Einmahl and Magnus (2007) Records in athretics through extreme-value theory. 04:35

( ) r-taka@maritime.kobe-u.ac.jp

Einmahl and Magnus (2007) Records in athretics through extreme-value theory. 04:35

0 2 4 6 8 10 12 14 102 104 106 108 Age Castillo, Hadi, Balakrishnan and Sarabia (2004) p.11 12. The yearly oldest ages at death in Sweden during the period from 1905 to 1958 for women. 54 101.50 107.90

15 20 25 30 1900 1920 1940 1960 1980 2000 (m/s) 1900 1999

Kumamoto Max. Daily Rainfall 100 200 300 400 1900 1920 1940 1960 1980 2000 Year (0.1mm) 1901 2005

Nara Daily Rainfall (0.1mm) 0 500 1000 1500 0 100 200 300 Day Nara Daily Rainfall (0.1mm) 0 500 1000 1500 0 100 200 300 Day (0.1mm) 1961 1962

Daily Maximum Temperature 100 200 300 0 500 1000 1500 Kyoto ( 0.1 C) 2000 1 1 2004 12 31

Coles A. B.

(block maximum) 1920 1940 Fréchet Fisher and Tippett von Mises Gnedenko

1960 Geffroy Tiago de Oliveira Sibuya Gumbel (1958)

Peaks Over Threshold (POT) (threshold exceedances Balkema and de Haan (1974) Pickands (1975) r Weissman (1978) Smith (1986) Tawn (1988)

X 1, X 2,..., X n,... F (x) = P (X i x), i = 1, 2,... X 1, X 2,..., X n X (1:n) X (2:n)... X (n:n) Z n = X (n:n) = max { X 1, X 2,..., X n } = max 1 i n X i

n Z n F x + Z n x + = sup{x : F (x) < 1}, n. Z n a n > 0, b n R (n = 1, 2,...) G(x) Z P ( Zn b n Z n b n a n d Z, n. a n ) x P (Z x) = G(x). G (extreme value distribution) F G (maximum domain of attraction) F MDA(G) (a n, b n )

(1) G(x) (2) F MDA(G) F (3) a n b n W n = min 1 i n X i W n = min { } { } X 1, X 2,..., X n = max X1, X 2,..., X n W G W (x) = 1 G( x)

P ( Zn b n a n ) x = P (Z n a n x + b n ) = P ( max 1 i n X i a n x + b n ) = P (X i a n x + b n, i = 1, 2,..., n) = n i=1 P (X i a n x + b n ) = F n (a n x + b n ).

lim n F n (a n x + b n ) = G(x), x R. a n x + b n x + = sup{x : F (x) < 1} y = a n x + b n ( ) P (Z n y) = F n y bn (y) G. a n Xi = µ + σx i i = 1, 2,..., n µ R σ > 0 = Z n = max 1 i n X i = µ + σ max 1 i n X i = µ + σz n.

0 < G(x) < 1 lim n F n (a n x + b n ) = G(x) lim n[1 F (a n n x + b n )] = log G(x). { F n (a n x+b n ) = 1 + n[1 F (a nx + b n )] n } n { } exp lim n[1 F (a n n x+b n )] ( x n x, n = lim 1 + x ) n n = e x. n n

Block Maximum from Exp(1) Block Maximum from N(0, 1) 0 10 20 30 40 0 10 20 30 40 4 6 8 maximum Block Maximum from Par(2) 1.5 2.0 2.5 3.0 3.5 4.0 maximum Block Maximum from U(0, 1) 0 20 40 60 80 100 0 20 40 60 80 120 0 20 40 60 80 100 maximum 0.94 0.96 0.98 1.00 maximum 100 200

F f(x) Exp(1) F (x) = 1 e x, f(x) = e x, x 0. F n (x + log n) = { { } 1 e (x+log n)} n { n = 1 e x e log n} n = 1 + e x n a n = 1 b n = log n e e x = exp( exp( x)), n. ( < x < ).

(α > 0) F (x) = 1 1 x α, f(x) = αx α 1, x 1. F n (n 1/α x) = { 1 1 (n 1/α x) α } n = { 1 + x α n } n a n = n 1/α b n = 0 e x α = exp( x α ), n. (x > 0).

(α > 0) F (x) = 1 (1 x) α, f(x) = α(1 x) α 1, 0 x 1. F n (n 1/α x + 1) = { { } n 1 ( n 1/α x) α} n = 1 + ( x)α n a n = n 1/α b n = 1 e ( x)α = exp( ( x) α ), n. (x 0).

[Fréchet (1927) Fisher and Tippett (1928) Gnedenko (1943); Trinity Theorem ] G(x) Λ(x) = exp( exp( x)), x R (Gumbel), Φ α (x) = exp( x α ), x 0, α > 0 (Fréchet), Ψ α (x) = exp( ( x) α ), x 0, α > 0 (Weibull). x R n N Λ n (x + log n) = Λ(x), Φ n α (n1/α x) = Φ α (x), Ψ n α (n 1/α x) = Ψ α (x).

s.t. G a n > 0, b n R (n = 1, 2,...) G n (a n x + b n ) = G(x), x R and n = 1, 2,... (max-stable)

von Mises-Jenkinson ((1936) (1955)) G γ (x) = exp{ (1 + γx) 1/γ }, 1 + γx > 0, < γ <. G γ (x) Z Z = (E γ 1)/γ, E Exp(1); γ 1 log(1 + γz) = Z 0 Λ. Λ(x) = G 0 (x), Φ α (x) = G 1/α (α(x 1)), Ψ α (x) = G 1/α (α(x 1)).

