MATH3/4/68181: EXTREME VALUES FIRST SEMESTER ANSWERS TO IN CLASS TEST

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1 MATH3/4/68181: EXTREME VALUES FIRST SEMESTER ANSWERS TO IN CLASS TEST ANSWER TO QUESTION 1 i If there re norming constnts n > 0, b n nd nondegenerte G such tht the cdf of normlized version of M n converges to G, i.e. Mn b n Pr n x = F n n x + b n Gx 1 s n then G must be of the sme tpe s cdfs G nd G re of the sme tpe if G x = Gx + b for some > 0, b nd ll x s one of the following three clsses: I : Λx = exp exp x}, x R; 0 if x < 0, II : Φ α x = exp x α } if x 0 for some α > 0; exp x III : Ψ α x = α } if x < 0, 1 if x 0 for some α > 0. ii The necessr nd sufficient conditions for the three extreme vlue distributions re: F t + xγt I : γt > 0 s.t. = exp x, x R, t wf F t F tx II : wf = nd t F t = x α, x > 0, III : wf < nd t 0 F wf tx F wf t = xα, x > 0. 3 mrks 3 mrks iii First, suppose tht G belongs to the mx domin of ttrction of the Gumbel extreme vlue distribution. Then, there must exist strictl positive function, s ht, such tht G t + xht = exp x t wg Gt for ever x,. But, using L Hopitl s rule, we note tht F t + xht t wf F t 1

2 t wf t wg t wg t wg t wg t wg 1 + xh t f t + xht 1 + xh t ft log [ G t + xht]} 2 g t + xht 1 + xh t g t + xht 1 + xh t 1 + xh t 1 + xh t log [ Gt]} 2 } log [ G t + xht] 2 log [ Gt] g t + xht G t 1 + xh 2 t g t + xht G t + xht g t + xht g t 1 + xh 2 t g t + xht 1 + xh t g t + xht g t + xht G t + xht t wg Gt = exp x for ever x,. So, it follows tht F lso belongs to the mx domin of ttrction of the Gumbel extreme vlue distribution with Pr n M n b n x} = exp exp x} n for some suitble norming constnts n > 0 nd b n. Second, suppose tht G belongs to the mx domin of ttrction of the Fréchet extreme vlue distribution. Then, there must exist β > 0, such tht Gtx Gt = x β for ever x > 0. But, using L Hopitl s rule, we note tht F tx F t xftx ft x log [ Gtx]} 2 gtx log [ Gt]} 2 } xgtx log [ Gtx] 2 log [ Gt] xgtx xgtx xgtx Gt Gtx xgtx } xgtx 2 xgtx } 2 2

3 Gtx Gt = x β for ever x > 0. So, it follows tht F lso belongs to the mx domin of ttrction of the Fréchet extreme vlue distribution with Pr n M n b n x} = exp n for some suitble norming constnts n > 0 nd b n. x β Third, suppose tht G belongs to the mx domin of ttrction of the Weibull extreme vlue distribution. Then, there must exist α > 0, such tht GwG tx GwG t = xα for ever x > 0. But, using L Hopitl s rule, we note tht F wf tx F wf t xfwf tx fwf t x log [ GwF tx]} 1 gwf tx log [ GwF tx]} 2 gwf t } xgwf tx log [ GwF tx] 2 gwf t log [ GwF t] xgwf tx gwf t xgwf tx gwf t xgwf tx gwf t GwF tx GwF t = x α. GwF t GwF tx xgwf tx gwf t gwf t } xgwf tx 2 xgwf tx gwf t } 2 So, it follows tht F lso belongs to the mx domin of ttrction of the Weibull extreme vlue distribution with Pr n M n b n x} = exp x α } n for some suitble norming constnts n > 0 nd b n. ANSWER TO QUESTION 2 i Note tht wf =. Then 4 mrks F t + xγt t F t t [ exp t xγt] 2 [ exp t] 2 3

4 t 2 exp t xγt 2 exp t = exp xγt = exp x if γt = 1. So, the exponentited exponentil cdf F x = [ exp x] 2 belongs to the Gumbel domin of ttrction. ii Note tht wf =. Then 2 mrks F t + xγt F t exp 2 t + xγt} exp 2 t + xγt} 2 exp 2t} exp 2t} 2 exp 2 t + xγt} exp 2t} exp 2 t + xγt} 2 exp 2t} 2 2 exp 2 t + xγt} 2 exp 2t} 2 exp 2 t + xγt 2 exp 2t exp 2xγt = exp x if γt = 0.5. So, the exponentited exponentil geometric cdf F x = to the Gumbel domin of ttrction. 1 exp 2x} exp 2x} 2 belongs so iii For the geometric distribution, F k = p k, PrX = k F k 1 p pk 1 = = p. p k 1 2 mrks Hence, there cn be no sequences n > 0 nd b n such tht M n b n / n hs non-degenerte iting distribution. 4

5 2 mrks iv Note tht wf =. Then F t + x F t Φ 2 t + x Φ 2 t 2Φ t + x φ t + x 2Φtφt Φ t + x φ t + x Φtφt φ t + x 1 + xg t φt [ 1 t + x2 exp 2π 2 [ ] 1 2π exp [ t + x2 exp 2 [ exp t xg t 1 + xg t ] 1 + xg t t2 2 ] 1 + xg t ] [ ] t 2 t + x2 exp 1 + xg t 2 2 exp 1 [t + x 2 t 2]} 1 + xg t 2 exp 1 [ ] } 2tx + x 2 g 2 t 1 + xg t 2 exp = exp x x x2 2t 2 } x t 2 if = 1/t. So, F belongs to the Gumbel domin of ttrction. 2 mrks v Note tht wf =. Then F t + x t F t exp exp t x} t exp exp t} 1 + exp t x t 1 + exp t exp t x t exp t exp t x t 5

6 = exp x if = 1. So, the cdf F x = exp exp x} belongs to the Gumbel domin of ttrction. ANSWER TO QUESTION 3 i The cdf of Y is F Y = PrY = Pr mx X 1,..., X α = Pr X 1,..., X α = Pr X 1 Pr X α [ K ] [ K ] = [ K ] α =. 2 mrks ii The corresponding pdf is K ] α 1 f Y = αk [ 2 mrks for K. iii The nth moment of Y cn be clculted s E Y n = αk K = αk K K ] α 1 [1 n d K n 1 x n 1 x α 1 K x 1 1 dx 1 = αk n x α 1 x n/ dx 0 = αk n B α, n provided tht n <, where we hve set x = K/. So, E Y = αkb α, 1. 2 mrks 3 mrks 6

7 iv So, V r Y = αk 2 B α, 2 α 2 K 2 B 2 α, 1. 3 mrks 7