Modeling Vehicle Insurance Loss Data Using a New Member of T-X Family of Distributions

Zubair Ahmad; Eisa Mahmoudi; Sanku Dey; Saima K. Khosa

doi:10.2991/jsta.d.200421.001

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Volume 19, Issue 2, June 2020, Pages 133 - 147

Modeling Vehicle Insurance Loss Data Using a New Member of T-X Family of Distributions

Authors

Zubair Ahmad¹, Eisa Mahmoudi¹^{, *}, Sanku Dey², Saima K. Khosa³

¹Department of Statistics, Yazd University, Yazd, Iran

²Department of Statistics, St. Anthonys College, Shillong, India

³Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan

^*Corresponding author. Email: emahmoudi@yazd.ac.ir

Corresponding Author

Eisa Mahmoudi

Received 16 October 2019, Accepted 14 January 2020, Available Online 28 May 2020.

DOI: 10.2991/jsta.d.200421.001 How to use a DOI?
Keywords: Heavy-tailed distributions; Weibull distribution; Insurance losses; Actuarial measures; Monte Carlo simulation; Estimation
Abstract: In actuarial literature, we come across a diverse range of probability distributions for fitting insurance loss data. Popular distributions are lognormal, log-t, various versions of Pareto, log-logistic, Weibull, gamma and its variants and a generalized beta of the second kind, among others. In this paper, we try to supplement the distribution theory literature by incorporating the heavy tailed model, called weighted T-X Weibull distribution. The proposed distribution exhibits desirable properties relevant to the actuarial science and inference. Shapes of the density function and key distributional properties of the weighted T-X Weibull distribution are presented. Some actuarial measures such as value at risk, tail value at risk, tail variance and tail variance premium are calculated. A simulation study based on the actuarial measures is provided. Finally, the proposed method is illustrated via analyzing vehicle insurance loss data.
Copyright: © 2020 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The usage of heavy tailed distributions for modeling insurance loss data is arguably an important subject matter for actuaries. Besides, one of the major goals of actuarial studies is to model the uncertainty pervading in the realm of insurance with respect to number of claims received in a particular period and the size of the claim severity. Generally speaking, insurance data sets are right-skewed [1], unimodal hump-shaped [2] and with heavy tails [3]. This type of heavy tailed distribution is apt for estimating insurance loss data and thereby helps in assessment of business risk level. Thus, due to its immense significance in actuarial sciences, this type of data has been studied extensively and several probability models have been proposed in actuarial literature. Among the variety of the models, the prominently used models related to insurance loss data, financial returns, file sizes on network servers, etc. are Pareto, lognormal, Weibull, gamma, log-logistic, Fréchet, Lomax, inverse Gaussian distributions; see Resnick [4], Hogg and Klugman [5], Qi [6], Hao and Tang [7], Gómez-Déniz and Calderín-Ojeda [8] and Calderín-Ojeda and Kwok [9] and Yang et al. [10], among others. For assessment of small size losses, distributions like log-normal, gamma, Weibull, Inverse Gaussian, etc. are used while for large insurance losses; commonly used distributions are Pareto, log-logistic, Fréchet, Lomax distributions, etc. Most of these aforementioned classical distributions are generalized for insurance loss data, financial returns etc. based on, but not limited to, the following five approaches: (i) transformation method, (ii) composition of two or more distributions, (iii) compounding of distributions, (iv) exponentiated distributions and (v) finite mixture of distributions, for further detail see, Miljkovic and Grün [11] and Bhati and Ravi [12].

In the recent past, Dutta and Perry [13] carried out an empirical study on loss distributions using Exploratory Data Analysis and empirical approaches to estimate the risk. However, due to lack of flexibility and poor results, they rejected the idea of using exponential, gamma and Weibull distributions and pointed out that “one would need to use a model that is flexible enough in its structure.”

Hence it is imperative to develop models either from the existing distributions or a new family of models to cater insurance loss data, financial returns, etc.

