Extreme Value Theory: An Introduction

Transcription

1 (correcto d Extreme Value Theory: A Itroducto by Laures de Haa ad Aa Ferrera Wth ths webpage the authors ted to form the readers of errors or mstakes foud the book after publcato. We also gve extesos for some materal the book. We ackowledge the cotrbuto of may readers. Chapter : Lmt Dstrbutos ad Domas of Attracto Chapter : Extreme ad Itermedate Order Statstcs Chapter 3: Estmato of the Extreme Value Idex ad Testg Chapter 4: Extreme Quatle ad Tal Estmato Chapter 6: Basc Theory hgher dmesoal space Chapter 7: Estmato of the Depedece Structure Chapter 8: Estmato of the Probablty of a Falure Set Chapter 9: Basc Theory C[,] Chapter : Estmato C[,] Appedx B: Regular Varato ad Extesos Further ad Updated Refereces Data fles /4 //

2 Chapter : Lmt Dstrbutos ad Domas of Attracto (correcto d page le error/uclear/mssg correcto 8 - (complemet Corollary..3A For x> lm atx ( t = x γ at ( 4 ' b = ( log / log log log(4 π - / (log -3 estmator of γ (Secto 3.5. Next we show. + ' b = ( log / log log + log(4 π - / (log estmator of γ (Secto 3.5. The ecessary ad suffcet codto for a dstrbuto fucto to belog to the doma of attracto of a extreme value dstrbuto s sometmes called the extreme value codto. Next we show. Chapter : Extreme ad Itermedate Order Statstcs page le error/uclear/mssg correcto 4 -, as we shall see. The followg result 4 k ku + k,, as we shall see. The asymptotc behavor of termedate order statstcs s mportat for statstcs of extreme values sce ay meagful estmator s based o (extreme ad termedate order statstcs (see Chapter 3. The followg result k U k +, k /4 //

3 Chapter 3: Estmato of the Extreme Value Idex ad Testg (correcto d page le error/uclear/mssg correcto 77 6 λ ρc λ ρc 4 Ψ ( ( + γ ρ ε, x x γ ρ, ' x x γ + Ψ ρ' γ ρ ε 7 The (3.6.5 becomes The (3.6.5 (multpled by f(t becomes ad (3.6.6 becomes 3 The egatve Hll estmator was proposed by Falk (995. ad (3.6.6 (multpled by f(t becomes The egatve Hll estmator was proposed by Smth ad Wessma (985. Chapter 4: Extreme Quatle ad Tal Estmato page le error/uclear/mssg correcto 8 label vertcal axs Fg.4.(b log (+ γ x / x the momet estmator of γ. We defe log (+ γ x / γ... the momet estmator of γ. I order to fd a estmator for the scale a (/k we use relato (3.5.3 for j = ad defe 34 - Theorem 4.3. Theorem 4.3. (de Haa ad Rootzé ( U( tx U( t U( tx U( t a( t a ( t 38 Theorem Theorem (Djk ad de Haa ( γ + γ + γ + γ 3 + 4γ + 5γ + γ = ( + γ ( + γ /4 //

4 (correcto d Chapter 6: Basc Theory hgher dmesoal space page le error/uclear/mssg correcto 9-8 U ([ ] etc. U ( to 9 (6..4 Remark 6..8 Relatos (6..5 does ot hold for all Borel sets A x,y. > ad ν ( A =, ad ay π / Ψ( dθ cosθ x sθ y. etc. Relatos (6..5 ca be wrtte c P( Axy, = exp{ ν ( Axy, } (6..5* Where P s the probablty measure wth dstrbuto fucto G. Relato (6..5* does ot ecessarly hold for Borel sets A ot of the form (6..6 >, ad ay (6..4 = x y r> cosθ sθ r dr Ψ( dθ 4 /4 // π / = Ψ( dθ cosθ x 7 - Defto 6..3 We call Defto 6..3 A dstrbuto fucto G s called a max-stable dstrbuto f there are costats A, C >, B ad D such that for all x,y ad =,, G (A x + B, C y + D = G(x,y. It s easy to see that ay dstbuto fucto G satsfyg (6,,5 s max-stable ad also G(αx + β, γ y + δ where α, γ are arbtrary postve costats ad β, δ arbtrary real costats. Sce ay max-stable dstrbuto s the class of lmt dstrbutos for (6.., we get all the max-stable dstrbutos ths way. The class of 4-8 ( ( ( Clearly G ad G Clearly G ad L U L ( cx sθ y. ( GU are max-stable dstrbutos wth margal dstrbutos exp where c s a geerc costat ad there