[Gnedenko (1943) de Haan (1970)] F MDA(Φ α ) F (x) < 1, x and lim x 1 F (tx) 1 F (x) = t α, t > 0. F MDA(Ψ α ) x + < and lim x x+ 1 F (x + (x + x)t) 1 F (x) = t α, t > 0. 1 F (x + ts(x)) F MDA(Λ) s( ) > 0 s.t. ( ) lim x x+ 1 F (x) ( ) = x+ x = s(x) = (1 F (y))dy <, x < x + x+ x = e t. (1 F (y))dy/(1 F (x)) satisfies ( ).

F MDA(Φ α ), F (a n ) = 1 1/n, b n = 0. F MDA(Ψ α ), F (x + a n ) = 1 1/n, b n = x +. F MDA(Λ), F (b n ) = 1 1/n, a n = s(b n ).

{X i } i=1 F MDA(G) N {X i } i=1 Po(λ) P (N = i) = e λ λi, i = 0, 1, 2,.... i! N 1 Z N = max{x 1,..., X N } F N (y) F N (y) = P (Z N y N 1) = P (max{x 1,..., X N } y, N 1) P (N 1) 1 ( = e λ{1 F (y)} 1 e λ e λ).

a n b n F y a [λ] x + b [λ] [λ] λ lim λ { 1 F (a [λ] x + b [λ] ) } λ = lim λ λ [λ] [λ] { 1 F (a [λ] x + b [λ] ) } = log G(x). λ F N (a [λ] x + b [λ] ) exp(log G(x)) = G(x). P (Z N y N 1) G y b [λ] a [λ].

(generalized extreme value) GEV(µ, σ, ξ) < µ < σ > 0 < ξ < G(z) = exp { [ 1 + ξ ( z µ σ )] 1/ξ } ( ) z µ = G ξ, 1+ξ(z µ)/σ > 0. σ G ξ GEV(0, 1, ξ) G ξ (z) = exp [ (1 + ξz) 1/ξ], 1 + ξz > 0, µ σ ξ

G(z) ξ < 0 Weibull z < µ σ/ξ ξ = 0 Gumbel < z < G 0 ((z µ)/σ) = lim ξ 0 G ξ ((z µ)/σ) = exp{ exp[ (z µ)/σ]} ξ > 0 Fréchet z > µ σ/ξ G ξ (z) { } (1 + ξ z) 1/ξ 1 exp (1 + ξ z) 1/ξ, 1 + ξz > 0, ξ 0, g ξ (z) = { } exp z exp( z), < z <, ξ = 0,

GEV( 2.5, 1, 0.4) 0 GEV(0, 1, 0) GEV(2.5, 1, 0.4) 0

(generalized Pareto) GP(σ, ξ) σ > 0 < ξ < H(y) = 1 ( 1 + ξ y ) 1/ξ ( ) y = Hξ, 1 + ξy/σ > 0. σ σ H ξ GP(1, ξ) H ξ (y) = 1 (1 + ξy) 1/ξ, 1 + ξy > 0, σ ξ

H(y) ξ < 0 0 < y < σ/ξ ξ = 0 0 < y < H 0 (y/σ) = lim ξ 0 H ξ (y/σ) = 1 e y/σ ξ > 0 0 < y < H ξ (y) (1 + ξ y) 1/ξ 1, 1 + ξy > 0, ξ 0, h ξ (y) = exp( y), 0 < y <, ξ = 0,

GP(1, ξ) ξ = 0.4, 0, 0.4

[von Mises (1936) Smith (1990) ] φ(x) := 1 F (x), φ (x) = d f(x) dx φ(x) lim φ (x) = ξ = F MDA(G x x ξ ). + d {log(1 F (x))} = f(x) dx 1 F (x) x = log(1 F (x)) = x = inf{x : F (x) > 0} [ ] [ x dt 1 F (x) = exp = exp [1 F (t)]/f(t) x x x x dt φ(t) f(t) 1 F (t) dt. ].

1 F (u + xφ(u)) 1 F (u) = exp [ u+xφ(u) = exp u < t < u + xφ(u), φ(u + sφ(u)) φ(u) u dt [ φ(t) ] u+xφ(u) x = exp [ dt φ(t) x 0 ] / exp ds [ u x φ(u + sφ(u))/φ(u) dt φ(t) 0 < s = (t u)/φ(u) < x φ φ(u + sφ(u)) φ(u) = 1+ φ(u) u < y < u + xφ(u) [ 1 F (u + xφ(u)) x ds = exp 1 F (u) 0 1 + sφ (y) s = 1+ 0 φ (u+wφ(u))dw = 1+sφ (y). ] = exp [ 1+xφ (y) 1 ]. ] dt φ (y)t ]

= exp [ ( t = 1 + sφ (y) ) 1 φ (y) log(1 + xφ (y)) ] = (1 + xφ (y)) 1/φ (y). n[1 F (b n )] = 1 a n = φ(b n ) b n x + φ (b n ) ξ n[1 F (b n + xφ(b n ))] n[1 F (b n )] (1 + xφ (b n )) 1/φ (b n ), lim n n[1 F (a nx + b n )] = (1 + ξx) 1/ξ = log G ξ (x). lim n F n (a n x + b n ) = G ξ (x). F (b n ) = 1 1/n, a n = [1 F (b n )]/f(b n ) = n[1 F (b n )]/nf(b n ) = 1/nf(b n )

ξ = 1 ξ = 1/α < 0 ξ = 0 ξ > 0

f(x) = cx c 1 exp ( x c), x 0, c > 0. F (x) = 1 exp( x c ) φ(x) = 1 cx c 1 φ (x) = 1 c cx c 0, x. Gumbel (ξ = 0)

Penultimate n[1 F (a n x + b n )] (1 + ξ n x) 1/ξn, ξ n = φ (b n ). F n (a n x + b n ) exp[ (1 + ξ n x) 1/ξn ] = G ξn (x). G ξn G ξ penultimate ξ = 0 ξ n > 0 ξ n < 0 ξ = 0 Gumbel Fisher and Tippett (1928) Penultimate Cohen (1982) Gumbel ξ n

