Statistical distributions play a vital role in modeling data in applied areas such as risk management, banking, economic, financial and actuarial sciences. However, the quality of the procedures primarily depends upon the assumed probability model of the phenomenon under consideration. Among applied fields, insurance data sets are usually positive (Klugman
Among available literature, the Pareto, Weibull, and gamma are the promising distributions used widely for modeling insurance data sets. Unfortunately, due to the monotonically decreasing shape, the Pareto distribution often does not provide a best fit to many data sets. Weibull distribution is suitable for modeling small losses, but fails to provide best fit to large losses. In addition, gamma distribution is also popular because it does not have a closed form expression of cumulative distribution function (cdf). Consequently, introducing new distributions to address these problems and cater heavy tailed data is an interesting research topic that is quite rich and growing rapidly. Therefore, researchers are often in search of finding more flexible distributions. This has been done through many different approaches such as (i) transformation method, (ii) composition of two or more distributions, (iii) compounding of distributions, and (iv) finite mixture of distributions, for details see Ahmad
Among the prominent methods, Mahdavi and Kundu (2017) recently proposed a new method for introducing statistical distributions via the cdf given by
where
Ahmad
The Ex-APT family is an extension of (
where,
The new pdf is most tractable when
The rest of this work is as follows. In Section 2, we introduce a special sub-case of (
In this section, we introduce a sub-model of the (
Let
The pdf corresponding (
The survival function (sf) and hazard function (hf) of NEx-APTW distribution are respectively, given by
Figure 1 sketches different plots for the density function of the NEx-APTW distribution.
Let
Alpha power transformed Weibull (APTW) distribution, if
Alpha power transformed exponential (APTE) distribution, if
Weibull distribution, if
Rayleigh distribution, if
Exponential distribution if
One parameter Weibull distribution, if
One parameter New extended alpha power transformed Weibull, if
New extended alpha power transformed Rayleigh, if
New extended alpha power transformed exponential, if
In this section, we derive some mathematical properties of the NEx-APT distribution, such as quantile function, moments and moment generating function.
The quantile function of the NEx-APT distribution, denoted by
where
In this sub-section, we intend to derive the moments and the moment generating function of the NEx-APT distribution. Let
and using (
The function
Taking
Using (
where
Furthermore, the moment generating function of the NEx-APT random variable
In this section, we present certain characterizations of the NEx-APT distribution in the following directions: (i) based on a simple relationship between two truncated moments and (ii) in terms of the reverse hazard function. It should be mentioned that for the characterization (i) the cdf is not required to have a closed form.
We present our characterizations (i)–(ii) in two subsections.
In this subsection, we present characterizations of NEx-APT distribution in terms of a simple relationship between two truncated moments. The first characterization result employs a theorem due to Glänzel (1987), see Theorem 4.1 below. Note that the result holds also when the interval
Let
and
and finally
Conversely, if
therefore,
Now, in view of Theorem 1,
The reverse hazard function,
In this subsection, we present a characterization of the NEx-APT distribution in terms of the reverse hazard function.
Is straightforward and hence omitted.
In this section, we estimate the parameters of the NEx-APT distribution via the method of maximum likelihood and provide Monti Carlo simulation to evaluate the performance of these estimators.
Let
where, Θ = (
and
Setting (
In this subsection, we derive the asymptotic confidence intervals of the unknown parameters of the NEx-APT distributions. The simplest large sample approach is to assume that the maximum likelihood estimators (
The second partial derivatives included in
and
The above approach is used to derive the (1 −
In order to evaluate the performances of the maximum likelihood estimators of the (
We generate
Initial values for the parameters are selected as given in Table 1.
Compute the biases and MSEs given by
One of the most important tasks of actuaries is to evaluate the exposure to market risk in a portfolio of instruments, which arise from changes in underlying variables such as prices of equity, interest rates or exchange rates. In this section, we calculate two well-known and important risk measures Value at Risk (VaR) and Tail VaR (TVaR) for the proposed distribution, which play a crucial role in portfolio optimization under uncertainty. Furthermore, based on these measures, a simulation study is performed and we show that the proposed distribution has heavier tails the Weibull and exponentiated Weibull (EW) distributions.