5 (correcto d coverges to - = - log G (x,y wth G from Theorem 6... r 3 q (θr, r(- θ. Let {r,j } d, j= be a matrx the radom vector (V d j= r,j V j,, V d j= r d,j V j two-dmesoal smple dstrbuto fucto ca be coverges to g(x,y : = - = (, s x gstdsdt t y { > } { > } for x,y. r 3 q (θr, r(- θ = q (θ, (- θ (cf. Coles ad Taw (99. Let {r,j } =,; j=,,,d be a matrx the radom vector (V d j= r,j V j, V d j= r,j V j two-dmesoal dstrbuto fucto wth Fréchet margals ca be -9 twce twce 33 6 complemet 6.4. Let (X,Y have a stadard sphercally symmetrc Cauchy dstrbuto. Show that the probablty dstrbuto of ( X, Y s the doma of attracto of a extreme value dstrbuto wth uform spectral measure Ψ. Show that the probablty dstrbuto of (X,Y s also a doma of attracto. Fd the lmt dstrbuto. Chapter 7: Estmato of the Depedece Structure page le error/uclear/mssg correcto a max-stable dstrbuto fucto.... a max-stable dstrbuto fucto. The margal dstrbuto are ( G ( x, = exp a ad G x (, y = exp a ( y for some postve a ad a ot ecessarly oe. Hece G s ot ecessarly smple max-stable (cf. 5 /4 //

6 (correcto d Defto , Not all equaltes have to hold, but at least oe of them Q( x,, Q(, y = 65 8 = x + y - L(x,y. Q( x, =, Q(, y =. = x + y - L(x,y = R (x,y. - does ot mply asymptotc depedece EW (x,, x d W (x,, x d = μ (R (x,, x d R (x,, x d 69 4 ad N s a stadard ormal radom varable. does ot mply asymptotc depedece. EW (x,, x d W (y,, y d = μ (R (x,, x d R (y,, y d ad N dcates a ormal probablty dstrbuto. Chapter 8: Estmato of the Probablty of a Falure Set page le error/uclear/mssg correcto 73 - complemet.e. Q = c S (assumpto, where 75 more detal below; cf. Theorems 8.. ad log ( cx log ( cx more detal secto 8..; cf. (8..7, (8,,8 ad (8,,5 k 6 /4 //

7 (correcto d Chapter 9: Basc Theory C[,] page le error/uclear/mssg correcto 3 - υ A =, ad ay a >, > ad ay a >, > ad ( 34 -, -, -3 ζ, ζ, ζ 3,... be a realzato of the pot process. Defe η : = ζ. = ( Z, π,( Z, π,( Z3, π 3,... + Z ad π C [ ] be a realzato of the pot process where ( ], η : = Zπ to9 These les should be deted (belog to ( Frst we ote that ths mples Frst we ote that ths assumpto mples -6-4 η Corollary d k ( s = ζ ( s = All smple max-stable processes C + [,] ca be geerated the followg way. stochastc processes V, V, C + [,] 7 /4 // η = Corollary (cf. M. Schlather ( k ( s = ζ ( s =,. Defe All smple max-stable processes η C + [,] ca be geerated the followg way. + stochastc processes V, V, C [, ]: = { f C[, ]: f } 38 6 Resck (977. Resck (977. Let W * be two-sded Browa moto: + * W ( s for s W (: s = W ( s for s< where W + ad W - are depedet Browa motos. I the rest of the example chage W to W *. 3-5 Theorem 9.5. Theorem 9.5. (de Haa ad L (

8 (correcto d 35 5 Theorem 9.6. (Resck ad Roy (99 3 of the theorem s easy. Next we tur Theorem 9.6. (Resck ad Roy (99 ad de Haa(984 of the theorem s easy. { Z T = Corollary 9.6.8A Let (, } be a realzato of a Posso pot process o (, ] wth mea measure ( dr r dλ ( λ smple max-stable process C + (, the there exsts a famly of fuctos f ( t ( s, t wth s / Lebesgue measure. If η s a. for each t we have a o-egatve cotuous fucto fs ( t : [,,. for each s f s ( t dt =, (9.6.7A 3. for each compact terval I ( sup f t dt <, s I such that η ( s d { s } = Z fs( T. (9.6.8A s = s Coversely every process of the form exhbted at the rght-had sde of (9.6.8A wth the stated codtos, s a smple max-stable process C + (. Proof. Let H be a probablty dstrbuto fucto wth a desty H` that s postve for all real x. Wth the fuctos f s from Theorem defe the fuctos f s( t : = fs( H( t H' ( t. Sce for ay s, s,..., s d ad x, x,..., x d postve f s ( t f ( s t max dt = max dt d x, d x 8 /4 //

9 4,6,,4 8,, 5 3 6, 8,,, 3, (#, -6, -8-6 [,] dstrbutos. W depedet Browa motos. (correcto d the represetato of the corollary follows easly from that of Theorem Next we tur dstrbutos. W * depedet two-sded Browa motos (cf. correcto to Example , 6 (3# (# 3 (3# (#, 3 x u y x e Φ u W x u y x e Φ + u W * (# (# (# x W W * y 9 /4 //