1 F (u + xφ(u)) 1 F (u) = [1 + xφ (y)] 1/φ (y), x > 0. y = xφ(u) σ = φ(u) P (X > u+y X > u) = 1 F (u + y) 1 F (u) = ( 1 + φ (y)y φ(u) ) 1/φ (y) ( 1 + ξ y σ) 1/ξ. F u (y) = P (X u y X > u) 1 (1 + ξy/σ) 1/ξ = H ξ (y/σ). F MDA(G ξ ) F G ξ ξ H ξ

tail index F G ξ F MDA(G ξ ) F H ξ ξ F F ξ Teugels ISI 1999 (F ) ξ < 0 ξ = 0 ξ > 0

R d {Y i } N(A) := i I A (Y i ), A R d, I A A I A (x) = { 1, x A, 0, x / A. N(A) A {Y i } N = N( ) (point process)

N Poisson (i) N(A) Po(Λ(A)) A Borel (ii) Borel B 1... B n N(B 1 )... N(B n ) Λ( ) Borel A Λ(A) = A λ(y)dy λ(y) Poisson λ(y) = Poisson

X 1, X 2,... i.i.d. Z n = max 1 i n X i a n > 0 b n R P { (Z n b n )/a n z } G(z), F MDA(G) { [ ( z µ G(z) = exp 1 + ξ σ )] 1/ξ } z z + G {( i N n = n + 1, X ) } i b n : i = 1,..., n a n u > z (0, 1) (u, z + ) A = [t 1, t 2 ] (z, z + ) Poisson [ ( )] z µ 1/ξ Λ(A) = (t 2 t 1 ) 1 + ξ. σ.

Point Process 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.0 0.2 0.4 0.6 0.8 1.0

A = [0, 1] (u, z + ) n A p u N n p = P { (X i b n )/a n > u } = 1 F (a n u + b n ) = n[1 F (a nu + b n )] n 1 n ( log G(u)) = 1 [ ( )] u µ 1/ξ 1 + ξ. n σ X i N n (A) B(n, p) n N n (A) Poisson N n (A) B(n, p) d Po(np), n. np = Λ(A) = [ 1 + ξ ( u µ σ )] 1/ξ.

A = [t 1, t 2 ] (u, z + ) [t 1, t 2 ] [0, 1] N n (A) Po(Λ(A)) Λ(A) = (t 2 t 1 ) [ 1 + ξ ( u µ σ )] 1/ξ. A

GEV A z = (0, 1) (z, ) {(Z n b n )/a n z} {N n (A z ) = 0} P { (Z n b n )/a n z } = P { N n (A z ) = 0 } P { N(A z ) = 0 } = exp{ Λ(A z )} = exp { [ 1 + ξ ( z µ σ )] 1/ξ }.

GP Λ(A z ) = Λ 1 ([t 1, t 2 ]) Λ 2 ((z, )) Λ 1 ([t 1, t 2 ]) = t 2 t 1 and Λ 2 ((z, )) = σ = σ + ξ(u µ) P { (X i b n )/a n > z (X i b n )/a n > u } [ 1 + ξ ( z µ σ )] 1/ξ. Λ 2(z, ) Λ 2 (u, ) = [1 + ξ(z µ)/σ] 1/ξ [1 + ξ(u µ)/σ] 1/ξ = [ 1 + ξ ( z u σ )] 1/ξ.

r r(> 1) r (X (n:n), X (n 1:n),..., X (n r+1:n) ) Z n = X (n:n) GEV(µ, σ, ξ) r (X (n:n), X (n 1:n),..., X (n r+1:n) ) z (1) z (2) z (r) ξ 0 { [ )] 1/ξ } r [ f ξ (z (1),..., z (r) ) = exp 1 + ξ ( z (r) µ σ k=1 1 + ξ(z (k) µ)/σ > 0 k = 1, 2,..., r 1 σ 1 + ξ ( )] z (k) 1/ξ 1 µ, σ

ξ = 0 { [ ( )]} z f 0 (z (1),..., z (r) (r) µ ) = exp exp σ r k=1 [ 1 σ exp rgev(µ, σ, ξ) ( )] z (k) µ. σ (µ, σ, ξ)

{X i } i=1 F i 1,..., i n k, n N = {1, 2,...} (X i1,..., X in ) d = (X i1 +k,..., X in +k). { X i } i=1 F Z n = max { X 1,..., X n }, Z n = max { X 1,..., X n },

i < j < i < j max { } X i,..., X j max { } X i,..., X j i j a n > 0 b n R lim n P { ( Z n b n )/a n x } = G(x) lim n P { (Z n b n )/a n x } = G θ (x), θ (0, 1]. θ (extremal index) X i (clustering) 1/θ

F = {f(x; θ) : θ Θ} f(x; θ) θ X n {X 1, X 2,..., X n } {x 1, x 2,..., x n } k θ = (θ 1, θ 2,..., θ k ) Θ R k

L(θ) = l(θ) = n i=1 n i=1 f(x i ; θ) log f(x i ; θ) θ l(θ) θ i = 0, i = 1, 2,..., k.

θ = ( θ 1, θ 2,..., θ k )

F n e ij (θ) = E I E (θ) = { 2 θ. N k (θ, I E (θ) 1 ). e 11 (θ) e 12 (θ) e 1k (θ) e 21 (θ). e 22 (θ). e 2k (θ). e k1 (θ) e k2 (θ) e kk (θ) l(θ) θ i θ j } = ne { 2, log f(x; θ) θ i θ j. I E (θ) }.

I E (θ) θ = θ 2 2 θ 2l(θ) 1 θ 1 θ l(θ) 2 θ 2 1 θ l(θ) k I O (θ) = θ 2 2 θ l(θ) 2 2 1 θ 2l(θ) 2 θ 2 θ l(θ) k.... θ 2 k θ l(θ) 2 1 θ k θ l(θ) 2 2 θ 2 l(θ) k 2 l(θ) = θ i θ j n l=1 2 θ i θ j log f(x l ; θ).

