HW2

RMSC 4006 Operational Risk Management Assignment 2 Due 23:59, Feb 23 (Upload to blackboard) 1) Find the support of the GEV distributions with parameter a) ( μ, σ, γ ) = (0 , 1 , 0 . 5) b) ( μ, σ, γ ) = (0 , 1 , 0) c) ( μ, σ, γ ) = (0 , 1 , - 0 . 5) 2) Let M n = max i =1 ,...,n X i . Find sequences { b n } , { a n } and the distribution D ( x ), such that P (( M n - b n ) /a n ) ≤ x ) → D ( x ), when a) X i iid ∼ exp( λ ), where λ > 0 b) X i iid ∼ F , where F ( x ) = e - λ/x , λ > 0 c) X i iid ∼ U (0 , b ), where b > 0 3) Show that GEV distribution G is max-stable by finding appropriate se- quences { c n } , { d n } such that G n ( x ) = G ( c n x + d n ). 4) Let X i be the yearly maximum of some quantity at the i -th year. Find a) the 0 . 05 upper-quantile of X i b) the 100 year return level c) the level such that the mean return period is 200 for each of the following cases: i) X i iid ∼ GEV (200 , 20 , 1) ii) X i iid ∼ GEV (200 , 20 , 0) iii) X i iid ∼ GEV (200 , 20 , - 1) 5) Find the support of the GPD distribution with parameter a) ( γ, σ ) = (0 . 5 , 1) b) ( γ, σ ) = (0 , 10) c) ( γ, σ ) = ( - 1 , 100) 6) Let X i iid ∼ F , and F [ u ] ( x ) = ( F ( x ) - F ( u )) / (1 - F ( u )). a) Identify the limiting distribution of F [ u ] ( x + u ) when i) X i iid ∼ exp( λ ), where λ > 0 ii) X i iid ∼ F , where F ( x ) = e - λ/x , λ > 0 iii) X i iid ∼ U (0 , λ ), where λ > 0 1