In the context of actuarial sciences, the measure VaR is widely used by practitioners as a standard financial market risk. It is also known as the quantile risk measure or quantile premium principle. VaR of a random variable
where
Another important measure is TVaR, also known as conditional tail expectation (CTE) used to quantify the expected value of the loss given that an event outside a given probability level has occurred. Let
Using (
Using the series
or
Based on actuarial measures, a simulation study is performed for the Weibull, exponentiated (EW) and proposed model for the selected parameters values. A model with higher values of the Risk measures (VaR and TVaR) is said to have a heavier tail. The simulated results provided in Tables 2 and 3 show that the proposed model has higher values of the risk measures than the other competitive distributions.
Figure 4 displays the simulation results provided in Table 2; in addition, Figure 5 displays the simulation results provided in Table 3.
Actuaries are looking for new distributions to provide an adequate fit to heavy tailed data in actuarial, financial sciences and related areas. In this section, we analyze a real data set from insurance sciences to demonstrate the flexibility of the NEx-APTW distribution. We also calculate actual measures of the Weibull, EW, and NEx-APTW distributions using a real data set.
The data set representing hospital costs in the state of Wisconsin is published by the Office of the Health Care Information, Wisconsin’s Department of Health and Human Resources. The data set is available at: https://www.dhs.wisconsin.gov/stats/index.htm. The comparison of the proposed method is made with the other ten (two, three and four parameters) well-known distributions. The cdf’s of the competitive distributions are:
Weibull distribution
Ex-APTW distribution
Lomax distribution
Burr-XII (B-XII) distribution
Pareto distribution
The alpha power transformed Weibull (APTW) distribution
The Marshall-Olkin (MOW) distribution
EW distribution
Kumaraswamy Weibull (Ku-W) distribution
The beta Weibull (BW) distribution
To determine the goodness-of-fit among the applied distributions, we consider certain goodness-of-fit measures such as Cramer-Von-Messes (CM) test statistic, Anderson Darling (AD) test statistic and Kolmogorov-Simonrove (KS) test statistic with corresponding
The AD test statistic
where
The CM test statistic
The KS test statistic is given by
where
A distribution with lower values of these analytical measures is considered as a good candidate model among the applied distributions for the underlying data sets. Based on the considered measures, the NEx-APTW distribution has the lowest values among all fitted models for the hospital cost insurance data. Table 4 reports parameter values with standard errors in parenthesis. In support of the numerical measures provided in Table 5, the empirical cdf and sf of the NEx-APTW are plotted in Figure 6. From Figure 6, we can see that the proposed model fit empirical cdf and sf very closely. In addition, the PP plot of the NEx-APTW distribution for the respective data set is plotted in Figure 7 and shows that the proposed provide best fit to the considered data. The box plot of the data set is also sketched in Figure 7 showing that the hospital cost insurance data is skewed to the right.
In this sub-section, we compute VaR and TVaR measures of Weibull, EW and the NEx-APTW distributions using estimated parameters values analyzed in Subsection 7.1. Table 6 reports the numerical results. A model with higher values of the risk measures possesses the heavier tails. The numerical results for the actuarial measures of the proposed and the other distributions show that the proposed distribution has a heavier tail than Weibull and EW distributions. In addition, it can be used as a good candidate model for modeling heavy tailed insurance data sets.
In this article, a new family of distributions called a new extended alpha power transformed family has been proposed. The proposed method examines a four-parameter special model of a new extended alpha power transformed Weibull distribution. Actuarial measures of the proposed model are also calculated and a simulation study is conducted to show the usefulness of the proposed method in actuarial sciences. A practical application to the heavy tailed insurance data is analyzed and the comparison of the proposed model with the other nine well-known competitors are presented. Actuarial measures based on a real data set is also calculated which shows that the proposed model may be a good candidate model to analyze actuarial data sets. We hope that the proposed method will attract a wider applications in actuarial sciences and related fields.