θ φ = g(θ) f(x; θ) φ R 1 θ θ φ = g(θ) φ φ = g( θ)

n θ θ V θ φ = g(θ) φ = g( θ). N(φ, V φ ) V φ = φ T V θ φ φ = [ φ,..., φ ] T θ 1 θ k θ

θ = (θ 1,..., θ k ) l(θ i, θ i ) θ i θ θ i θ i (profile log-likelihood) l p (θ i ) = max θ i l(θ i, θ i ) θ 2 (θ (1), θ (2) ) θ (1) k 1 θ (2) k k 1 θ (1) l p (θ (1) ) = max θ (2) l(θ(1), θ (2) ).

θ k θ = (θ (1), θ (2) ) θ (1) θ k 1 n D p (θ (1) ) = 2 { } l( θ) l p (θ (1). ) χ 2 k1

1 θ i (1 α) } { } {θ i : D p (θ i ) χ 2 1 (α) = θ i : max l(θ i, θ i ) l( θ) χ 2 1 θ (α)/2 i χ 2 1 (α) χ2 1 α θ M 1 M 0 θ k 1 0 θ = (θ (1), θ (2) ) M 0 k 1 0 D p (θ (1) ) > χ 2 k (α) 1 α M 1

PWM F (x; θ) X E(X p ) V (X) θ θ 2 x s 2 x = 1 n n i=1 x i, s 2 = 1 n n i=1 (x i x) 2.

PWM F (x) = F (x; θ) X (probabilityweighted moment PWM) p r s M p, r, s = E [X p {F (X)} r {1 F (X)} s ] = 1 0 (F 1 (u)) p u r (1 u) s du PWM M 1, r, s u r (1 u) s u 1 u M 1, r, 0 M 1, 0, s α r := M 1, r, 0 = E [X{F (X)} r ], r = 0, 1, 2,... β s := M 1, 0, s = E [X{1 F (X)} s ], s = 0, 1, 2,... r = s = 0 α 0 = β 0 = E(X)

α r β s x (1) x (2) x (n) a r = 1 n b s = 1 n n i=1 n i=1 (i 1)(i 2) (i r) (n 1)(n 2) (n r) x (i), r = 1, 2,... (n i)(n i 1) (n i s + 1) x (n 1)(n 2) (n s) (i), s = 1, 2,... γ δ α r = 1 n β s = 1 n n i=1 n i=1 p r i,n x (i), r = 0, 1,... (1 p i,n ) s x (i), s = 0, 1,... p i,n = i + γ, i = 1, 2,..., n. n + δ

{x 1, x 2,..., x n } F n x (1) x (2) x (n) ˆF F F F (x) = 1 n + 1 n i=1 I (, x] (x i )

0.0 0.2 0.4 0.6 0.8 1.0 True Estimate 0.0 0.5 1.0 1.5 2.0 2.5 H 0 GP(1, 0)

(probability plot {( ˆF (x (i) ), i n + 1 ) PP plot) } : i = 1,..., n. (quantile plot QQ plot) {( ˆF 1 ( i n + 1 ), x (i) ) } : i = 1,..., n. ˆF 1 (i/(n + 1)) x (i) F i/(n + 1)

GEV) G(z) = exp { {z 1, z 2,..., z n } GEV(µ, σ, ξ) [ 1 + ξ ( z µ σ )] 1/ξ } ( ) z µ = G ξ, 1+ξ(z µ)/σ > 0, σ (µ, σ, ξ)

GEV(µ, σ, ξ) 1 p z p ( ) zp µ G ξ = 1 p σ µ + σ [{ log(1 p) } ξ ]/ 1 ξ, ξ 0, z p = µ + σ [ log { log(1 p) }], ξ = 0. z p (return period) 1/p (return level) 1/p = 200 z 1/200 200 1 p 1/n

y p = log(1 p) z p = µ + σ ( y ξ p 1 )/ ξ, ξ 0, µ + σ ( log y p ), ξ = 0, { ( log yp, z p ); 0 < p < 1 } (return level plot) ξ < 0 (concave) p 0 ( log y p ) µ σ/ξ ξ = 0 ξ > 0 (convex) p 0

GEV(0, 1, ξ) ξ = 0.2, 0, 0.2

GEV(µ, σ, ξ) ξ 0 l(µ, σ, ξ) = n log σ (1 + 1/ξ) n [ ( )] zi µ log 1 + ξ σ i=1 n [ ( )] 1/ξ zi µ 1 + ξ, σ i=1 1 + ξ(z i µ)/σ > 0, i = 1,..., n, ξ = 0 l(µ, σ) = n log σ n i=1 ( zi µ σ ) n i=1 exp { ( zi µ σ )}. ( µ, σ, ξ) ( µ, σ)

GEV(µ, σ, ξ) (Prescott and Walden, 1980) n σ 2 ξ 2 ξ 2 p ξ { p 2 Γ(2 + ξ) } σξ 1 2 Γ(2 ξ) + p σ σ 2 [ 1 γ + π2 6 + ( ( q p ξ ) 1 Γ(2 + ξ) ξ 1 γ + 1 ξ q + p ξ ) 2 2q ξ + p ] ξ 2 (µ, σ, ξ) Γ( ) Ψ(r) = d log Γ(r)/dr p = (1+ξ) 2 Γ(1+2ξ) q = Γ(2+ξ){Ψ(1+ξ)+(1+ξ)/ξ} γ = 0.5772157... Euler