In the premises of the above, we are motivated to search for more flexible probability distributions which provide greater accuracy in data fitting. Hence, in this paper, we introduce a new member of the T-X family [14], called the weighted T-X(WT-X) family of distributions for modeling insurance losses. With this idea, we construct a new model named the weighted T-X Weibull (WT-XW) distribution which is more flexible than the Weibull distribution. We later prove empirically that the WT-XW model provides better fits than the well-known competitive distributions in terms of different measures of model validation by means of vehicle insurance loss data. Two of the competitive distributions have one extra model parameter, and the others have the same number of parameters.

We hope that the new distribution will attract wider applications in insurance loss data, financial returns, etc. Finally, estimation of the WT-XW model parameters using the method of maximum likelihood estimation have been carried out. Further, some actuarial measures such as value at risk (VaR), tail value at risk (TVaR), tail variance (TV) and tail variance premium (TVP) are also calculated.

The rest of the paper is organized as follows: In Section 2, we discuss the proposed approach based on the T-X family of distributions. In Section 3, the proposed approach is applied to the classical Weibull distribution to derive the WT-XW distribution. In Section 4, the maximum likelihood estimators (MLEs) of the parameters are derived and simulation study is carried out in the same section. Afterward, in Section 5, some of actuarial measures are obtained and numerical studies of the risk measures are provided. In Section 6, a numerical application is illustrated based upon the vehicle insurance loss data. Here, the weighted T-X Weibull distribution is compared with the Weibull and other (i) two-parameter distributions such as Pareto, Lomax, Burr X-II (BX-II), Log-normal and (ii) well-known three-parameter models such as Dagum and Marshall-Olkin Weibull (MOW) distributions under different discrimination and goodness of fit measures. Finally, some concluding remarks are given in the last section.

2. PROPOSED METHOD

Let vt be the probability density function (pdf) of a random variable, say T, where T∈m, n for −∞≤m<n<∞, and let WFx be a function of cumulative distribution function (cdf) of a random variable, say X, satisfying the conditions given below:

WFx∈m,n
WFx is differentiable and monotonically increasing
WFx→m as x→−∞ and WFx→n as x→∞

The cdf of the T-X family of distributions is defined by

Gx=∫mWF xv(t) dt, x∈ℝ,(1)

where, WFx satisfies the conditions stated above. The pdf corresponding to (1) is

gx=∂∂xWFxvWFx, x∈ℝ.

Using the T-X family idea, several new classes of distributions have been introduced in the literature. Table 1 provides some WFx functions for some members of the T-X family.

WFx	Range of X	Members of T-X Family
Fx	[0,1]	Beta-G [15]
−log1−Fx	0,∞	Gamma-G Type-2 [16]
−logFx	0,∞	Gamma-G Type-1 [17]
F(x)1−F(x)	0,∞	Gamma-G Type-3 [18]
−log[1−Fα(x)]	0,∞	Exponentiated T-X [19]
log{F(x)1−F(x)}	−∞,∞	Logistic-G [20]
log−log1−Fx	−∞,∞	The Logistic-X Family [21]
[−log{1−F(x)}]1−F(x)	0,∞	New Weibull-X Family [22]
−log(1−F(x)eF(x))	0,∞	Weighted T-X Family(Proposed)

Table 1

Some members of the T-X family.

Now, we introduce the proposed family. Let T∼exp(1), then its cdf is given by

Vt=1−e−t, t≥0.(2)

The density function corresponding to (2) is

vt=e−t, t>0.(3)

If vt follows (3) and setting W[F(x)]=−log(1−F(x)eF(x)) in (1), we define the cdf of the weighted T-X family by

Gx=1−1−FxeF x, x∈ℝ.(4)

The density function corresponding to (4) is

gx=fxeF x2−Fx, x∈ℝ.(5)

The key motivations for using the WT-XW distribution in practice are the following:

A very simple and convenient method to modify the existing distributions.
To improve the characteristics and flexibility of the existing distributions.
To introduce the extended version of the baseline distribution having closed form of distribution function.
To provide best fit to the heavy-tailed data in financial sciences and other related fields.
Another most important motivation of the proposed approach is to introduce new distributions without adding additional parameter results in avoiding re-scaling problems.

