tailieunhanh - Operational Risk Modeling Analytics phần 6

Nó là thuận tiện để viết chỉ số trên của đuôi phụ thuộc về chức năng phụ thuộc A (w). Kết quả làNếu một copula được xác định thông qua A (w), sau đó chỉ số phụ thuộc đuôi phía trên có thể dễ dàng tính toán. Có một số nổi tiếng copulas trong lớp này. | STABILITY OF THE MAXIMUM OF THE EXTREME VALUE DISTRIBUTION 213 Example Example continued Suppose in addition that the individual losses are exponentially distributed with Fx x 1 - exp X 0. Then the distribution of the maximum loss for a fc-year period has df Fmn Ơ0 1 X 0. Example Example continued Suppose instead that the individual losses are Pareto distributed with df F z 1 - a 0 a I 7 X 0 a 0 0. y u J Then the distribution of the maximum loss for a fc-year period has df FMnW L S 1-Fx x STABILITY OF THE MAXIMUM OF THE EXTREME VALUE DISTRIBUTION The Gumbel Frechet and Weibull distributions have another property called stability of the maximum or max-stabilty that is very useful in extreme value theory. This is already hinted at in Examples and . First for the standardized Gumbel distribution we note that Go a Inn exp 71 exp X Inn exp exp a Go x . Equivalently Go a Go x Inn . This shows that the distribution of the maximum of n observations from the standardized Gumbel distribution has itself a Gumbel distribution after a shift of location of In n. Including location and scale parameters yields 214 EXTREME VALUE THEORY THE STUDY OF JUMBO LOSSES Go M e x n Go 0 x-ụ Go I ------Inn V J X JI fflnn G V------ 0 ---- 0o Go p ớ where JI ụ 0 In n. Similarly for the standardized Frechet distribution Gi a n1 a r 1 0 - Equivalently Gi Q a ra Gi a ĩ t This shows that the distribution of the maximum of n observations from the standardized Frechet distribution after a scale change has itself a Frechet distribution. Including location and scale parameters yields Gi Q M e a n - Gi a where 0 ớn1 . The key idea of this section is that the distribution of the maximum after a location or scale normalization for each of the extreme value EV distributions also has the same EV distribution. Section shows that these EV distributions are also approximate distributions of the maximum for almost any distribution. THE FiSHER-TIPPETT THEOREM We now examine