ. With the condition v2x≠0, if κ1x and κ2x are differentiable, then the quotient κ1x/κ2x is differentiable for all x∈ℝ and(11)ddxκ1xκ2x=κ1′xκ2x−κ1xκ2′xκ22x.Proof.
See “Proof of Quotient Rule.”
Taking into account the right continuity ofMx;β,Δ, we have the following theorem.Theorem 2.
IfHx;Δ is a right continuous function, so is Mx;β,Δ.Proof.
Assume thatlimx⟶a+Hx;Δ=Ha;Δ for all a∈supHx;Δ. Hence, we have(12)limx⟶a+eHβx;Δ+Hβx;Δ−1=eHβa;Δ+Hβa;Δ−1,limx⟶a+eHβx;Δ=eHβa;Δ.
SincesupMx;β,Δ=supHx;Δ,(13)limx⟶a+Mx;β,Δ=Ma;β,Δ,for each a∈supMx;β,Δ. Therefore, Mx;β,Δ is right continuous.
As we mentioned earlier that the regularly varying tail behavior (RVTB) is a very crucial property to characterize the HT distributions, now we provide the mathematical treatment of RVTB of the NE-X family.Theorem 3.
IfH¯x;Δ is a regularly varying distribution, so is M¯x;β,Δ.Proof.
Supposelimx⟶∞1−Hβbx;β,Δ/1−Hβx;β,Δ=gb is finite for all b>0. Then, using equation (8), we have(14)limx⟶∞M¯bx;β,ΔM¯x;β,Δ=limx⟶∞1−Hβbx;Δ1−Hβx;Δ×eHβx;ΔeHβbx;Δ=limx⟶∞1−Hβbx;Δ1−Hβx;Δ×eHβ∞;ΔeHβb.∞;Δ.
Sincelimx⟶∞Hx;Δ=1, we have(15)limx⟶∞M¯bx;β,ΔM¯x;β,Δ=limx⟶∞1−Hβbx;Δ1−Hβx;Δ×ee=limx⟶∞1−Hβbx;Δ1−Hβx;Δ=gb,for b>0; thus, M¯x,β,Δ is a regularly varying distribution.
Linking this to equation (8), the PDF represented by mx;β,Δ is(16)mx;β,Δ=βhx;ΔHβ−1x;ΔeHβx;Δ2−Hβx;Δ,x∈ℝ.
The RV with PDF (16) is represented by X∼NE-Xx;β,Δ.
The SF,M¯x;β,Δ, and HRF (hazard rate function), wx;β,Δ, of X are(17)M¯x;β,Δ=1−Hβx;ΔeHβx;Δ,x,∈ℝ,wx;β,Δ=βhx;ΔHβ−1x;Δ1−Hβx;Δ2−Hβx;Δ,x∈ℝ,respectively. In the next section, we discuss a subcase of the NE-X family, called the new exponential-Weibull (NE-Weibull) distribution.


## 2. NE-Weibull Distribution

This section deals with the NE-Weibull distribution as a subcase of the NE-X family. Suppose that X∼Weibullα,γ with CDF, Hx;Δ=1−exp−γxα, and PDF, hx;Δ=αγxα−1exp−γxα, where Δ=α,γ. By inserting the CDF of the Weibull model in equation (8), the CDF and PDF of the NE-Weibull distribution take the forms(18)Mx;β,Δ=1−1−1−exp−γxαβexp1−exp−γxαβ,x≥0,α,γ,β>0,mx;β,Δ=αγβxα−1exp−γxα1−exp−γxαβ−1exp1−exp−γxαβ2−1−exp−γxαβ.Some possible PDF behaviors for the NE-Weibull distribution are shown in Figure1. From Figure 1, it is obvious that as the values of α and β increase, the NE-Weibull model becomes a HT distribution.Figure 1
Visual display ofmx;β,Δ.As mentioned above, different PDF shapes of the NE-Weibull model for various values ofβ, γ=1, and α are sketched in Figure 1. When α,β<1, the NE-Weibull distribution behaves similar to the exponential distribution expγ. For β>1, the NE-Weibull distribution captures the characteristics of Weiα,γ. The NE-Weibull model, however, has certain benefits over the Weibull distribution. For instance, it has heavier tails than the Weibull distribution and provides closer fit to data in the finance sector.Some possible HRF behaviors for the NE-Weibull distribution are shown in Figure2. The NE-Weibull distribution provides increasing, unimodal, decreasing, and bathtub HRF shapes.Figure 2
Different HRF plots of the NE-Weibull model.


