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 , , 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 Illustration of convolution calculations Suppose individual losses follow the distribution given in Table given in units of 1000 . Table Loss distribution for Example X fx x 1 2 3 4 5 6 7 8 9 10 Furthermore the frequency distribution is given in Table . Table Frequency distribution for Example n Pn 0 1 2 3 4 5 6 7 8 168 AGGREGATE LOSS MODELS Table Aggregate probabilities for Example 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 .