{GEV(µ, σ, ξ), < µ <, σ > 0, < ξ < } ξ > 0.5 Smith, 1985 ξ 0.5 0.5 < ξ < 0.5 (µ, σ, ξ)

z p y p = log(1 p) ẑ p = µ + σ { y ξ p 1 }/ ξ, ξ 0, µ + σ { log y p }, ξ = 0. V (ẑ p ) zp T V z p V ( µ, σ, ξ) z T p = = [ zp µ, z p σ, z p ξ [ 1, (y ξ p ( µ, σ, ξ) ] ] 1)/ξ, σyp ξ ( log y p)/ξ σ(yp ξ 1)/ξ 2

ξ < 0 z 0 = µ σ/ξ ẑ 0 = µ σ/ ξ z T p = [ 1, 1/ξ, σ/ξ 2 ]

ξ z p ξ ξ = ξ 0 l(µ, σ, ξ 0 ) µ σ ξ 95% { ξ : 2 { l( µ, σ, ξ) } max µ, σ l(µ, σ, ξ)} χ 2 1 (0.05) = 3.841 { = ξ : max µ, σ l(µ, σ, ξ) l( µ, σ, ξ) } 1.921

z p µ = z p σ [ y ξ p 1 ] /ξ (µ, σ, ξ) (z p, σ, ξ) n [ )] n [ )] 1/ξ l(z p, σ, ξ) = n log σ (1+1/ξ) i=1 log y ξ p + ξ ( zi z p σ i=1 y ξ p + ξ ( zi z p z p 95% { z p : 2 { l(ẑ p, σ, ξ) max l(z p, σ, ξ) } } χ 2 1 (0.05) σ, ξ { = z p : max l(z p, σ, ξ) l(ẑ p, σ, ξ) } χ 2 1 (0.05)/2 σ, ξ σ

PWM PWM (Hosking et al., 1985) α r := M 1, r, 0 = E [X{G(X)} r ], r = 0, 1, 2,... { { G 1 µ + σ ( log u) (u) = ξ 1 } /ξ, ξ 0, µ + σ log( log u), ξ = 0, ξ 0 PWM α r = 0 G 1 (u)u r du = µ + σ { (r + 1) ξ Γ(1 ξ) 1 } /ξ, ξ < 1. r + 1 1

α 0 = µ + σ { Γ(1 ξ) 1 } /ξ, 2α 1 α 0 = σ(2 ξ 1)Γ(1 ξ)/ξ, (3α 2 α 0 )/(2α 1 α 0 ) = (3 ξ 1)/(2 ξ 1). 1 2 σ = (2α 1 α 0 ) ξ Γ(1 ξ)(2 ξ 1), µ = α 0 + σ { } 1 Γ(1 ξ). ξ 3 1/2 < ξ < 1/2 ξ = 7.8590 c 2.9554 c 2, c = log 2 log 3 2α 1 α 0 3α 2 α 0 0.0009

z (1) z (2) z (n) n ( ) i 0.35 r z (i), r = 0, 1, 2 α r = 1 n i=1 n PWM s α r, r = 0, 1, 2 PWM ξ σ µ ξ = 7.8590 c 2.9554 c 2, c = log 2 log 3 2 α 1 α 0 3 α 2 α 0,

H 0 : ξ = 0 vs. H 1 : ξ 0 {z 1, z 2,..., z n } Gumbel PWM ξ H 0 : ξ = 0 N(0, 0.5633/n) ξ 0.5633/n. N(0, 1).

10% ξ / 0.5633/n > 1.645 H 0 reject Hosking et al. (1985) Hosking (1984) Bartlett

GEV F MAD(G ξ ) {( Ĝ(z (i) ), Ĝ(z (i) ) = exp i n + 1 ) [ 1 + ξ } : i = 1,..., n ( z(i) µ σ )] 1/ ξ.

{( Ĝ 1 ( i n + 1 ), z (i) ) } : i = 1,..., n Ĝ 1 ( i n + 1 ) = µ + σ { log ( )} i ξ n + 1 1 / ξ. 0 < p < 1 ẑ p z p { } ( log y p, ẑ p ) : 0 < p < 1

(block maximum method)?

GEV Z t GEV(µ(t), σ(t), ξ(t)), t = 1, 2,..., m. µ(t) µ(t) = β 0 + β 1 t µ(t) = β 0 + β 1 t + β 2 t 2 σ(t) σ(t) = exp(β 0 + β 1 t)

GEV Z t GEV(µ(t), σ(t), ξ(t)), t = 1, 2,..., m. µ(t) σ(t) ξ(t) β L(β) = m t=1 1 σ(t) g ξ(t) g ξ (z) GEV(0, 1, ξ) ( zt µ(t) σ(t) ). β β

M 0 M 1 { } 2 l 1 (M 1 ) l 0 (M 0 ) χ 2 k 1 l i (M i ) M i (i = 0, 1) k 1 M 1 M 0 1 2

GEV Z t GEV(µ(t), σ(t), ξ(t)), t = 1, 2,..., m, Z 0 = 1 { ( )} Zt ξ(t) log µ(t) 1 + ξ(t) σ(t) GEV(0, 1, 0) : Gumbel P (Z 0 z) = G 0 (z) = exp( exp( z)), < z <. z 1, z 2,..., z m µ(t) σ(t) ξ(t) β z t = 1 { ( )} ξ(t) log 1 + ξ(t) zt µ(t), t = 1, 2,..., m σ(t) Gumbel z (1) z (2) z (m)

{( G 0 ( z (i) ), i m + 1 ) } : i = 1,..., m G 0 (z (i) ) = exp( exp( z (i) )). ( i {( G 1 0 m + 1 G 1 0 ( i m + 1 ) ), z (i) ) } : i = 1,..., m = log( log(i/(m + 1))).

