A Conditional Approach to Modeling Multivariate Extremes

Transcription

1 A Approach to ing Multivariate Extremes By Heffernan & Tawn Department of Statistics Purdue University s April 30, 2014

2 Outline s s

3 Multivariate Extremes s A central aim of multivariate extremes is trying to describe the structure of tail dependence

4 : componentwise maxima approach Consider X = (X 1,, X d ) be a d-dimensional random vector with distribution function F, X i = (X i1,, X id ), 1 i n be random sample from F. Define M nj = max i=1,,n X ij, if there exist sequences a nj > 0, b nj R, j = 1,, d such that ( Mn1 b n1 P a n1 x 1,, M nd b nd a nd ) x d G(x) s weakly as n, and marginal distributions G 1,, G d of G are non-degenerate, we say that F is in the domain of attraction of G Such G is called an multivariate extreme value distribution which is the limiting distribution function of the vector component-wise maxima of (X 1,, X n )

5 : componentwise maxima approach Consider X = (X 1,, X d ) be a d-dimensional random vector with distribution function F, X i = (X i1,, X id ), 1 i n be random sample from F. Define M nj = max i=1,,n X ij, if there exist sequences a nj > 0, b nj R, j = 1,, d such that ( Mn1 b n1 P a n1 x 1,, M nd b nd a nd ) x d G(x) s weakly as n, and marginal distributions G 1,, G d of G are non-degenerate, we say that F is in the domain of attraction of G Such G is called an multivariate extreme value distribution which is the limiting distribution function of the vector component-wise maxima of (X 1,, X n )

6 : componentwise maxima approach Consider X = (X 1,, X d ) be a d-dimensional random vector with distribution function F, X i = (X i1,, X id ), 1 i n be random sample from F. Define M nj = max i=1,,n X ij, if there exist sequences a nj > 0, b nj R, j = 1,, d such that ( Mn1 b n1 P a n1 x 1,, M nd b nd a nd ) x d G(x) s weakly as n, and marginal distributions G 1,, G d of G are non-degenerate, we say that F is in the domain of attraction of G Such G is called an multivariate extreme value distribution which is the limiting distribution function of the vector component-wise maxima of (X 1,, X n )

7 Componentwise maxima approach s + Extensive literature along this direction max-stable models asymptotic independent variables as exactly independent (Ledford & Tawn, 1996) Only allows investigation of joint tail

8 Componentwise maxima approach s + Extensive literature along this direction max-stable models asymptotic independent variables as exactly independent (Ledford & Tawn, 1996) Only allows investigation of joint tail

9 Componentwise maxima approach s + Extensive literature along this direction max-stable models asymptotic independent variables as exactly independent (Ledford & Tawn, 1996) Only allows investigation of joint tail

10 Motivating Problem: air quality data s Problem of interest: to study the extremal dependence structure among air pollutants and to estimate critical (extreme) functionals

11 Outline s s

12 Basic idea s 1. condition on one variable being extreme 2. examine behavior of remaining variables conditional on having am extreme component 3. Rationale: it is less likely that component-wise maxima occurs together e.g. NO and O 3 (in winter)

13 Characterizing the tail of a multivariate distribution s Set-up X = (X 1,, X d ); X i = (X 1,, X i 1, X i+1,, X d ) n i.i.d. observations of X from unknown distribution F To characterize the tail it requires P(X i > u i ) needs a univariate threshold X i X i > u i needs a marginal model X i X i > u i needs a dependence model

14 Marginal 1. GPD model for threshold exceedances, empirical distribution for non-extremes P (X i > x + u i X i > u i ) = P (X i u i ) = ˆF Xi (x) ( 1 + ξ i x ) β i + 1 ξ i for x > 0 for x u i 2. Apply probability [ integral ( )] transform Y i = log log ˆF Xi (x) to obtain approximately Gumbel distribution i.e. P (Y i y) exp (e y ) 3. Henceforth assume marginal distributions are exactly Gumbel and concentrate on dependence among Y 1,, Y d s

15 Marginal 1. GPD model for threshold exceedances, empirical distribution for non-extremes P (X i > x + u i X i > u i ) = P (X i u i ) = ˆF Xi (x) ( 1 + ξ i x ) β i + 1 ξ i for x > 0 for x u i 2. Apply probability [ integral ( )] transform Y i = log log ˆF Xi (x) to obtain approximately Gumbel distribution i.e. P (Y i y) exp (e y ) 3. Henceforth assume marginal distributions are exactly Gumbel and concentrate on dependence among Y 1,, Y d s