3. SUB-MODEL DESCRIPTION

In this section, we introduce a special sub-model of the proposed family, called the WT-XW distribution. Let Fx;ξ be the cdf of the Weibull distribution given by F(x;ξ)=1−e−γxα,x≥0, α,γ>0, where ξ=α,γ. Then, the cdf of the weighted T-X Weibull has the following expression:

Gx;α,γ=1−e−γxαe1−e−γxα, x≥0 α,γ>0.(6)

The pdf and hazard rate function (hrf) of the WT-XW model are given, respectively, by

gx;α,γ=αγxα−1e−γxαe1−e−γxα1+e−γxα, x>0,(7)

and

hx;α,γ=αγxα−11+e−γxα, x>0.

For different values of the model parameters, plots of the density function of the WT-XW are sketched in Figure 1.

In Figure 1, we plotted different shapes for the density of WT-XW distribution for fixed values of γ and different values of α. When α<1, the proposed model behaves like exponential distribution. But, as the value of the α increases the proposed model captures the characteristics of the Weibull distribution. However, the proposed model has certain advantages over the Weibull distribution. For examples, it has heavier tails than the Weibull distribution as shown (in Section 5) through simulated study of the risk measures including VaR, TVaR, TV and TVP. Also, the proposed model provides best fit to the heavy tailed insurance loss data as shown in (Section 6). The hrf is plotted in Figure 2. The hrf of the proposed model is very flexible in accommodating different shapes namely, decreasing, increasing and unimodal and hence the WT-XW distribution becomes an important model to fit several real-lifetime data in applied areas such as reliability, survival analysis, economics and finance.

4. ESTIMATION AND SIMULATION STUDY

Several methods for parameter estimation have been introduced in the literature. Among these approaches, the maximum likelihood method is the most commonly employed. The maximum likelihood estimations enjoy several desirable properties and can be used for constructing confidence intervals. In this section, the method of maximum likelihood estimation is used to estimate the model parameters α,γ. For assessing the performance of the MLEs, we provide a comprehensive Monte Carlo simulation study to evaluate the behavior of these estimators.

4.1. Maximum Likelihood Estimation

Here, we discuss the MLEs of the model parameters of the WT-XW distribution. Let x1,x2,…,xn be the observations from pdf (7) with parameters α and γ. Then, the log-likelihood function corresponding to (7) is given by

Lxi,α,γ=nlogα+nlogγ+α−1∑i=1nlogxi−γ∑i=1nxiα−∑i=1n1−e−γxiα(8)

+∑i=1nlog1+e−γxiα.

The log-likelihood function can be maximized either directly or by solving the nonlinear likelihood function obtained by differentiating (8). We used the goodness of fit function in R with “Nelder-Mead” algorithm to obtain the MLEs. The first order partial derivatives of (8) with respective to the parameters are given, respectively, by

∂∂αLxi,α,γ=nα+∑i=1nlogxi−γ∑i=1nlogxixiα−γ∑i=1nlogxixiαe−γxiα(9)

−γ∑i=1nlogxixiαe−γxiα1+e−γxiα,

and

∂∂γLxi,α,γ=nγ−∑i=1nxiα−∑i=1nxiαe−γxiα−∑i=0nxiα1+e−γxiα.(10)

Setting ∂∂αlogLxi,α,γ and ∂∂γlogLxi;α,γ equal to zero and solving numerically these expressions simultaneously yield the MLEs of α,γ.

4.2. Simulation Study

In order to assess the performances of the maximum likelihood estimates of the parameters of the proposed distribution, a comprehensive simulation study is carried out. The process is carried out as follows:

The number of Monte Carlo replications was made 500 times each with sample sizes n=25,50,…500.
Initial values for the parameters are selected such as (i) for set 1, α=1.5 and γ=1, (ii) for set 2, α=0.5 and γ=1, and (iii) and for set 3, α=0.7 and γ=1.
Formulas used for calculating Bias and mean square error (MSE) are given by Bias(α^)=1500∑i=1500(αi^−α) and MSE(α^)=1500∑i=1500(αi^−α)2, respectively.
Step (3) is also repeated for γ.