## 3. Properties

Here, we provide a concise treatment of the mathematical properties of NE-X distribution. These properties include QF (quantile function), moments, SK (skewness), and KUR (kurtosis). Furthermore, different plots for the SK and KUR are also provided.


### 3.1. Quantile Function

SupposeX denotes the NE-X family with CDF presented in equation (8); then, the QF of NE-X distributions, Qu, is(19)xu=Qu=M−1u=H−1t,where t is the solution of Hx;Δ+Hβx;Δ+log1−u−1.


### 3.2. Moments

Now, we introduce the moments of the NE-X distributions which can further be used to obtain other characteristics.Therth moment of the NE-X family reduces to(20)μr′=∫−∞∞xrmx;β,Δdx.By putting equation (16) in (20), we obtain(21)μr′=β∫−∞∞xrhx;ΔHβ−1x;ΔexpHβx;Δ2−Hβx;Δdx=β∑i=0n21−i−1ini∫−∞∞xrhx;ΔHβ+iβ−1x;ΔexpHβx;Δdx=β∑i=0n∑j=0∞21−i−1i+jniκr,i,j,β,where κr,i,j,β=∫−∞∞xrhx;ΔHβ+iβ+jβ−1x;Δdx. For r=1,2,3,4, we obtain the first four moments of the NE-X family.Using the expressions of the moments, we obtain the mathematical form of the SK and KUR measures. The SK and KUR of the NE-Weibull distribution can be calculated, respectively, via the expressions(22)SK=μ3μ23/2,KUR=μ4μ22.The effects ofβ, α, and γ on the SK, KUR, variance, and mean of the NE-Weibull distribution are displayed in Figures 3–5.Figure 3
Plots of SK, KUR, variance, and mean of the NE-Weibull distribution forγ=2 and different values of α and β.Figure 4
Plots of SK, KUR, variance, and mean of the NE-Weibull distribution forβ=2 and different values of α and γ.Figure 5
Plots of SK, KUR, variance, and mean of the NE-Weibull distribution forα=2 and different values of β and γ.


## 3.1. Quantile Function

SupposeX denotes the NE-X family with CDF presented in equation (8); then, the QF of NE-X distributions, Qu, is(19)xu=Qu=M−1u=H−1t,where t is the solution of Hx;Δ+Hβx;Δ+log1−u−1.


## 3.2. Moments

Now, we introduce the moments of the NE-X distributions which can further be used to obtain other characteristics.Therth moment of the NE-X family reduces to(20)μr′=∫−∞∞xrmx;β,Δdx.By putting equation (16) in (20), we obtain(21)μr′=β∫−∞∞xrhx;ΔHβ−1x;ΔexpHβx;Δ2−Hβx;Δdx=β∑i=0n21−i−1ini∫−∞∞xrhx;ΔHβ+iβ−1x;ΔexpHβx;Δdx=β∑i=0n∑j=0∞21−i−1i+jniκr,i,j,β,where κr,i,j,β=∫−∞∞xrhx;ΔHβ+iβ+jβ−1x;Δdx. For r=1,2,3,4, we obtain the first four moments of the NE-X family.Using the expressions of the moments, we obtain the mathematical form of the SK and KUR measures. The SK and KUR of the NE-Weibull distribution can be calculated, respectively, via the expressions(22)SK=μ3μ23/2,KUR=μ4μ22.The effects ofβ, α, and γ on the SK, KUR, variance, and mean of the NE-Weibull distribution are displayed in Figures 3–5.Figure 3
Plots of SK, KUR, variance, and mean of the NE-Weibull distribution forγ=2 and different values of α and β.Figure 4
Plots of SK, KUR, variance, and mean of the NE-Weibull distribution forβ=2 and different values of α and γ.Figure 5
Plots of SK, KUR, variance, and mean of the NE-Weibull distribution forα=2 and different values of β and γ.


