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Anderson-Darling test is one of the modifications of K-S test. The test statistic is a weighted measure of the area between the empirical and fitted CDF’s. It is calculated according to:

Figure 1 Anderson-Darling test is one of the modifications of K-S test. The test statistic is a weighted measure of the area between the empirical and fitted CDF’s. It is calculated according to:

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Tab.1 shows summary statistics for X. These statistics and Box-and-Whisker plot confirm the skew nature of the claims data. Also by density trace by (5) for X in Fig. 2 can be concluded that loss distribution in our case is skew and long or heavy tailed.
Tab.1 shows summary statistics for X. These statistics and Box-and-Whisker plot confirm the skew nature of the claims data. Also by density trace by (5) for X in Fig. 2 can be concluded that loss distribution in our case is skew and long or heavy tailed.
Table 2 Estimated parameters of the fitted distributions Table 3 Anderson-Darling Goodness-of-Fit Tests for X
Table 2 Estimated parameters of the fitted distributions Table 3 Anderson-Darling Goodness-of-Fit Tests for X
Table 4 Chi-Squared test with lognormal distribution
Table 4 Chi-Squared test with lognormal distribution
Table 3 shows the results of tests run to determine whether X can be adequately modelled by gamma, lognormal or Weibull distributions. P-values less than 0,01 would indicate that X does not come from the selected distributions with 99% confidence.
Table 3 shows the results of tests run to determine whether X can be adequately modelled by gamma, lognormal or Weibull distributions. P-values less than 0,01 would indicate that X does not come from the selected distributions with 99% confidence.
Fig. 4 Quantile-Quantile plot of selected distributions Fig. 3 Histogram and estimated loss distributions
Fig. 4 Quantile-Quantile plot of selected distributions Fig. 3 Histogram and estimated loss distributions
From Fig. 3 it seems that lognormal distribution follows the data best. It is hard to compare the tail fit, but clearly the all distributions have discrepancies at middle claims intervals.
From Fig. 3 it seems that lognormal distribution follows the data best. It is hard to compare the tail fit, but clearly the all distributions have discrepancies at middle claims intervals.
Fig. 5 Quantile-Quantile plot for lognormal distribution of X weight in upper tail than the lognormal distribution provides. Figure 5 and Tab. 4 lead to the conclusion that the fitted lognormal model undervalues the losses which exceed the 4000 CZK.. We need to find a distribution with rather more weight in upper tail than the lognormal distribution provides.
Fig. 5 Quantile-Quantile plot for lognormal distribution of X weight in upper tail than the lognormal distribution provides. Figure 5 and Tab. 4 lead to the conclusion that the fitted lognormal model undervalues the losses which exceed the 4000 CZK.. We need to find a distribution with rather more weight in upper tail than the lognormal distribution provides.
There are 34 values ranging from 4000 to 22 820 thousand CZK. Table 5 and Table 6 show the results of fitting a 2- parameters Pareto distribution (16) to the data on X4o09. The estimated parameters of the fitted distribution using maximum likelihood method are shown in Table 5. The results of K-S test whether the 2-parameter Pareto distribution fits the data adequately is shown in Table 6. Since the p-value = 0,895778 is greater than 0,05, we cannot reject the null hypothesis that sampling values of the variable X49. comes from a 2- parameters Pareto distribution with 95% confidence. Table 5 Estimated parameters by ML
There are 34 values ranging from 4000 to 22 820 thousand CZK. Table 5 and Table 6 show the results of fitting a 2- parameters Pareto distribution (16) to the data on X4o09. The estimated parameters of the fitted distribution using maximum likelihood method are shown in Table 5. The results of K-S test whether the 2-parameter Pareto distribution fits the data adequately is shown in Table 6. Since the p-value = 0,895778 is greater than 0,05, we cannot reject the null hypothesis that sampling values of the variable X49. comes from a 2- parameters Pareto distribution with 95% confidence. Table 5 Estimated parameters by ML
Now we can check whether the Pareto distribution with maximum likelihood estimators provides an adequate fit to the data using y’ goodness-of-fit test. The ’ statistic is computed by (7). The result of goodness of fit test presents tab. 7. quantile x>,, =11,0705 on 5 degrees of freedom (since there are
Now we can check whether the Pareto distribution with maximum likelihood estimators provides an adequate fit to the data using y’ goodness-of-fit test. The ’ statistic is computed by (7). The result of goodness of fit test presents tab. 7. quantile x>,, =11,0705 on 5 degrees of freedom (since there are
Fig. 6 Quantile-Quantile plot for Pareto distribution of X4o99 In the Fig. 6 we can see that the Pareto model (16) for the variable Xy4o99 gives an better fit and is a considerable improvement over lognormal model. Pareto distribution has been used to define the X-axis at Fig.6. The fact that the points lay close to the diagonal line confirms the fact that this distribution provides a good model for the claim amounts data above 4 million CZK and we can assume that a good model for losses above 4 million CZK can be Pareto distribution with PDF expressed by the formula (17).
Fig. 6 Quantile-Quantile plot for Pareto distribution of X4o99 In the Fig. 6 we can see that the Pareto model (16) for the variable Xy4o99 gives an better fit and is a considerable improvement over lognormal model. Pareto distribution has been used to define the X-axis at Fig.6. The fact that the points lay close to the diagonal line confirms the fact that this distribution provides a good model for the claim amounts data above 4 million CZK and we can assume that a good model for losses above 4 million CZK can be Pareto distribution with PDF expressed by the formula (17).
Table 10 K-S goodness-of-fit tests for Xgoo0 Table 10 shows the results of tests run to determine whether Xgoo0 can be adequately modelled by a 2-parameter Pareto distribution (16) with ML estimated parameters from Table 9. Since the p-value = 0,339047 is greater than 0,05, we cannot reject the null hypothesis that values of Xgo99 come from the 2- parameter Pareto distribution with 95% confidence.
Table 10 K-S goodness-of-fit tests for Xgoo0 Table 10 shows the results of tests run to determine whether Xgoo0 can be adequately modelled by a 2-parameter Pareto distribution (16) with ML estimated parameters from Table 9. Since the p-value = 0,339047 is greater than 0,05, we cannot reject the null hypothesis that values of Xgo99 come from the 2- parameter Pareto distribution with 95% confidence.
We use results of parts V. and VI. to simulate 10 the largest claim amounts by (28) in case of number 1000 claim amounts. Table 11 contains the results of simulation step by step by part VI using formulas (25) and (26).
We use results of parts V. and VI. to simulate 10 the largest claim amounts by (28) in case of number 1000 claim amounts. Table 11 contains the results of simulation step by step by part VI using formulas (25) and (26).
Fig. 7 Graphical result of simulation of 10 the largest claim amounts Table 11 Steps of simulation of 10 the largest claim amounts
Fig. 7 Graphical result of simulation of 10 the largest claim amounts Table 11 Steps of simulation of 10 the largest claim amounts
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