The simulation results are provided in Table 2 and displayed graphically in Figures 3–5.

		Set 1: α = 0.5, γ = 1			Set 2: α = 1.5, γ = 1			Set 3: α = 0.7, γ = 1
n	Par	MLE	Bais	MSE	MLE	Bais	MSE	MLE	Bais	MSE
25	α	0.5387	0.03879	7.7×10−03	1.5829	0.0829	0.0771	0.7399	0.0399	0.0164
	γ	1.0674	6.7×10−02	9.1×10−02	1.0596	0.0596	0.0966	1.0943	0.0943	0.0926
50	α	0.5197	0.01972	2.8×10−03	1.5534	0.0534	0.0350	0.7205	0.0205	0.0066
	γ	1.0166	1.6×10−02	2.7×10−02	1.0196	0.01961	0.0310	1.0288	0.0288	0.0337
100	α	0.5104	0.01048	8.8×10−04	1.5264	0.0264	0.0163	0.7113	0.0113	0.0027
	γ	1.0032	3.2×10−03	7.0×10−03	1.0248	0.0248	0.0179	1.0087	0.0087	0.0132
200	α	0.5031	0.00319	1.8×10−04	1.5111	0.0111	0.0067	0.7083	0.0083	0.0012
	γ	1.0028	2.8×10−03	1.5×10−03	1.0065	0.0065	0.0066	1.0052	0.0052	0.0057
300	α	0.5012	0.00125	4.7×10−05	1.5025	0.0025	0.0041	0.7040	0.0040	0.0006
	γ	1.0001	1.4×10−04	5.1×10−04	1.0043	0.0043	0.0045	1.0027	0.0027	0.0033
400	α	0.5004	0.00042	1.6×10−05	1.5030	0.0030	0.0031	0.7039	0.0039	0.0004
	γ	0.9998	-1.0×10−04	5.3×10−05	0.9995	−0.0004	0.0032	1.0031	0.0031	0.0019
500	α	0.5003	0.00036	1.3×10−05	1.5016	0.0016	0.0026	0.7024	0.0024	0.0002
	γ	1.0006	6.1×10−04	5.7×10−05	0.9995	−0.0008	0.0030	0.9998	−0.0001	0.0014

WT-XW, weighted T-X Weibull; MLE, maximum likelihood estimator; MSE, mean square error.

Table 2

The simulation results of the WT-XW distribution.

5. ACTUARIAL MEASURES

One of the most important tasks of actuarial sciences institutions 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 some important risk measures (VaR, TVaR, TV, TVP) for the proposed distribution, which play a crucial role in portfolio optimization under uncertainty.

5.1. 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. The VaR is always specified with a given degree of confidence say q (typically 90%, 95% or 99%), and represent the percentage loss in portfolio value that will be equalled or exceeded only X percent of the time. VaR of a random variable X is the qth quantile of its cdf, see Artzner [23]. If X follows (7), then the VaR of the X is derived as

xq=F−1t,(11)

where t is the solution of the equation log1−q+t=log1−t.

5.2. Tail Value at Risk

Another important measure is TVaR, also known as conditional tail expectation (CTE) or tail conditional expectation (TCE), used to quantifies the expected value of the loss given that an event outside a given probability level has occurred. Let X follows the proposed family, then TVaR of X is defined as

TVaRqX=11−q∫VaRq∞x gxdx.(12)

Using (7) in (12), we get

TVaRqX=αγ1−q∫VaRq∞x α+1−1e−γxαe1−e−γxα1+e−γxαdx,

TVaRqX=αγ1−q∑i=0∞∑j=0i−1i+ji−j!j!∫VaRq∞x α+1−1e−γ j+1xαdx+∫VaRq∞x α+1−1e−γ j+2xαdx.(13)

Recall, the definition of incomplete gamma function in the form Γα,x=∫x∞tα−1e−tdt, so, from (13), we get

TVaRqX=11−q∑i=0∞∑j=0i−1i+ji−j!j!γ1/αΓ1α+1,γ j+1VaRqα j+11α+1+Γ1α+1,γ j+2VaRqα j+21α+1.(14)

5.3. Tail Variance

The TV is one of the most important actuarial measures which pay attention to the TV beyond the VaR. The TV of the WT-XW distributed random variable is derived as

TVqX=EX2|X>xq−TVaRq2.(15)

Consider

EX2|X>xq=αγ1−q∫VaRq∞xα+2−1e−γxαe1−e−γxα1+e−γxαdx.