## 4. Estimation

Within this section, we derive the maximum likelihood estimators (MLEs) of the NE-Weibull parameters. Considerx1,x2,…,xn as the values of a sample from the NE-Weibull distribution with parameters β and Δ. The log-likelihood (LL) function ℓβ,Δ of the NE-Weibull distribution takes the form(23)ℓβ,Δ=klogβ+∑i=0kloghxi;Δ+β−1∑i=0klogHxi;Δ−∑i=0kHβxi;Δ+∑i=0klog2−Hβxi;Δ.The partial derivatives of the LL function are(24)∂∂βℓβ,Δ=nβ+∑i=0klogHxi;Δ−∑i=0klogHβxi;ΔHβxi;Δ−∑i=0klogHβxi;ΔHβxi;Δ2−Hβxi;ξ,∂∂Δℓβ,Δ=β−1∑i=0k∂Hxi;Δ/∂ΔHxi;Δ−β∑i=0kHβ−1xi;Δ∂Hxi;Δ∂Δ+∑i=0k∂hxi;Δ/∂Δhxi;Δ−β∑i=0kHβ−1xi;Δ∂Hxi;Δ/∂Δ2−Hβ−1xi;Δ.Equating∂/∂βℓβ,Δ and ∂/∂Δℓβ,Δ to zero and simultaneously solving them yield the MLEs β^,Δ^ of β,Δ.


## 5. Simulation Study

Here, we implement the Monte Carlo simulation approach to address the MLE behavior in estimating the NE-Weibull parameters. The NE-Weibull distribution can be simulated by using equation (8). Let U follow the standard uniform distribution; hence, the quantile function reduces to(25)Hx;Δ+Hx;Δθ+log1−u−1.The simulation is done for (i)α=0.5,β=1.2,γ=1, (ii) α=1.2,β=0.8,γ=0.9, and (iii) α=0.8,β=0.5,γ=1.1.The simulation results are obtained by utilizing theR software with the algorithms root Solve and “LBFGS−B” with optim. The results obtained are based on K=500 replications for samples of size k, where k=10,20,…,500. Statistical tools such as biases and mean square errors (MSEs) are obtained as assessing tools. These tools are calculated as follows:(26)biasΛ=1500∑i=1500Λ^−Λ,MSEΛ=1500∑i=1500Λ^−Λ2,where Λ=β,Δ.The summary measures (SMs) of the three simulated sets of data are provided in Table1, whereas the histograms, box plots, kernel density estimator, and estimated CDF of the NE-Weibull model are provided in Figures 6–8. The simulation results are visually displayed in Figures 9–11.Table 1
SMs of the simulated data sets.
Simulated dataMin.1st Qu.MedianMean3rd Qu.Max.Data set 10.00000.03900.19261.11910.971763.2960Data set 20.00000.13880.34750.54690.71883.4976Data set 30.00000.00700.07070.33640.31259.1907Figure 6
Kernel density estimator, box plot, histogram, and fitted CDF of the NE-Weibull distribution for simulated data set 1.Figure 7
Kernel density estimator, box plot, histogram, and fitted CDF of the NE-Weibull distribution for simulated data set 2.Figure 8
Kernel density estimator, box plot, histogram, and fitted CDF of the NE-Weibull distribution for simulated data set 3.Figure 9
Graphical display of different measures for simulated data set 1.Figure 10
Graphical display of different measures for simulated data set 2.Figure 11
Graphical display of different measures for simulated data set 3.


## 6. Data Modeling in the Finance Sector

Here, we illustrate the importance of the NE-Weibull distribution in modeling insurance claims data from the financial sector. We also calculated the risk measures such as VaR (value at risk) and TVaR (tail value at risk) for this data.


