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Operational Risk Modeling Analytics phần 5

"tối đa-stabilty" đó là rất hữu ích trong lý thuyết giá trị cực đoan. Điều này đã được ám chỉ trong các ví dụ 7.1, 7.2, và 7,3. Đầu tiên, sự phân bố Gumbel tiêu chuẩn hóa, chúng tôi lưu ý rằngthiết lập (w) trên của nó bị ràng buộc . Ở thái cực khác, nếu , sau đó là mối tương quan hoàn hảo, và vì thế hoàn hảo u) phụ thuộc với C . | THE COMPOUND MODEL FOR AGGREGATE LOSSES 167 The normal distribution provides a good approximation when E 2V is large. In particular if N has the Poisson binomial or negative binomial distribution a version of the central limit theorem indicates that as À m or r respectively goes to infinity the distribution of s becomes normal. In this example E 2V is small so the distribution of s is likely to be skewed. In this case the lognormal distribution may provide a good approximation although there is no theory to support this choice. Example 6.3 Illustration of convolution calculations Suppose individual losses follow the distribution given in Table 6.1 given in units of 1000 . Table 6.1 Loss distribution for Example 6.3 X fx x 1 0.150 2 0.200 3 0.250 4 0.125 5 0.075 6 0.050 7 0.050 8 0.050 9 0.025 10 0.025 Furthermore the frequency distribution is given in Table 6.2. Table 6.2 Frequency distribution for Example 6.3 n Pn 0 0.05 1 0.10 2 0.15 3 0.20 4 0.25 5 0.15 6 0.06 7 0.03 8 0.01 168 AGGREGATE LOSS MODELS Table 6.3 Aggregate probabilities for Example 6.3 X fx fx fx fx fx fx y 6 fx f 8 JX fs x 0 1 0 0 0 0 0 0 0 0 .05000 1 0 .150 0 0 0 0 0 0 0 .01500 2 0 .200 .02250 0 0 0 0 0 0 .02338 3 0 .250 .06000 .00338 0 0 0 0 0 .03468 4 0 .125 .11500 .01350 .00051 0 0 0 0 .03258 5 0 .075 .13750 .03488 .00270 .00008 0 0 0 .03579 6 0 .050 .13500 .06144 .00878 .00051 .00001 0 0 .03981 7 0 .050 .10750 .08569 .01999 .00198 .00009 .00000 0 .04356 8 0 .050 .08813 .09750 .03580 .00549 .00042 .00002 .00000 .04752 9 0 .025 .07875 .09841 .05266 .01194 .00136 .00008 .00000 .04903 10 0 .025 .07063 .09338 .06682 .02138 .00345 .00031 .00002 .05190 u 0 0 .06250 .08813 .07597 .03282 .00726 .00091 .00007 .05138 12 0 0 .04500 .08370 .08068 .04450 .01305 .00218 .00022 .05119 13 0 0 .03125 .07673 .08266 .05486 .02062 .00448 .00060 .05030 14 0 0 .01750 .06689 .08278 .06314 .02930 .00808 .00138 .04818 15 0 0 .01125 .05377 .08081 .06934 .03826 .01304 .00279 .04576 16 0 0 .00750 .04125 .07584 .07361 .