On solving, we get

EX2|X>xq=11−q∑i=0∞∑j=0i−1i+ji−j!j!γ2/αΓ2α+1,γ j+1VaRqα j+12α+1+Γ2α+1,γ j+2VaRqα j+22α+1.(16)

Using (14) and (16) in (15), we get the expression for the TV of WT-XW distribution.

5.4. Tail Variance Premium

The TVP is another important measure play an essential role in insurance sciences. The TVP of WT-XW distributed random variable is derived as

TVPqX=TVaRq+δTVq,(17)

where 0<δ<1. Using the expressions (14) and (15) in (17), we get the TVP of the proposed distribution.

5.5. Numerical Study of the Risk Measures

In this section, we simulate the risk measures described above to show the suitability of the proposed model. The simulation is performed for the Weibull and proposed model for the values of parameters. A model with higher values of the risk measures is said to have heavier tail. The simulated results provided in Tables 3 and 4 show that the proposed model has higher values of the risk measures than the traditional Weibull distribution. The simulation results are graphically displayed in Figures 6–9, which show that the proposed model has heavier tail than the Weibull distribution.

	Set 1: α = 0.7, γ = 0.5
Dist	Level of Significance	VaR	TVaR	TV	TVP
	0.700	3.5226	8.8297	39.3627	18.6704
	0.750	4.3088	9.8157	41.3919	20.1637
	0.800	5.3327	11.070	43.8447	22.0318
Weibull	0.850	6.7450	12.761	46.9629	24.5026
	0.900	8.8951	15.276	51.2849	28.0976
	0.950	12.954	19.898	58.5092	34.5257
	0.975	17.440	24.889	65.5638	41.2805
	0.999	42.732	52.080	96.7712	76.2729
	0.700	1.7264	6.9464	65.3076	23.2733
	0.750	2.2581	7.9401	72.4397	26.0500
	0.800	3.0158	9.2719	81.6688	29.6892
WT-XW	0.850	4.1748	11.179	94.3033	34.7551
	0.900	6.1745	14.233	113.307	42.5602
	0.950	10.623	20.431	148.193	57.4798
	0.975	16.363	27.802	185.051	74.0656
	0.999	57.437	75.031	370.190	167.579

WT-XW, weighted T-X Weibull; VaR, value at risk; TVaR, tail value at risk; TV, tail variance; TVP, tail variance premium.

Table 3

Simulation results of VaR, TVaR, TV and TVP for n=100.

	Set 1: α = 1.3, γ = 1
Dist	Level of Significance	VaR	TVaR	TV	TVP
	0.700	1.4266	5.7649	43.6476	16.6768
	0.750	1.8914	6.5885	48.3030	18.6643
	0.800	2.5493	7.6857	54.3512	21.2735
Weibull	0.850	3.5421	9.2447	62.7181	24.9243
	0.900	5.2180	11.718	75.5996	30.6186
	0.950	8.8325	16.697	100.570	41.8403
	0.975	13.392	22.622	129.265	54.9384
	0.999	46.962	62.527	311.306	140.354
	0.700	0.4150	4.6755	119.152	34.4637
	0.750	0.6220	5.5085	138.819	40.2133
	0.800	0.9619	6.6914	166.526	48.3229
WT-XW	0.850	1.5702	8.5105	208.787	60.7074
	0.900	2.8319	11.708	282.440	82.3183
	0.950	6.4148	19.165	452.656	132.329
	0.975	12.300	29.521	688.040	201.531
	0.999	81.598	124.89	2925.11	856.172

WT-XW, weighted T-X Weibull; VaR, value at risk; TVaR, tail value at risk; TV, tail variance; TVP, tail variance premium.