### 6.1. Insurance Claims Data

The insurance claims data represents the initial claims of the unemployment insurances per month from 1971 to 2018 [20]. The data set can also be retrieved from https://data.worlddatany-govns8z-xewg.The summary statistics for the insurance claims data are reported in Table2. Corresponding to this data, the histogram, box plot, and total time test (TTT) plots are provided in Figure 12. Figure 12 shows that the insurance claims data is unimodal, right-skewed, and HT.Table 2
SMs of the insurance claims data.
Min.1st Qu.MedianMean3rd Qu.Max.49.2682.71100.02109.18129.74308.35Figure 12
Histogram, TTT, and box plots for the insurance claims data.The NE-Weibull distribution is applied to fit the financial data, and it is compared with the Weibull, W-Loss (Weibull-Loss), NHT-Weibull (new heavy-tailed Weibull), Ku-Weibull (Kumaraswamy Weibull), and B-Weibull (beta Weibull) distributions. The CDFs of these models are as follows:(i)
Weibull model:(27)Hx=1−exp−γxα,x≥0.(ii)
W-Loss model:(28)Hx=1−βexp−γxαβ+γxα,x≥0.(iii)
NHT-Weibull model:(29)Hx=β1−exp−γxαβ−exp−γxα2,x≥0.(iv)
Ku-Weibull model:(30)Hx=1−1−1−exp−γxαab,x≥0.(v)
B-Weibull model:(31)Hx=I1−e−γxαx;a,b,x≥0.To decide about the best fitting of the applied distributions, certain criteria are taken into account. These criteria are as follows:(i)
The AIC is(32)2p−2ℓΘ.(ii)
The BIC is(33)plogk−2ℓΘ.(iii)
The HQIC is(34)2ploglogk−2ℓΘ.(iv)
The CAIC is(35)2pkk−p−1−2ℓΘ.Here,ℓΘ denotes the LL function, Θ is a parametric space, p represents the model parameters, and k is the number of selected samples. In addition to the criteria measures, three goodness-of-fit tests with its corresponding p values are also considered. These tests are given by the following:(i)
The Anderson–Darling (AD) test statistic is(36)AD=−k−1n∑j=1k2j−1logHxj+log1−Hxk−j+1,wherexj is the jth observation in k samples, calculated after sorting the data in the ascending order.(ii)
The Cramer–von Mises (CM) test statistic is(37)CM=112k+∑j=1k2j−12k−Hxj2.(iii)
The Kolmogorov–Smirnov (KS) test statistic is(38)KS=supxHnx−Hx,wheresupx represents the supremum of the set of distances and Hnx indicates the empirical CDF.The LL function is optimized, and the goodness-of-fit measures are calculated using the algorithm method “BFGS” via optim()R-function [31]. The MLEs of the fitted models with standard errors are listed in Table 3. Tables 4 and 5 show the goodness-of-fit values for the considered models.Table 3
Estimated parameters for the insurance claims data.
DistributionαγβabNE-Weibull1.283 (0.012)0.005 (0.001)8.689 (0.610)Weibull1.486 (1.096)0.001 (0.095)W-Loss1.564 (1.075)0.041 (0.076)12.765 (0.963)NHT-Weibull1.316 (0.964)0.025 (0.049)10.498 (0.875)31.713 (3.791)Ku-Weibull0.984 (0.076)0.052 (0.017)64.933 (0.063)0.657 (0.167)B-Weibull1.090 (0.058)0.031 (0.000)40.704 (10.392)0.592 (0.139)Table 4
Discrimination criteria of the competitive models for the insurance claims data.
DistributionAICBICCAICHQICNE-Weibull5519.4605532.5235519.5025524.555Weibull6169.5746178.2906169.6346172.975W-Loss5585.7385596.7085585.9845590.767NHT-Weibull5546.0075564.7565546.0835553.865Ku-Weibull5601.7345619.1525601.8055608.527B-Weibull5602.5555602.6255619.9725609.348Table 5
Goodness-of-fit criteria of the competing distributions for insurance claims data.
DistributionCMADKSp valueNE-Weibull0.1460.6740.0160.668Weibull0.9742.0940.8960.129W-Loss0.1960.8950.0270.538NHT-Weibull0.1750.7950.0230.618Ku-Weibull0.2001.0370.0370.383B-Weibull0.2211.1210.0350.471The fitted PDF and CDF of the NE-Weibull model are provided in Figure13, showing that the right-skewed HT data can be adequately fitted by the NE-Weibull distribution. The Kaplan–Meier survival (KMS) and PP (probability-probability) plots of the NE-Weibull distribution are sketched in Figure 14.Figure 13
Estimated density and distribution functions of the NE-Weibull distribution.Figure 14
KMS and PP plots of the NE-Weibull distribution.