Gumbel

0 2 4 6 8 10 12 14 102 104 106 108 Age The yearly oldest ages at death in Sweden during the period from 1905 to 1958 for women. 101.50 107.90

95.904 µ = 103.966 (0.209), σ = 1.3834 (0.147), ξ = 0.221 (0.0886). ξ µ σ/ ξ = 103.966 + 1.3834/0.221 = 110.226 107.90

Probability Plot Quantile Plot Empirical 0.0 0.2 0.4 0.6 0.8 1.0 Empirical 102 104 106 108 0.0 0.2 0.4 0.6 0.8 1.0 Model Return Level Plot 102 103 104 105 106 107 Model Density Plot Return Level 102 104 106 108 f(z) 0.0 0.10 0.20 0.1 1.0 10.0 100.0 1000.0 Return Period 102 104 106 108 z

Profile Log-likelihood -98.0-97.5-97.0-96.5-96.0-0.4-0.3-0.2-0.1 0.0 Shape Parameter ξ 95% [ 0.384, 0.023]. ξ = 0.221 (0.0886)

Profile Log-likelihood -101-100 -99-98 -97-96 107 108 109 110 111 Return Level 100 95% [107.263, 109.929] ẑ 1/100 = 107.96

1900 1999 (m/s) (1982) (2002) GEV (2004)

15 20 25 30 1900 1920 1940 1960 1980 2000 (m/s) 1900 1999 33.3(m/s)

GEV(µ, σ, ξ) ( µ, σ, ξ) = (15.349, 2.550, 0.111) 257.78 V = 0.08235 0.03024 0.00686 0.03024 0.04703 0.00231 0.00686 0.00231 0.00567 ( µ, σ, ξ) µ = β 0 + β 1 257.352 µ σ µ σ

Probability Plot Quantile Plot Empirical 0.0 0.2 0.4 0.6 0.8 1.0 Empirical 15 20 25 30 0.0 0.2 0.4 0.6 0.8 1.0 Model Return Level Plot 15 20 25 30 Model Density Plot Return Level 15 20 25 30 35 40 f(z) 0.0 0.04 0.08 0.12 0.1 1.0 10.0 100.0 1000.0 Return Period 10 15 20 25 30 z

ξ = 0.111 > 0 Fréchet ξ 95% ξ 0.111 ± 1.96 0.00567 = [ 0.037, 0.259]. 95% [ 0.021, 0.274]

Profile Log-likelihood -260.0-259.5-259.0-258.5-258.0 0.0 0.1 0.2 0.3 Shape Parameter ξ ξ = 0.111 > 0 95% [ 0.021, 0.274] [ 0.037, 0.259]

200 z 1/200 33.74 95% z 1/200 ẑ 1/200 ± 1.96 z T 1/200 V z 1/200 = [25.75, 41.73]. 95% [28.29, 46.90]

Profile Log-likelihood -260.0-259.5-259.0-258.5-258.0 30 35 40 45 Return Level 200 ẑ 1/200 = 33.74 95% [28.29, 46.90] ([25.75, 41.73]

Kumamoto Max. Daily Rainfall 100 200 300 400 1900 1920 1940 1960 1980 2000 Year (0.1mm) 1901 2005

GEV Z t GEV(µ(t), σ(t), ξ), t = 1, 2,..., m. 1 µ(t) = µ, σ(t) = σ. 2 µ(t) = β 0 + β 1 t, σ(t) = σ. 3 µ(t) = β 0 + β 1 t + β 2 t 2, σ(t) = σ. 4 µ(t) = β 0 + β 1 t, σ(t) = exp(β 2 + β 3 t). 5 µ(t) = β 0 + β 1 t + β 2 t 2, σ(t) = exp(β 3 + β 4 t).

1 578.204 χ 2 1 (0.05) = 3.841 2 576.973 χ 2 2 (0.05) = 5.991 3 576.527 4 573.376 5 573.290 4 µ(t) = 123.85 + 23.33t, σ(t) = exp(3.820 + 0.377t), ξ = 0.112. t = (year 1953)/52

Probability Plot Quantile Plot Empirical 0.0 0.2 0.4 0.6 0.8 1.0 Empirical 100 200 300 400 0.0 0.2 0.4 0.6 0.8 1.0 Model Return Level Plot 100 200 300 400 Model Density Plot Return Level 100 300 500 700 f(z) 0.0 0.002 0.006 0.1 1.0 10.0 100.0 1000.0 Return Period 0 100 200 300 400 500 z 1 1901 2005

Residual Probability Plot Residual Quantile Plot (Gumbel Scale) Empirical 0.0 0.2 0.4 0.6 0.8 1.0 Model -2 0 2 4 0.0 0.2 0.4 0.6 0.8 1.0 Model -1 0 1 2 3 4 Empirical 4 1901 2005

kumamoto 100 200 300 400 1900 1920 1940 1960 1980 2000 year

(GP) H(y) = 1 {y 1, y 2,..., y n } GP(σ, ξ) ( 1 + ξ y ) 1/ξ ( ) y = Hξ, 1 + ξy/σ > 0, σ σ

Y GP(σ, ξ) ξ < 1 E(Y ) = y+ y+ (1 H(y))dy = 0 0 y + = sup{y : H(y) < 1} ( 1 + ξ y ) 1/ξ σ dy = σ 1 ξ. ξ < 1/2 V (Y ) = σ 2 (1 ξ) 2 (1 2ξ).

U U(0, 1) ( 1 + ξ Y ) 1/ξ = U = 1 ( σ ξ log 1 + ξ Y ) σ = log U, Y = σ(u ξ 1). ξ Y v Y > v GP(σ + ξv, ξ) (v > 0) P (Y v > y Y > v) = = 1 H((y + v)/σ) = 1 H(v/σ) ( y 1 + ξ σ + ξv ) 1/ξ. ( 1 + ξ(y + v)/σ ) 1/ξ ( 1 + ξv/σ ) 1/ξ

e(v) = E(Y v Y > v) Y (mean excess) ẽ(v) Y (median excess) P (Y v ẽ(v) Y > v) = 1/2 Y v Y > v GP(σ + ξv, ξ) ẽ(v) = e(v) = σ + ξv 1 ξ = σ 1 ξ + ξ 1 ξ v, σ(2 ξ 1)/ξ + (2 ξ 1)v, ξ 0, σ log 2, ξ = 0.

