Multivariate Pareto distributions: properties and examples

Transcription

1 Multivariate Pareto distributions: properties and examples Ana Ferreira 1, Laurens de Haan 2 1 ISA UTL and CEAUL, Portugal 2 Erasmus Univ Rotterdam and CEAUL EVT2013 Vimeiro, September 8 11

2 Univariate Pareto distribution: P(X > x) = ( 1+γ x µ ) 1/γ, σ 0 < x µ σ < (0 ( γ)) 1, σ > 0, µ,γ R; read e (x µ)/σ if γ = 0. MDA (Balkema and de Haan (1974), Pickands (1975)) F (d.f.) is in the (maximum) domain of attraction of some Generalized extreme value distribution GEV γ if and only if s(t) > 0 such that lim t x P (X > t +x s(t) X > t) = (1+γx) 1/γ, 1+γx > 0, (1) where x = sup{x : F(x) < 1}; that is, above the high threshold t we get a Pareto tail in the limit.

3 Multivariate Pareto distributions Definition 1. (Falk et al. (2004); Michel (2008)) A multivariate generalized Pareto distribution is any multivariate distribution function that can be represented by 1+logGEV in a neighborhood of the right endpoint of GEV. Definition 2. (Rootzén and Tajvidi (2006)) A distribution function H is a multivariate generalized Pareto distribution if H(x) = 1 loggev(0) log GEV(x) GEV(x 0) for some GEV with non-degenerate margins and with 0 < G(0) < 1. (...)

4 Simple Multivariate Pareto Let, W = (W 1,...,W d ) T random vector (r.vect.) in R d + = [0, ) d, W = max 1 i d W i, ω 0 > 0, threshold parameter. Theorem 1. The following three statements are equivalent: 1.(a) E (W i / W ) > 0 for all i = 1,...,d, 1.(b) P ( W /ω 0 > x) = x 1, for x > 1 (stand. Pareto distr.), 1.(c) ( ) ( ) ω0 W P W B ω0 W W > rω 0 = P W B, (2) for all r > 1 and B B ( D + ω 0 ) with D + ω 0 := {w R d + : w = ω 0 }.

5 Simple Multivariate Pareto Theorem 1. (cont.) 2.(a) P ( W > ω 0 ) = 1, 2.(b) E (W i / W ) > 0 for all i = 1,...,d, 2.(c) P(W ra) = r 1 P(W A), (3) for all r > 1 and A B ( D + ω 0 ) with D + ω 0 := {w R d + : w ω 0 }. Recall stand. Pareto r.var.: P(Y > ry) = r 1 P(Y > y), y,r>1.

6 Simple Multivariate Pareto Theorem 1. (cont.) W = (W 1,...,W d ) = Y(V 1,...,V d ) = YV verifying: 3.(a) V R d + r. vect., V = ω 0 a.s. and EV i > 0 for all i = 1,...,d, 3.(b) Y is a stand. Pareto r.var., P(Y y) = 1 1/y, y > 1, 3.(c) Y and V are independent. Definition 1. W R d + characterized in Theorem 1. is simple Pareto with threshold parameter ω 0.

7 Some distribution formulas: If w > 0, ( P(W w) = E max 1 i d V i w i ω 0 ) ( ) V i E max 1 i d w i (which corresponds to Def.1. from Rootzén and Tajvidi (2006)). If w > 0 and w > ω 0, ( P(W > w) = E If E (min 1 i d V i ) > 0, for x R, min 1 i d ) V i. w i P(W > x W > ω 0 ) = { 1, x ω0 ω 0 /x, x > ω 0. For x R and for each i = 1,...,d, { 1, x ω0 P(W i > x W i > ω 0 ) = ω 0 /x, x > ω 0.

8 Definition 2: Multivariate GP r.vect. W µ,σ,γ R d Let W simple Pareto r.vect. γ = (γ 1,...,γ d ) R d extreme value index vector, µ = (µ 1,...,µ d ) R d mean value vector, σ = (σ 1,...,σ d ) > 0 scale vector. Define, W µ,σ,γ = µ+σ Wγ 1. γ

9 Domain of attraction Let X = (X 1,...,X d ) R d with continuous d.f. F, γ = (γ 1,...,γ d ) R d, a t = (a t,1,...,a t,d ) > 0, b t = (b t,1,...,b t,d ) R d, ( ) 1/γ T t X = 1+γ X bt a. t + Theorem 2. F is in the maximum domain of attraction of some GEV with normalizing constants a t > 0 and b t R d iff, 1(a) tp ((X i b t,i )/a t,i > x) (1+γ i x) 1/γ i, i, 1(b) P ( T t X > x T t X > 1) x 1, x > 1, 1(c) ( ) Tt X P T t X B T t X > 1 ρ(b), (4) B B( D + 1 ), ρ( B) = 0, ρ some probability measure on D + 1.

10 Example Let W = (W 1,W 2 ) = (YB,Y(1 B)) with B Bernoulli (1/2) independent of Y stand. Pareto. Then, The spectral measure, i.e. the probability measure of (V 1,V 2 ) = (B,1 B) is concentrated on {(0,1),(1,0)} with prob. 1/2 at each atom. Its d.f. is in the max-domain of atraction of some GEV with independent components, hence verifies asymptotic independence. Note that E(V1 V 2 ) = 0. For W R d, let λ i,j = P(W i > u W j > u) = E(V i V j ) E(V j ) and bivariate asymptotic independence holds if λ i,j = 0. That is, whenever E(V i V j ) = 0 we get asymptotic independence.

11 The Peaks-over-threshold method Corollary 1. Under the max-domain of attraction conditions lim P ( T t X A T t X > 1 ) = P (W A), (5) t A B(D 1 + ), P(W A) = 0, W simple Pareto r.vect. Recall univariate POT: ( X t lim t x P s(t) > x X > t Example of application: ) = (1+γx) 1/γ, 0 < x < (0 ( γ)) 1. Rainfall data at L locations (rain gauge stations) s 1,...,s L, described by some r.vect. X R L verifying the domain of attraction condition. Estimate probabilities: given the relations with max-stable processes, we can use known statistical methods e.g. estimation of the exponent measure. (6)

12 The POT method - Application Estimation of P ( T n/k X A Tn/k X > 1 ) on the basis of n i.i.d. observations of X. 1. Obtain ˆγ n/k,â n/k,ˆb n/k. 2. Obtain the normalized processes ( X T n/k X j = 1+ ˆγ j ˆb n/k n/k â n/k 3. ) 1/ˆγ n/k P ( T n/k X A T n/k X > 1 ) = +, j = 1,...,n. n j=1 I ( T n/k X j A & X j ˆb n/k > 0) ( ). n j=1 I X j ˆb n/k > 0

13 References Balkema, A.A. and de Haan, L. (1974) Residual life time at great age. Ann. Probab. 2, Ferreira, A. and de Haan, L. (2013) The generalized Pareto process; with a view towards application and simulation. Bernoulli: to appear. Falk, M., Hüsler, J. and Reiss, R.-D. (2010) Laws of Small Numbers: Extremes and Rare Events, Birkhäuser, Springer. Michel, R. (2008) Some notes on multivariate generalized Pareto distributions. J. Multiv. Analysis 99, Pickands, J. III (1975) Statistical inference using extreme order statistics. Ann. Statist. 3, Rootzén, H. and Tajvidi, N. (2006) Multivariate generalized Pareto distributions. Bernoulli 12,