### 6.2. VaR and TVaR for Insurance Claims Data

Now, we calculate the VaR and TVaR measures for the Weibull and proposed NE-Weibull distributions based on the MLEs and the data provided in Section6.1. The results of the VaR and TVaR of the NE-Weibull and Weibull distributions are presented in Table 6. We also display these results graphically in Figure 15.Table 6
VaR and TVaR results for the NE-Weibull and Weibull distributions.
DistributionParametersLevel of significanceVaRTVaRNE-Weibullα=1.283γ=0.005β=8.6890.700145.8537153.96590.750153.5376179.21750.800168.5360198.76540.850192.3638218.71890.900209.8965231.98760.950224.0768249.07860.975239.7853268.02980.999253.6358287.0948Weibullα=1.486γ=0.0010.70070.346289.75680.75077.7643101.85390.80083.8743115.79430.85098.3245135.87640.900120.5643156.98750.950142.6750179.98760.975176.7656201.98750.999202.6752230.9786Figure 15
Graphical presentation for the values of the VaR and TVaR.A model having higher values for the TVaR and VaR is considered a HT distribution. The numerical results in Table6 and their graph in Figure 15 show that the NE-Weibull is a HT model.


## 6.1. Insurance Claims Data

The insurance claims data represents the initial claims of the unemployment insurances per month from 1971 to 2018 [20]. The data set can also be retrieved from https://data.worlddatany-govns8z-xewg.The summary statistics for the insurance claims data are reported in Table2. Corresponding to this data, the histogram, box plot, and total time test (TTT) plots are provided in Figure 12. Figure 12 shows that the insurance claims data is unimodal, right-skewed, and HT.Table 2
SMs of the insurance claims data.
Min.1st Qu.MedianMean3rd Qu.Max.49.2682.71100.02109.18129.74308.35Figure 12
Histogram, TTT, and box plots for the insurance claims data.The NE-Weibull distribution is applied to fit the financial data, and it is compared with the Weibull, W-Loss (Weibull-Loss), NHT-Weibull (new heavy-tailed Weibull), Ku-Weibull (Kumaraswamy Weibull), and B-Weibull (beta Weibull) distributions. The CDFs of these models are as follows:(i)
Weibull model:(27)Hx=1−exp−γxα,x≥0.(ii)
W-Loss model:(28)Hx=1−βexp−γxαβ+γxα,x≥0.(iii)
NHT-Weibull model:(29)Hx=β1−exp−γxαβ−exp−γxα2,x≥0.(iv)
Ku-Weibull model:(30)Hx=1−1−1−exp−γxαab,x≥0.(v)
B-Weibull model:(31)Hx=I1−e−γxαx;a,b,x≥0.To decide about the best fitting of the applied distributions, certain criteria are taken into account. These criteria are as follows:(i)
The AIC is(32)2p−2ℓΘ.(ii)
The BIC is(33)plogk−2ℓΘ.(iii)
The HQIC is(34)2ploglogk−2ℓΘ.(iv)
The CAIC is(35)2pkk−p−1−2ℓΘ.Here,ℓΘ denotes the LL function, Θ is a parametric space, p represents the model parameters, and k is the number of selected samples. In addition to the criteria measures, three goodness-of-fit tests with its corresponding p values are also considered. These tests are given by the following:(i)
The Anderson–Darling (AD) test statistic is(36)AD=−k−1n∑j=1k2j−1logHxj+log1−Hxk−j+1,wherexj is the jth observation in k samples, calculated after sorting the data in the ascending order.(ii)
The Cramer–von Mises (CM) test statistic is(37)CM=112k+∑j=1k2j−12k−Hxj2.(iii)
The Kolmogorov–Smirnov (KS) test statistic is(38)KS=supxHnx−Hx,wheresupx represents the supremum of the set of distances and Hnx indicates the empirical CDF.The LL function is optimized, and the goodness-of-fit measures are calculated using the algorithm method “BFGS” via optim()R-function [31]. The MLEs of the fitted models with standard errors are listed in Table 3. Tables 4 and 5 show the goodness-of-fit values for the considered models.Table 3
Estimated parameters for the insurance claims data.
DistributionαγβabNE-Weibull1.283 (0.012)0.005 (0.001)8.689 (0.610)Weibull1.486 (1.096)0.001 (0.095)W-Loss1.564 (1.075)0.041 (0.076)12.765 (0.963)NHT-Weibull1.316 (0.964)0.025 (0.049)10.498 (0.875)31.713 (3.791)Ku-Weibull0.984 (0.076)0.052 (0.017)64.933 (0.063)0.657 (0.167)B-Weibull1.090 (0.058)0.031 (0.000)40.704 (10.392)0.592 (0.139)Table 4
Discrimination criteria of the competitive models for the insurance claims data.
DistributionAICBICCAICHQICNE-Weibull5519.4605532.5235519.5025524.555Weibull6169.5746178.2906169.6346172.975W-Loss5585.7385596.7085585.9845590.767NHT-Weibull5546.0075564.7565546.0835553.865Ku-Weibull5601.7345619.1525601.8055608.527B-Weibull5602.5555602.6255619.9725609.348Table 5
Goodness-of-fit criteria of the competing distributions for insurance claims data.
DistributionCMADKSp valueNE-Weibull0.1460.6740.0160.668Weibull0.9742.0940.8960.129W-Loss0.1960.8950.0270.538NHT-Weibull0.1750.7950.0230.618Ku-Weibull0.2001.0370.0370.383B-Weibull0.2211.1210.0350.471The fitted PDF and CDF of the NE-Weibull model are provided in Figure13, showing that the right-skewed HT data can be adequately fitted by the NE-Weibull distribution. The Kaplan–Meier survival (KMS) and PP (probability-probability) plots of the NE-Weibull distribution are sketched in Figure 14.Figure 13
Estimated density and distribution functions of the NE-Weibull distribution.Figure 14
KMS and PP plots of the NE-Weibull distribution.


