*4.5. Simulation Study*

By using the result defined in Equation (25), we evaluate the sensitivity of the method of estimations using the MLEs of OGE2Fr distribution parameters by Monte Carlo simulation technique. The simulation study is conducted for sample sizes *n* = 50, 100, 200, 300, 500, 600 and parameter combinations, denoted by *<sup>ω</sup>*(.), are:


We use Equation (20) to generate the random observations. For each *<sup>ω</sup>*(.) , the empirical bias and MSE values are the average of the values from *N* = 1000 simulated samples for given sample size *n*. The formula to evaluate the mean squared error (MSEs) and the average bias (Bias) of each parameter, is given below

$$MSE(\hat{\Theta}) = \sum\_{i=1}^{N} \frac{(\hat{\Theta}\_i - \Theta)^2}{N} \qquad \text{and} \qquad Bias(\hat{\Theta}) = \sum\_{i=1}^{N} \frac{\hat{\Theta}\_i}{N} - \Theta.$$

We report the results of the AE, Bias and MSE for the parameters *α*, *β*, *a* and *b* in Table 2. The MSE of the estimators increases when the assumed model deviates from the genuine model, as anticipated. When the sample size grows larger and the symmetry degrades, the MSE shrinks. Generally speaking, the MSE decreases when the kurtosis grows. Similarly, when the asymmetry rises, the bias grows, and vice versa. As the kurtosis grows, the bias becomes smaller. In conclusion, it is apparent that the MSEs and Biases decrease when the sample size *n* increases. Thus, we can say that the MLEs perform satisfactorily well in estimating the parameters of the OGE2Fr distribution.

**Table 2.** AEs, MSEs and Biases for *ω*1, *ω*2 & *ω*3.


### **5. Application of OGE2Fr to Premium Data**

Most skewed distributions are suitable to measure risk measures associated with actuarial data. The risks involve credit, portfolio, capital, premiums losses, and stocks prices among others. We focus our attention on the stakes based on premiums. Premiums are the payments for insurance that the customer pay to the company to which they are insured. In this section, we apply the OGE2Fr lifetime model for the statistical analysis of two real life data sets both of which include premium losses. Our aim is to compare the fits of the OGE2Fr model with other well-known generalizations of the Fréchet (Fr) models given in Table 3.

The first premium data set, designated as PD1, is derived from complaints upheld against vehicle insurance firms as a proportion of their overall business over a two-year period. The study was conducted by DFR (Darla Fry Ross) insurance and investment company (2009–2016), registered in New York state. The most common complaints are over delays in the settlement of no-fault claims and non-renewal of insurance. Top of the list are insurers with the fewest upheld complaints per million USD of premiums. The companies with the greatest complaint ratios are at the bottom of the list. The data understudy is from the year 2016. The second premium data, denoted by PD2, signifies the net premiums written (in billions of USD) to insurers which, under Article 41 of the New York Insurance Law, are required to meet minimum financial security requirements. Table 4: Descriptives statistics of PD1 and PD2.


**Table 3.** The comparative fitted models.

**Table 4.** The descriptive statistics related to PD1 & PD2.


The OGE2Fr model is validated through the discriminatory criterions (DCs) we considered for each data set. It includes the negative log-likelihood ( −<sup>ˆ</sup>) of the model taken at the corresponding MLEs, the Akaike Information Criterion (AICs), Bayesian Information Criterion (BICs), Anderson-Darling (AD), Cramér–von Mises (CvM), and Kolmogrov-Smirnov (KS) as well as the *p*-value (P-KS) of the related KS test. We use the method of maximum likelihood estimation to estimate the unknown parameters as presented in Section 4.2. For each criterion (except *p*-value (KS)) with highest value), the smallest values is gained by the OGE2Fr model, indicating the best fit among its competitive models.

Some descriptive statistics related to these data are given in Table 4. The skewness and kurtosis are indicative of exponentially tailed data (reversed-J shape). The TTT plots for the both data sets are given in Figure 4. In particular, the TTT plots show largely decreasing hrf, permitting to fit OGE2Fr model on these data sets. The estimated hrf in Figure 5 matches Figure 4. In Table 5, we present the estimates (MLEs) along with their respective standard errors(SEs) while the DCs are listed in Table 6 for PD1 & PD2, respectively. For a more visual view, the estimated pdf, cdf, sf and Q-Q plots of the OGE2Fr model for two data sets are displayed in Figures 6 and 7. Furthermore, the PP-plots of OGE2Fr and its three other competitive 4-parameter models for PD1 and PD2 are displayed in Figures 8 and 9. The log-likelihood function profiles for PD1 and PD2, respectively, are provided in Figures 10 and 11 to highlight the universality of the MLEs of Θ vector. The graphical visualizations are indicative of nice fits for the OGE2Fr model.

**Figure 4.** TTT plots of (**a**) PD1 and (**b**) PD2.


**Table 5.** Estimates and standard errors for PD1 & PD2.

**Figure 5.** Estimated hazard rate plots of (**a**) PD1 and (**b**) PD2 of OGE2Fr.

**Table 6.** The statistics ˆ , AIC, CAIC, BIC, HQIC, AD, CvM, KS and *p*-value (KS) for the PD1 & PD2.


**Figure 6.** *Cont*.

**Figure 6.** Estimated (**a**) density, (**b**) cdf, (**c**) sf, and (**d**) QQ-plot for PD1.

**Figure 7.** Estimated (**a**) density (**b**) cdf (**c**) sf, and (**d**) QQ-plot for PD2.

**Figure 8.** PP-plots of (**a**) OGE2Fr alongside competitive (**b**) BFr, (**c**) KwFr and (**d**) EGFr (4-parameter models) for PD1.

**Figure 9.** *Cont*.

**Figure 9.** PP-plots of (**a**) OGE2Fr alongside competitive (**b**) BFr, (**c**) KwFr and (**d**) EGFr (4-parameter models) for PD2.

**Figure 10.** Profiles of the log-likelihood function for the parameters *α*, *β*, *a* and *b*, respectively, of the OGE2Fr for the PD1.