Table 4

Simulation results of VaR, TVaR, TV and TVP for n=150.

The simulation process is described below:

Random samples of sizes n = 100 and 150 are generated from the Weibull and WT-XW models and parameters have been estimated via maximum likelihood method.
1000 repetitions are made to calculate the VaR, TVaR, TV and TVP for these distributions.

6. A REAL-LIFE APPLICATION

The main applications of the heavy tail models are the so-called extreme value theory or insurance loss phenomena. We consider a data set from insurance sciences. In this section, we illustrate the WT-XW model by analyzing vehicle insurance loss data to show how the proposed method works in practice. Furthermore, we calculate the actual measures of the Weibull and WT-XW distributions using the real data set.

6.1. Application to the Vehicle Insurance Loss Data

In this sub-section, we illustrate the proposed method using a real data set representing the vehicle insurance losses available at http://www.businessandeconomics.mq.edu.au/our_departments/Applied_Finance_and_Actuarial_Studies/research/books/GLMsforInsuranceData. First, we check whether the considered data set actually comes from the WT-XW or not by goodness of fit test and compare the fits with the other heavy tailed distributions including the two and three parameters distributions. This procedure is based on the Anderson Darling (AD) test statistic, Cramer–von-Mises (CM) test statistic and Kolmogorov–Smirnov (KS) statistic with the corresponding p-values. Note that, the AD, CM and KS statistic to be used only to verify the goodness-of-fit and not as a discrimination criteria. Therefore, we consider four discrimination criteria based on the log-likelihood function evaluated at the maximum likelihood estimates. The criteria are Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan-Quinn information criterion (HQIC), Corrected Akaike information criterion (CAIC). A model with lowest values for these statistics could be chosen as the best model to fit the data. The analytical measures are calculated are calculated as follows:

The AD test statistic is given by
AD=−n−1n∑i=1n2i−1logGxi+log1−Gxn−i+1,

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

The CM test statistic is given by
CM=112n+∑i=1n2i−12n−Gxi2.

The KS test statistic is given by
KS=supxGnx−Gx,

where Gnx is the empirical cdf and supx is the supremum of the set of distances. The discrimination criteria are calculated using the formulas:

The AIC is given by
AIC=2k−2ℓ.
The BIC is given by
BIC=klogn−2ℓ.
The HQIC is given by
HQIC=2kloglogn−2ℓ.
The CAIC is given by
CAIC=2nkn−k−1−2ℓ,

where ℓ denotes the log-likelihood function evaluated at the MLEs, k is the number of model parameters and n is the sample size. The distribution functions of the competitive models are

Weibull
Gx;α,γ=1−e−γxα, x≥0, α,γ>0.
Pareto
Gx;α,γ=1−γxα, x≥0, α,γ>0.
Lomax
Gx;α,γ=1−1+xγ−α, x≥0, α,γ>0.
B-XII
Gx;c,k=1−1+xc−k, x≥0, c,k>0.
Log-normal
Gx;μ,σ=12+12erflnx−μ2σ, σ>0, μ∈ℝ, x∈ℝ+.
Dagum
Gx;α,γ,θ=1+xγ−α−θ, x≥0, α,γ,θ>0.
MOW
Gx;α,γ,σ=1−e−γxασ+1−σ1−e−γxα, x≥0, α,γ,σ>0.

The values of MLEs of the parameters along with the corresponding standard errors (in parenthesis) are provided in Table 5. The discrimination criteria are presented in Table 6. Whereas, the goodness of fit measures are reported in Table 7. A model with lowest values for these statistics is considered as a best candidate model. As we see, the results (Tables 6 and 7) show that the WT-XW distribution provides better fit than the other considered competitors. Hence, the proposed model can be used as a best candidate model for modeling insurance losses. Furthermore, in support of Tables 6 and 7, the estimated cdf and pdf of the proposed model are plotted in Figure 10. The Kaplan–Meier survival plot and PP plot are provided in Figure 11. These plots also reveal that the WT-XW distribution provides the best fit to data compared to the other models.