GP(σ, ξ) ξ 0 l(σ, ξ) = n log σ (1 + 1/ξ) n i=1 1 + ξ y i /σ > 0, i = 1, 2,..., n, log(1 + ξ y i /σ), ξ = 0 l(σ) = n log σ 1 σ n i=1 y i. ( σ, ξ) σ

ξ > 0.5 (Smith, 1985) ξ > 0.5 ( σ, ξ) (σ, ξ) V = 1 n ( 2σ 2 (1 + ξ) σ(1 + ξ) σ(1 + ξ) (1 + ξ) 2 ) ξ 0.5

{x 1, x 2,..., x no } F n o F u y i = x [i] u i = 1, 2,..., n u GP(σ, ξ) u Smith (1987) F MDA(G ξ ) F ξ < 0 ξ > 0 ξ = 0 ξ < 0

m (σ, ξ) GP(σ, ξ) u X u X > u GP(σ, ξ) P (X > x X > u) = [ 1 + ξ ( )] x u 1/ξ, x > u. σ ζ u = P (X > u) P (X > x) = ζ u [1 + ξ ( x u σ )] 1/ξ,

x m m 1 ( )] xm u 1/ξ 1 ζ u [1 + ξ = σ m x m = u + σ ξ [(mζ u ) ξ 1 ]. ξ = 0 x m = u + σ log(mζ u ). m x m > u x m m (m-observation return level)

m { } (log m, x m ); x m > u m ξ < 0 ξ = 0 ξ > 0 N n y m = N n y N z N = u + σ [ (Nny ζ u ) ξ 1 ]/ ξ, ξ 0, u + σ log ( Nn y ζ u ), ξ = 0. σ ξ ζ u σ ξ ζ u = n/n o

u B(n o, ζ u ) ζ u V ( ζ u ) ζ u (1 ζ u )/n o ( ζ u, σ, ξ) [ ζ V = u (1 ζ)/n o 0 T ]. 0 V 0 T = (0, 0) V ( σ, ξ)

x m V ( x m ) x T m V x m. x m = x m ζ u x m σ x m ξ = σm ξ ζ ξ 1 u { (mζu ) ξ 1 } /ξ ( ζ u, σ, ξ) σ(mζ u ) ξ log(mζ u )/ξ σ{(mζ u ) ξ 1}/ξ 2

m x m σ = (x m u)ξ (mζ u ) ξ 1, ξ 0, x m u log(mζ u ), ξ = 0, l(x m, ξ) x m max ξ l(x m, ξ) ζ u

GP(σ, ξ) ξ < 1/2 σ 1 ξ = y, σ 2 (1 ξ) 2 (1 2ξ) = s2 σ M = 1 2 y(y2 /s 2 + 1), ξ M = 1 2 (1 y2 /s 2 ). y s 2

PWM PWM (Hosking et al., 1987) β s = M 1, 0, s = E[Y {1 H(Y )} s ] = 1 0 H 1 (u)(1 u) s du = ξ < 1 s = 0, 1 β 0 = σ 1 ξ, β 1 = σ 2(2 ξ) σ (s + 1)(s + 1 ξ). σ = 2β 0β 1 β 0 2β 1, ξ = 2 β 0 β 0 2β 1.

β 0 = y, y (1) y (2) y (n) β 1 = 1 n n i=1 ( 1 i 0.35 n ) y (i) PWM s β 0 β 1 PWM σ ξ

(threshold method u u u

(mean excess plot) u x [1], x [2],..., x [nu ] x max u < x max u, 1 n u n u i=1 (x [i] u) : u < x max u u Y GP(σ, ξ) E(Y v Y > v) = σ 1 ξ + ξ 1 ξ v

(median excess plot) u < x max { {(u, median 1 i nu x[i] u }) } : u < x max u u ẽ(v) = σ(2 ξ 1)/ξ + (2 ξ 1)v, ξ 0, σ log 2, ξ = 0.

ξ σ u < x max {x [i] u} n u i=1 GP(σ u, ξ) σ u ξ u (u, σ ) (u, ξ) σ = σ u ξu u σ ξ u Y v Y > v GP(σ + ξv, ξ) (v > 0), σ v = σ + ξv.

Pareto Hill ξ > 0 m x (1) x (2) x (m) {( ( ) ) } i log, log x (i), i = 1, 2,..., m m + 1 x (k+1) x (k+1) Hill (Hill, 1975) ξ H (k) = 1 k k i=1 ( log x (i) log x (k+1) ) k ξ H (k) k x (k+1)

GP(σ, ξ) GP y (1) y (2)... y (n) {( Ĥ(y (i) ), i n + 1 Ĥ(y (i) ) = 1 ( ) } : i = 1,..., n 1 + ξ y (i) σ ) 1/ ξ.

{( Ĥ 1 ( i n + 1 ), y (i) ) } : i = 1,..., n Ĥ 1 ( i n + 1 ) = σ ( n + 1 i n + 1 ) ξ 1 / ξ. { m (log m, x m ) } [ (m x m = u + σ ζ ]/ ) ξ u 1 ξ.