Simulation results for different combination of parameters values of new extended alpha power transformed Weibull distribution
Parameters | MLE | Bias | MSE | |
---|---|---|---|---|
25 | 2.905 | 1.705 | 7.485 | |
2.197 | 1.697 | 6.860 | ||
4.098 | 3.098 | 12.38 | ||
1.718 | 0.818 | 3.805 | ||
50 | 2.550 | 1.350 | 6.123 | |
1.914 | 1.414 | 5.932 | ||
3.569 | 2.569 | 10.25 | ||
1.682 | 0.782 | 3.525 | ||
100 | 1.945 | 0.745 | 3.829 | |
1.565 | 1.065 | 4.843 | ||
2.825 | 1.825 | 7.272 | ||
1.603 | 0.703 | 3.020 | ||
150 | 1.792 | 0.592 | 2.957 | |
1.225 | 0.725 | 3.381 | ||
2.400 | 1.400 | 5.543 | ||
1.382 | 0.482 | 2.127 | ||
250 | 1.398 | 0.198 | 1.127 | |
0.844 | 0.344 | 1.718 | ||
1.695 | 0.695 | 2.767 | ||
1.084 | 0.184 | 0.951 | ||
350 | 1.270 | 0.070 | 0.545 | |
0.739 | 0.239 | 1.192 | ||
1.452 | 0.452 | 1.790 | ||
1.035 | 0.865 | 0.681 | ||
450 | 1.245 | 0.045 | 0.309 | |
0.590 | 0.090 | 0.495 | ||
1.266 | 0.266 | 1.044 | ||
0.948 | 0.048 | 0.268 | ||
550 | 1.216 | 0.016 | 0.157 | |
0.578 | 0.078 | 0.410 | ||
1.169 | 0.169 | 0.671 | ||
0.929 | 0.029 | 0.196 | ||
650 | 1.226 | 0.026 | 0.143 | |
0.549 | 0.049 | 0.244 | ||
1.183 | 0.083 | 0.331 | ||
0.934 | 0.014 | 0.101 | ||
750 | 1.194 | 0.005 | 0.031 | |
0.520 | 0.040 | 0.189 | ||
1.052 | 0.052 | 0.209 | ||
0.920 | 0.010 | 0.096 |
MLE = maximum likelihood estimators; MSE = mean squared error.
Simulation results for VaR abd TVaR of the NEx-APTW and other fitted distributions
Distribution | Parameters | Level of significance | VaR | TVaR |
---|---|---|---|---|
Weibull | 0.700 | 3.695 | 11.208 | |
0.750 | 4.674 | 12.617 | ||
0.800 | 5.995 | 14.447 | ||
0.850 | 7.885 | 16.970 | ||
0.900 | 10.889 | 20.828 | ||
0.950 | 16.885 | 28.199 | ||
0.975 | 23.886 | 36.494 | ||
0.999 | 67.955 | 85.868 | ||
EW | 0.700 | 1.414 | 3.076 | |
0.750 | 1.700 | 3.381 | ||
0.800 | 2.060 | 3.758 | ||
0.850 | 2.534 | 4.250 | ||
0.900 | 3.217 | 4.949 | ||
0.950 | 4.408 | 6.155 | ||
0.975 | 5.614 | 7.369 | ||
0.999 | 11.272 | 13.033 | ||
NEx-APTW | 0.700 | 36.029 | 46.156 | |
0.750 | 37.733 | 48.016 | ||
0.800 | 39.859 | 50.332 | ||
0.850 | 42.665 | 53.380 | ||
0.900 | 46.743 | 57.792 | ||
0.950 | 54.038 | 65.646 | ||
0.975 | 61.731 | 73.881 | ||
0.999 | 102.339 | 116.836 |
VaR = Value at Risk; TVaR = Tail VaR; EW = exponentiated Weibull; NEx-APTW = new extended alpha power transformed Weibull.
Simulation results for VaR abd TVaR of the NEx-APTW and other fitted distributions.
Distribution | Parameters | Level of significance | VaR | TVaR |
---|---|---|---|---|
Weibull | 0.700 | 3.601 | 15.695 | |
0.750 | 4.831 | 17.998 | ||
0.800 | 6.593 | 21.084 | ||
0.850 | 9.288 | 25.500 | ||
0.900 | 13.905 | 32.569 | ||
0.950 | 24.059 | 46.976 | ||
0.975 | 37.119 | 64.354 | ||
0.999 | 137.146 | 185.066 | ||
EW | 0.700 | 1.052 | 2.439 | |
0.750 | 1.286 | 2.694 | ||
0.800 | 1.582 | 3.010 | ||
0.850 | 1.976 | 3.424 | ||
0.900 | 2.548 | 4.015 | ||
0.950 | 3.554 | 5.039 | ||
0.975 | 4.578 | 6.072 | ||
0.999 | 9.398 | 10.902 | ||
NEx-APTW | 0.700 | 56.131 | 78.680 | |
0.750 | 59.724 | 82.844 | ||
0.800 | 64.265 | 88.080 | ||
0.850 | 70.352 | 95.060 | ||
0.900 | 79.371 | 105.338 | ||
0.950 | 95.977 | 124.064 | ||
0.975 | 114.069 | 144.267 | ||
0.999 | 217.212 | 257.047 |
VaR = Value at Risk; TVaR = Tail VaR; EW = exponentiated Weibull; NEx-APTW = new extended alpha power transformed Weibull.