Dist.	α̂	γ̂	σ̂	ĉ	k̂	θ̂	μ̂
WT-XW	0.868 (0.038)	0.006 (0.001)	–	–	–	–	–
Weibull	0.759 (0.041)	0.019 (0.005)	−	−	−	−	−
Pareto	0.959 (0.291)	1.385 (0.491)	–	–	–	–	–
Lomax	1.689 (0.357)	1.256 (0.648)	–	–	–	–	–
B-XII	–	–	–	6.672 (11.259)	0.032 (0.055)	–	–
Log-normal	–	–	0.965 (0.325)	–	–	–	0.765 (0.101)
Dagum	0.968 (0.040)	0.903 (0.103)	–	–	–	0.698 (0.795)	–
MOW	0.594 (0.098)	0.072 (0.289)	2.515 (0.904)	–	–	–	–

WT-XW, weighted T-X Weibull; MOW, Marshall-Olkin Weibull.

Table 5

The estimated values of the parameters with standard errors in parenthesis) of the fitted distributions.

Dist.	AIC	BIC	CAIC	HQIC
WT-XW	2234.262	2240.603	2234.331	2236.833
Weibull	2243.208	2249.549	2243.277	2245.780
Pareto	2260.071	2276.518	2266.090	2264.057
Lomax	2272.037	2279.289	2274.980	2273.885
B-XII	2499.052	2505.393	2499.121	2501.624
Lognormal	2236.643	2243.980	2237.873	2239.098
Dagum	2238.876	2245.084	2239.764	2240.709
MOW	2256.673	2266.184	2256.812	2260.530

WT-XW, weighted T-X Weibull; MOW, Marshall-Olkin Weibull; AIC, Akaike information criterion; BIC, Bayesian information criterion; HQIC, Hannan-Quinn information criterion; CAIC, Corrected Akaike information criterion.

Table 6

Analytical measures of the proposed and other competitive models.

Dist.	CM	AD	KS	p-value
WT-XW	0.595	3.659	0.090	0.152
Weibull	0.724	4.399	0.103	0.123
Pareto	1.103	5.709	0.478	0.101
Lomax	1.290	5.774	0.573	0.098
B-XII	0.809	5.107	0.408	0.118
Log-normal	0.635	3.971	0.094	0.139
Dagum	0.6409	4.094	0.095	0.134
MOW	0.903	5.431	0.421	0.109

WT-XW, weighted T-X Weibull; MOW, Marshall-Olkin Weibull; Anderson Darling (AD) test statistic, CM, Cramer von-Mises; KS, Kolmogorov-Smirnov.

Table 7

Analytical measures of the proposed and other competitive models.

6.2. Calculation of the Actuarial Measures Using the Vehicle Insurance Loss Data

In this sub-section, we calculate the actuarial measures of the Weibull and the WT-XW distribution using the estimated values of the parameters for the insurance loss data set. The numerical results are reported in Table 8.

Dist	Parameters	Level of Significance	VaR	TVaR	TV	TVP
		0.700	255.920	590.031	142719.084	36269.802
		0.750	308.244	651.818	148310.999	37729.568
	α̂=0.759	0.800	375.326	729.705	154963.790	39470.652
Weibull	γ̂=0.018	0.850	466.263	833.533	163268.542	41650.669
		0.900	602.016	985.866	174526.532	44617.499
		0.950	851.889	1260.832	192776.813	49455.035
		0.975	1121.071	1552.087	210014.124	54055.618
		0.999	2564.815	3075.459	281148.054	73362.473
		0.700	258.291	639.741	210718.724	53319.422
		0.750	312.536	710.832	222490.325	56333.413
	α̂=0.868	0.800	383.825	801.937	236506.473	59928.555
WT-XW	γ̂=0.006	0.850	483.543	925.725	253773.605	64369.126
		0.900	638.454	1111.378	276269.591	70178.776
		0.950	938.583	1454.882	309257.870	78769.349
		0.975	1275.602	1824.653	335843.672	85785.571
		0.999	3111.461	3740.299	413793.771	107188.742

WT-XW, weighted T-X Weibull; VaR, value at risk; TVaR, tail value at risk; TV, tail variance; TVP, tail variance premium.

