In this section, we present two numerical illustrations: the first one is on simulated data, and the second one is on a real data set.
5.1. Numerical Illustration Using Simulated Data
In this section, we used simulated data to check the performance of the first estimation procedure (Method 1) proposed in
Section 4.3. The true values of the parameters were selected such that they satisfied all the continuity conditions given in Proposition 2: Gumbel:
Pareto:
, while
. Note that due to the heavy-tailedness of the Pareto distribution (
), there is no expected value for this particular distribution (its pdf is plotted in
Figure 2).
With the aim of studying the properties of Method 1, using the two simulation methods described in
Section 4.2, we generated 100 samples of size
and
1000, respectively, for the two methods. For each such sample, in the first step, we performed marginal estimation by imposing the continuity condition for each marginal (which restricts the parameters
r, as stated in Proposition 2). As a consequence,
and
a are estimated twice (for each marginal), and because of the differences in these estimations, we cannot rely only on marginal estimation. However, marginal estimation provides starting values for performing full MLE, and even better, gives an idea of where to look for the thresholds. More precisely, we restricted the search to about 40 intervals for each
, i.e., we took
. Thus, the computing time was significantly reduced compared to the threshold search through all data.
Finally, we estimated the Mean Square Error and the Mean Absolute Error , where and represent the true and estimated parameters, respectively.
With the estimated parameters obtained from the 100 replicas generated with each simulation method, we obtained the MSE and the MAE that are shown in
Table 1. The results indicate that both error criteria decrease when the sample size increases. Some differences between the two simulation methods can be observed (e.g., the MSE of
is larger for simulation Method II than for simulation Method I, while the MSE of
a is smaller for simulation Method II than for simulation Method I), but we believe that these differences are due to the randomness of the results, where some samples fall more in the Pareto part or in the exponential part; further simulation investigation is worthwhile, assuming that the estimation method can be modified to reduce the computing time.
Concerning Method 2, as already noticed, it is a more analytical procedure for a specific sample, and therefore, it cannot be standardized and we cannot perform several iterations to calculate MSE and MAE.
All the computations were preformed in R software using an optimization function with constraints to implement the continuity restrictions. The code is available upon request from the authors.
5.2. Numerical Illustration with Real Data
In this section, we fit our proposed bivariate Gumbel–Pareto distribution to a random sample of
motor insurance claims that include bodily injury. For these claims, we separately know the cost of property damage including third-part liability (variable
) and the cost of exceptional medical expenses not covered by public social security (variable
). The data were provided by a major insurer in Spain in the year 2002 and correspond to claims that occurred in the year 2000. These data were studied in previous works (see [
5,
6,
12]).
In
Table 2, we display the descriptive statistics of the original data divided by 1000; this change of scale is convenient, and it facilitates the MLE of the parameters. These descriptive statistics show that both variables have a strong right skewness. Furthermore, the left plot in
Figure 4 shows the scatterplot of both cost variables in the original scale divided by 1000, where the existence of extreme values in both variables can be noticed. When we have right-skewed variables with extreme values, the MLE of a simple distribution as, e.g., the exponential, the Weibull, or the log-normal, tends to underestimate the probability on the right tail.
Figure 5 displays the univariate exponential pdf fitted by MLE to each marginal variable; with these densities, we also plotted the observed costs: on top the costs of property damage, including third-part liability, and on bottom the costs of exceptional medical expenses not covered by public social security. For better visibility, the domains of the cost variables were divided in two parts, resulting in two plots for each marginal.
Figure 5 shows how the density reaches zero in the part of the domain where there are still sample observations; so clearly, this model assumes a zero probability where it should not. Similar results are obtained using univariate Weibull and log-normal densities.
Therefore, the composite model with a Pareto right tail is a good way to improve the MLE fit for both univariate and bivariate data. Moreover, graphical analysis (e.g., the Hill plot) indicates that both variables have a Pareto tail with a shape parameter very close to 1, i.e., we have heavy-tailed marginal distributions. Thus, we can conclude that their distributions have only the first-order moment finite, or they do not have finite moments at all. In the left scatterplot of
Figure 4, we can note that the sample information on extreme values is scarce; this is a difficulty in samples from heavy-tailed or Pareto distributions.
To asses the joint behavior of
and
, we calculated the Pearson linear correlation and the Kendall and the Spearman rank correlation coefficients, displayed in
Table 3. These results show a strong dependence between the two cost variables. However, as can be seen from
Figure 4, which presents the data scatterplot in both original and natural logarithm scales, the dependence is not linear. As shown in [
12], these data exhibit extreme value dependence, i.e., the higher the costs, the stronger the dependency. This behavior can also be observed in
Figure 4. Furthermore, [
10] shows that when the bivariate Pareto parameter
a is
, as is the case with our cost data, the theoretical variance and covariance do not exist or cannot be calculated. Therefore, the Pearson linear correlation cannot be interpreted.
Further, from the right plot in
Figure 4, it can be observed that for small values of both variables, the shape of the point cloud is spherical, i.e., the dependence is almost zero; however, for larger values, the shape indicates positive dependence between both variables. Clearly, this denotes a change of the joint distribution between the smaller and the larger costs.
In
Table 4, we present the MLE parameters for Gumbel’s bivariate exponential distribution described in
Section 2.2.1 and for the Gumbel–Pareto distribution from
Section 4. The estimated parameters of the latter were obtained with Method 2 described in
Section 4.3, imposing all continuity conditions (Method 1 yielded similar results). The initial values of the thresholds were taken from the Hill plots, and in this case,
, resulting in
, i.e.,
and
; also,
. Comparing the AICs, BICs, and CAICs given in
Table 4 indicates that the bivariate Gumbel–Pareto clearly outperforms Gumbel’s bivariate exponential distribution. Moreover, from MLE, the dependence parameter of Gumbel’s bivariate exponential distribution,
, is zero, and it is close to zero for the Gumbel–Pareto distribution, which is coherent with the scatterplot in
Figure 4.
In
Figure 6, we also plotted a partial histogram of the data alongside the corresponding Gumbel–Pareto pdf with the estimated parameters, while in
Figure 7, we plotted the marginal histograms with the fitted pdfs.
Finally, as a risk management application, we estimated the total risk of loss for the aggregate cost random variable
using Monte Carlo simulation, and based on it, we calculated the Value-at-Risk (VaR) measure. VaR is equivalent to an extreme quantile of the distribution, i.e.,
, where
is close to 1. In
Table 5, we present the VaR results with
for: the empirical distribution of the original data, the distribution of
S simulated from Gumbel’s bivariate exponential distribution, and the distribution of
S simulated from the Gumbel–Pareto distribution. Furthermore, we added the VaR obtained for the bivariate log-normal distribution fitted to the data; note that this distribution underestimates the risk in a way similar to that of Gumbel’s bivariate exponential.
When data follow a heavy-tailed distribution, the empirical VaR depends on the maximum data observed, and it is not an efficient estimator. The Gumbel–Pareto distribution provides an estimation that extrapolates beyond the observed maximum cost and takes into account the long and heavy bivariate tail with dependent marginal distributions.