GP GP X 1, X 2,... t s(t) X t u s(t) X t > u s(t) GP(σ s(t), ξ s(t) ). u s(t) s(t) σ ξ s(t) s(t)

GP Y t GP(σ(t), ξ(t)) Y 0 = 1 { ξ(t) log 1 + ξ(t) Y } t σ(t) GP(1, 0) : P (Y 0 y) = H 0 (z) = 1 exp( y), 0 < y <. y t1, y t2,..., y tk σ(t) ξ(t) β ỹ tk = 1 ξ(t) log { 1 + ξ(t) y t i σ(t) }, i = 1, 2,..., k

ỹ (1) ỹ (2) ỹ (k) {( 1 exp( ỹ (i) ), {( ( log 1 i k + 1 i k + 1 ) ), ỹ (i) ) } : i = 1,..., k } : i = 1,..., k

Anderson & Coles (2002) Beirlant et al. (2004) clean bearing steel 5µm 112 0 10 20 30 40 50 6 8 10 12 Diameter

Mean Excess 0 1 2 6 8 10 12 u

Modified Scale -5 0 5 10 15 5 6 7 8 9 Threshold Shape -1.0-0.5 0.0 0.5 5 6 7 8 9 Threshold σ ξ

GP 5 ξ = 0.085 (0.0914), σ = 1.68 (0.220) 160.6 PP QQ

Probability Plot Quantile Plot Empirical 0.0 0.2 0.4 0.6 0.8 1.0 Empirical 4 6 8 10 12 14 0.0 0.2 0.4 0.6 0.8 1.0 Model 4 6 8 10 12 14 Model

0.1mm GP 1961 1 1 2004 12 31

Daily Rainfall 0 500 1000 1500 0 5000 10000 15000 Nara (0.1mm) 1961 1 1 2004 12 31 1000 500

Mean Excess 0 100 200 300 400 0 500 1000 1500 u

Modified Scale 0 100 200 300 400 500 100 200 300 400 500 600 Threshold Shape -0.3-0.1 0.1 0.3 100 200 300 400 500 600 Threshold σ ξ

u 300 Coles (2001) u σ ξ u = 300 453 (σ, ξ) 2839.646 σ = 180.65 (12.541), ξ = 0.072 (0.0512). GP

Probability Plot Quantile Plot Empirical 0.0 0.2 0.4 0.6 0.8 1.0 Empirical 400 800 1200 1600 0.0 0.2 0.4 0.6 0.8 1.0 Model Return Level Plot 400 600 800 1000 1200 1400 1600 Model Density Plot Return level 1000 2000 3000 4000 5000 f(x) 0.0 0.002 0.004 0.1 1.0 10.0 100.0 1000.0 Return period (years) 400 600 800 1000 1200 1400 1600 x

ξ ξ ξ 95% [ 0.020, 0.183] 100 ẑ 1/100 = 1925 95% [1567, 2633]

Profile Log-likelihood -2842.0-2841.5-2841.0-2840.5-2840.0 0.0 0.05 0.10 0.15 0.20 Shape Parameter ξ ξ = 0.072 (0.0512) 95% [ 0.020, 0.183]

Profile Log-likelihood -2842.5-2841.5-2840.5 1600 1800 2000 2200 2400 2600 Return Level z 1/100 ẑ 1/100 = 1925 95% [1567, 2633]

GP GP(σ, ξ) 1961 1962 300? 404 453 49 GP

DailyRainfall 0 500 1000 1500 0 100 200 300 Nara DailyRainfall 0 200 400 600 0 100 200 300 Nara (0.1mm) 1961 ( 1962 (

300 2548.728 σ = 189.54 (13.901), ξ = 0.064 (0.0540). ξ z 1/100 ξ z 1/100 95% [ 0.032, 0.182] [1559, 2649] declustering Coles (2001) 5.3.2

Probability Plot Quantile Plot Empirical 0.0 0.2 0.4 0.6 0.8 1.0 Empirical 400 800 1200 1600 0.0 0.2 0.4 0.6 0.8 1.0 Model Return Level Plot 400 600 800 1000 1200 1400 1600 Model Density Plot Return level 1000 2000 3000 4000 5000 f(x) 0.0 0.002 0.004 0.1 1.0 10.0 100.0 1000.0 Return period (years) 400 600 800 1000 1200 1400 1600 x

Profile Log-likelihood -2551.0-2550.5-2550.0-2549.5-2549.0 0.0 0.05 0.10 0.15 0.20 Shape Parameter ξ ξ = 0.064 (0.0540) 95% [ 0.032, 0.182]. [ 0.020, 0.183]

Profile Log-likelihood -2551.5-2550.5-2549.5 1600 1800 2000 2200 2400 2600 2800 Return Level z 1/100 z 1/100 95% [1559, 2649] [1567, 2633]

(PP) u (0, 1) (u, ) N n Poisson N A = (0, 1) (u, ) { (t1, x 1 ),..., (t N(A), x N(A) ) } A N n N A = [0, 1] (u, ) Λ(A) = [ 1 + ξ ( z µ σ )] 1/ξ

n y L A (µ, σ, ξ; x 1,..., x n ) = exp { Λ(A) } N(A) i=1 λ(t i, x i ) exp { n y [ 1 + ξ ( u µ σ )] 1/ξ } N(A) i=1 1 σ [ 1 + ξ ( xi µ σ )] 1/ξ 1 ( µ, σ, ξ)

PP 0.1 C PP 2000 1 1 2004 12 31 Coles (2001) u(t) = 245 + 120 sin(2π(t 110)/365.25)

Daily Maximum Temperature 100 200 300 0 500 1000 1500 Kyoto (0.1 C) 2000 1 1 2004 12 31

Coles (2001) µ(t) = β 0 + β 1 sin(2πt/365.25) + β 2 cos(2πt/365.25), log σ(t) = β 3 + β 4 sin(2πt/365.25) + β 5 cos(2πt/365.25). ξ Coles S-Plus 149.03 β 0 = 287.29 (1.100), β 1 = 28.38 (1.377), β 2 = 103.07 (2.033), β 3 = 1.233 (0.1335), β 4 = 0.233 (0.0465), β 5 = 0.277 (0.0684), ξ = 0.451 (0.0473)

Residual Probability Plot Residual quantile Plot (Exptl. Scale) Empirical 0.0 0.2 0.4 0.6 0.8 1.0 Empirical 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 Model 0 1 2 3 4 5 Model