Estimated values with standard error (in parenthesis) of the proposed and other competitive models for the hospital cost insurance data
Distribution | |||||||
---|---|---|---|---|---|---|---|
NEx-APTW | 3.419 (2.751) | 0.696 (0.028) | 1.469 (0.231) | 0.521 (1.613) | |||
Ex-APTW | 3.893 (2.751) | 0.865 (0.183) | 1.665 (0.069) | ||||
Weibull | 0.674 (0.022) | 1.918 (0.084) | |||||
Lomax | 1.569 (0.207) | 0.365 (0.074) | |||||
B-XII | 2.885 (0.129) | 0.813 (0.026) | |||||
Pareto | 2.970 (0.420) | 1.909 (2.895) | |||||
APTW | 0.185 (2.910) | 1.229 (0.403) | 0.762 (0.945) | ||||
MOW | 0.808 (0.955) | 1.186 (0.863) | 0.387 (0.145) | ||||
EW | 1.944 (0.459) | 2.564 (0.261) | 0.487 (0.063) | ||||
Ku-W | 0.847 (0.805) | 1.086 (1.290) | 2.567 (2.906) | 4.583 (1.095) | |||
BW | 0.441 (0.089) | 0.828 (0.928) | 2.097 (0.761) | 3.804 (1.092) |
NEx-APTW = new Ex-APTW; Ex-APTW = extended APTW; B-XII = Burr-XII; APTW = alpha power transformed Weibull; MOW = Marshall-Olkin; EW = exponentiated Weibull; Ku-W = Kumaraswamy Weibull; BW = beta Weibull.
Goodness-of-fit measures of the proposed and other competitive models for the hospital cost insurance data
Distribution | CM | AD | KS | |
---|---|---|---|---|
NEx-APTW | 0.093 | 0.890 | 0.031 | 0.695 |
Ex-APTW | 0.109 | 0.932 | 0.034 | 0.585 |
Weibull | 0.112 | 0.936 | 0.035 | 0.035 |
Lomax | 0.464 | 2.623 | 0.078 | 0.002 |
B-XII | 0.250 | 1.426 | 0.045 | 0.229 |
Pareto | 0.560 | 1.708 | 0.058 | 0.247 |
APTW | 0.152 | 0.957 | 0.038 | 0.414 |
MOW | 0.155 | 0.961 | 0.037 | 0.447 |
EW | 0.168 | 0.997 | 0.049 | 0.150 |
Ku-W | 0.159 | 0.944 | 0.039 | 0.396 |
BW | 1.948 | 1.912 | 0.197 | 0.206 |
CM = Cramer-Von-Messes; AD = Anderson Darling; KS = Kolmogorov-Simonrove; NEx-APTW = new Ex-APTW; Ex-APTW = extended APTW; B-XII = Burr-XII; APTW = alpha power transformed Weibull; MOW = Marshall-Olkin; EW = exponentiated Weibull; Ku-W = Kumaraswamy Weibull; BW = beta Weibull.
Results for the actuarial measures using health care insurance data
Distribution | Parameters | Level of significance | VaR | TVaR |
---|---|---|---|---|
Weibull | 0.700 | 0.710 | 0.914 | |
0.750 | 1.047 | 1.275 | ||
0.800 | 1.323 | 1.709 | ||
0.850 | 1.793 | 1.907 | ||
0.900 | 2.134 | 2.086 | ||
0.950 | 2.600 | 2.395 | ||
0.975 | 2.906 | 2.706 | ||
0.999 | 3.145 | 2.875 | ||
EW | 0.700 | 0.023 | 0.021 | |
0.750 | 0.108 | 0.067 | ||
0.800 | 0.290 | 0.153 | ||
0.850 | 0.787 | 0.490 | ||
0.900 | 0.973 | 0.630 | ||
0.950 | 1.122 | 0.959 | ||
0.975 | 1.645 | 1.141 | ||
0.999 | 1.760 | 1.397 | ||
NEx-APTW | 0.700 | 0.925 | 1.563 | |
0.750 | 1.754 | 2.901 | ||
0.800 | 1.895 | 3.396 | ||
0.850 | 2.365 | 5.303 | ||
0.900 | 3.933 | 5.925 | ||
0.950 | 4.884 | 6.964 | ||
0.975 | 5.510 | 7.313 | ||
0.999 | 6.339 | 7.890 |
VaR = Value at Risk; TVaR = Tail VaR; EW = exponentiated Weibull; NEx-APTW = new extended alpha power transformed Weibull.