10 (correcto d 36-6 Hece for s < < s Hece for s < < s ad fact for all real s, s Let W be Browa moto depedet of Y. Cosder the process Let W * be two-sded Browa moto: + * W ( s for s W (: s = W ( s for s< where W + ad W - are depedet Browa motos. Let Y ad W * be depedet. Cosder the process I the rest of the example chage W to W * a> a> { } s Example 9.8. Remove the text of the example Example 9.8. (exteso of Brow ad Resck (977 Let X ( s Orste- Uhlebeck process,.e., ( s ( u s s/ u/ + ( = { } + ( X s e N e dw u s + + s/ / { } e N e dw s ( u < be a wth N, W + ad W - depedet, N a stadard ormal radom varable ad W + ad W - stadard Browa motos. Sce for s t the radom vector (X(s, X(t s multvarate ormal wth correlato coeffcet less tha, Example 6..6 tells us that relato (9.5. ca ot hold for ay max-stable process C[,]: sce Y has cotuous sample paths, Y(s ad Y(t ca ot be depedet. Hece we compress space order to create more depedece,.e., we cosder the covergece of s b X b = b (9.8.4 s C[ s, s] for arbtrary s >, where X, X, are depedet ad detcally dstrbuted copes of X ad the b are the approprate ormalzg costats for the stadard oe-dmesoal ormal dstrbuto, e.g., (cf. Example..7 ( log log log log ( 4 / b = π. The /4 //

11 (correcto d s b X b b s / ( b s / b u / s / ( b e b( N b b e dw ± = ( u e + + b ± + where W ( s s W ( s for s ad W ( s for s <. Note that uformly for s s s/ ( b e = + O. b u / Further, sce e = + O( / b for u < s / b, s / b u / ± ± s b e dw ( u = O bw. + b b Fally, for s s, s / ( b s e b = + O. b It follows that s b X b b ± s s = + O b ( N b + bw + O. b b b ± W s : = bw s / b. The W s also Browa moto. We have We wrte ( ( /4 // = s b X b b s ( s = + O b N b W( s O. b + + = b

12 (correcto d Hece the lmt of ( s the same as that of ( ( s b N b + W( s. = s (9.8.5 The rest of the proof rus as the prevous example. Oe fds that the sequece of processes (9.8.5 coverges weakly C s, s hece C, to [ ] ( 38-7 depedet of V depedet of Y 39 u> x> ( log Z W( s s + = s wth {Z }the pot process from (9.8.. Note that the pot process {Z } ad the radom processes W are depedet. Chapter : Estmato C[,] page le error/mssg correcto 33 - Theorem.. Theorem.. (de Haa ad L ( :. F s, ( x = { X ( s > x} F ( s, x ( j= : = { }. X j s > x Theorem.4. Theorem.4. (de Haa ad L ( ζ -k+, ζ -k, 35 6 υ (S υ (S /4 // j=

13 (correcto d Appedx B: Regular Varato ad Extesos page le error correcto ad -3 (B..6 (B exp (tegral} exp(tegral 37 - f ( t = exp[ logt] f ( t = exp{ [ logt] } (B..3 (B..4 (B..4 (B.. (B..3 (B Hece f(t s bouded for t t. Hece f(t s locally bouded for t t f ( f ( t = f ( f ( t 38 9 x δ δ (left sde 38-8 From Remark B..4( t follows x δ δ From part 3 of the preset proposto t follows Further ad Updated Refereces S. Coles ad J. Taw: Modelg extreme multvarate evets. J. Royal Statst. Soc. Ser. B 53, (99. V. Djk ad L. de Haa: O the estmato of the exceedace probablty of a hgh level. Order Statstcs ad o-parametrcs: theory ad applcatos. P.K. Se ad I.A. Salama (Edtors, Elsever, Amsterdam (99 L. de Haa ad T. L: O covergece towards a extreme value dstrbuto C[,]. A. Prob. 9, (. L. de Haa ad T. L: Weak cosstecy of extreme value estmators C[,]. A. Statst. 3, 996- (3. L. de Haa ad T.T. Perera: Spatal Extremes: Models for the statoary case. A. Statst., 34, (6. L. de Haa ad H. Rootzé: O the estmato of hgh quatles. J. Statst. Plag ad Iferece, 35, -3 (993. M. Schlather: Models for statoary max-stable radom felds. Extremes, 5, (. 3 /4 //

14 (correcto d R.L.Smth ad I.Wessma: Maxmum lkelhood estmato of the lower tal of a probablty dstrbuto. J. Royal Statst. Soc. Ser. B 47, (985. Data fles Last updated: November,. 4 /4 //