Table 8

Simulation results of VaR, TVaR, TV and TVP for n=150.

As we have mentioned earlier that a model with higher values of the risk measures is said to posses the heavier tails. From the numerical results for the actuarial measures of the proposed and Weibull distributions provided in Table 6, it is clear that the proposed distribution has heavier tail than the Weibull distribution and can be used as good candidate model for modeling heavy tailed insurance data sets.

7. CONCLUDING REMARKS

In the present work, we have proposed a versatile two parameter heavy tailed weighted T-X family of the Weibull distribution. The distribution has closed-form expressions for some insurance measures such as VaR, TVaR, TV and TVP. We have also studied some of its basic properties. Although the method has only been applied to the classical Weibull distribution, yet, this procedure can be extended by using other probabilistic families as parent distribution. The motivation for conducting this study is to find out whether the model can be applied in respect of vehicle insurance loss data. Numerical results show that the WT-XW distribution outperforms other existing long-tail distributions under the different measures of model assessment considered in respect of vehicle insurance loss data. This new method, which has a promising approach for data modeling in the actuarial field, may be very useful for practitioners who handle large claims and thereby it can be deemed as an alternative to the Weibull distribution.

AUTHORS' CONTRIBUTIONS

(i) Zubair Ahmad wrote the initial draft of the paper and did the simulation study and analysis, (ii) Eisa Mahmoudi helped in the simulation study and superivised overall work, and (iii) Sanku Dey and Saima K. Khosa helped in writing and improving the Introduction Section.

FUNDING STATEMENT

The work is sponsored by the Department of Statistics, Yazd University, Iran.

DATA AVAILABILITY STATEMENT

This work is mainly a methodological development and has been applied on secondary data related to the Vehicle Insurance Loss Data, but if required, data will be provided.

ACKNOWLEDGMENT

The authors are grateful to the Editor and referees for many of their valuable comments and suggestions which lead to this improved version of the manuscript. The first two authors also acknowledge the support of the Yazd University, Iran.

DEDICATION

This article is drafted from the Ph.D work of the first Author. The author would like to dedicate this article to the memory of his late parents.

APPENDIX

R Code for calculating the analysis measures in this manuscript.

############################################################
Important Note:
The data is saved in the csv file in our PC, so we used the commond
“read.csv” to load the data in to R. The data is saved in column 11,
that's why we have used the commond “data=data[,11]” to call the
data from the 11th column of the csv file. In the propogram, pm is
used for the proposed model.
############################################################
data<-read.csv(file.choose(), header=TRUE)
data=data[,11]
data=data[!is.na(data)]
data=data/5
data
############################################################
##### The density of the proposed distribution
############################################################
pdf_pm <- function(par,x)
{
alpha=par[1]
gamma=par[2]
alpha*gamma*(x^(alpha-1))*exp(-gamma*x^alpha)*(1+exp (-gamma*x^alpha))
*(1/(exp(1-exp(-gamma*x^alpha))))
}
############################################################
##### The distribution function of the proposed distribution
############################################################
cdf_pm <- function(par,x)
{
alpha= par[1]
gamma= par[2]
1-(exp(-gamma*x^alpha)/(exp(1-exp(-gamma*x^alpha))))
}
set.seed(0)
goodness.fit(pdf=pdf_pm, cdf=cdf_pm,
starts = c(0.5,0.5), data = data,
method=“Nelder-Mead”, domain=c(0,Inf),mle=NULL)

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 19 - 2
Pages: 133 - 147
Publication Date: 2020/05/28
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.200421.001 How to use a DOI?
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TY  - JOUR
AU  - Zubair Ahmad
AU  - Eisa Mahmoudi
AU  - Sanku Dey
AU  - Saima K. Khosa
PY  - 2020
DA  - 2020/05/28
TI  - Modeling Vehicle Insurance Loss Data Using a New Member of T-X Family of Distributions
JO  - Journal of Statistical Theory and Applications
SP  - 133
EP  - 147
VL  - 19
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.200421.001
DO  - 10.2991/jsta.d.200421.001
ID  - Ahmad2020
ER  -

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