## 6.2. VaR and TVaR for Insurance Claims Data

Now, we calculate the VaR and TVaR measures for the Weibull and proposed NE-Weibull distributions based on the MLEs and the data provided in Section6.1. The results of the VaR and TVaR of the NE-Weibull and Weibull distributions are presented in Table 6. We also display these results graphically in Figure 15.Table 6
VaR and TVaR results for the NE-Weibull and Weibull distributions.
DistributionParametersLevel of significanceVaRTVaRNE-Weibullα=1.283γ=0.005β=8.6890.700145.8537153.96590.750153.5376179.21750.800168.5360198.76540.850192.3638218.71890.900209.8965231.98760.950224.0768249.07860.975239.7853268.02980.999253.6358287.0948Weibullα=1.486γ=0.0010.70070.346289.75680.75077.7643101.85390.80083.8743115.79430.85098.3245135.87640.900120.5643156.98750.950142.6750179.98760.975176.7656201.98750.999202.6752230.9786Figure 15
Graphical presentation for the values of the VaR and TVaR.A model having higher values for the TVaR and VaR is considered a HT distribution. The numerical results in Table6 and their graph in Figure 15 show that the NE-Weibull is a HT model.


## 7. Final Comments

During the last couple of years, the study and modeling of extreme value data has gained increased interest in numerous areas, particularly in the hydrology and finance sector. Recent studies have shown the importance and potential of the HT distributions in various areas for analyzing real data sets, especially in the finance sector. In this article, a new interesting HT extension of the Weibull distribution, called the NE-Weibull distribution, is proposed using the NE-X approach. The MLEs of the NE-Weibull parameters β and Δ are obtained. To show the applicability of the NE-Weibull model, a real-life insurance claims data from the financial sector is considered. Based on certain tests, it is showed that the NE-Weibull model offers an adequate fit to the insurance claims data than the Weibull, W-Loss, NHT-Weibull, Ku-Weibull, and B-Weibull. On the other hand, the numerical results for the VaR and TVaR measures showed that the NE-Weibull model is a HT distribution as compared to the classical Weibull distribution.



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*Source: 8759055-2021-10-21.xml*