16 Marginal 1. GPD model for threshold exceedances, empirical distribution for non-extremes P (X i > x + u i X i > u i ) = P (X i u i ) = ˆF Xi (x) ( 1 + ξ i x ) β i + 1 ξ i for x > 0 for x u i 2. Apply probability [ integral ( )] transform Y i = log log ˆF Xi (x) to obtain approximately Gumbel distribution i.e. P (Y i y) exp (e y ) 3. Henceforth assume marginal distributions are exactly Gumbel and concentrate on dependence among Y 1,, Y d s

17 Existing techniques Most existing extreme value methods with Gumbel margins reduce to P (Y t + A) e t η Y P (Y A) for some η Y (0, 1] Ledford-Tawn classification η Y = 1: asymptotic dependence 1 d < η Y < 1: positive extremal dependence 0 < η Y < 1 d : negative extremal dependence η Y = 1 d : near extremal independence Disadvantage: doesn t work for extreme sets that are not simultaneously extreme in all components s

18 asymptotics for multivariate extremes s Define Y i = (Y 1,, Y i 1, Y i+1,, Y d ). For d-dimensional Y, consider for each i = 1,, d P (Y i y i Y i = y i ) as y i

19 Asymptotic assumptions s Assume there exist vector normalizing functions a i (y i ) : R R d 1 b i (y i ) : R R d 1 such that for any fixed z i and any sequence y i ( ) Y i lim P a i (y i ) z i Y i = y i = G y i b i (y i ) i (z i ) where G i has non-degenerate margins

20 Normalizing functions The authors examined a wide range of multivariate Extremal dependence η a(y) b(y) Perfect pos. dependence 1 y 1 Bivariate EVD 1 y 1 1+ρ Bivariate Normal (ρ > 0) 2 ρ 2 y y 1 2 Inverted logistic (α (0, 1]) 2 α 0 y 1 α 1 Independent ρ Bivariate Normal (ρ < 0) 2 log(ρ 2 y) y 1 2 Perfect neg. dependence 0 log(y) 1 s

21 Normalizing functions cont d s They found a i (y) and b i (y) fall in the parametric family: a i (y)= a i y + 1 {a i =0,b i <0}(c i d i log y) b i (y)= y b i There is no general form for G i or for its marginal distributions G j i

22 assumptions Asymptotic structure assumed to hold exactly above high threshold ( ) Y i a i (y i ) P z i Y i = y i = G b i (y i ) i (z i ) for y i > u i s Then Z i = Y i a i (y i ), for y i > u i b i (y i ) are assumed to follow G i and be independent of Y i Remarks: parametric forms for a i and b i nonparametric model for G i

23 Estimation of marginal parameters s Let ψ = (β = (β 1,, β d ), ξ = (ξ 1,, ξ d )) denotes parameters for individual GPD s Maximize log L(ψ) = n uxi d log ˆf Xi (x i i,k ) i=1 k=1 where u Xi is the number of threshold exceedances in i th component and ˆf Xi (x i i,k ) is the GP density evaluated at the k th exceedance.

24 Single conditional: estimation of θ i = (a i (y i ), b i (y i )) s The problem: don t know the distribution of Z i The solution: assume Z i has two finite marginal moments. This leads to µ i (y) = a i (y) + µ i b i (y) σ i (y) = σ i b i (y) Estimating equation Q i = n uyi [ log σ j i (y i i,k ) + 1 { } yj i,k µ j i (y i i,k ) 2 ] 2 σ j i k=1 j i (y i i,k )

25 All conditionals s To estimate all a i (y i ) and b i (y i ) jointly maximize Remarks: Q = d i=1 Assume independence between conditional distributions Analogous with pseudolikelihood estimation (Besag 1975) Q i

26 Uncertainty s Uncertainty comes from semiparametric marginal models parametric normalization functions of conditional dependence structure nonparametric models for Z i Bootstrap

27 Outline s s

28 monitoring s Data: daily values of five air pollutants (O 3, NO 2, NO, SO 2, PM 10 ) in Leeds, U.K., during Two seasons: winter (NDJF) and early summer (AMJJ) Problem of interest: to study the extremal dependence among air pollutants and to estimate critical functionals

29 Transforming data to known margins s Transform variable X to have standard Gumbel marginal distributions

30 data extrapolation PM 10 and NO: s

31 approach: Summary s accommodates any functional of multivariate extreme events handles asymptotic dependence and asymptotic independence including negative dependence not just model bivariate extremes

32 s

33 s Besag, J. Statistical analysis of non-lattice data The statistician, , 1975 Heffernan J. E. & Tawn J. A. A conditional approach for multivariate extreme values (with Discussion) J. R. Statist. Soc. B, , 2004 Ledford, A. W., & Tawn, J. A. Statistics for near independence in multivariate extreme values Biometrika, , 1996

34 Asymptotic (in)dependent s Definition A bivariate random variable (X 1, X 2 ) is called asymptotically independent if λ = lim x x F P (X 2 > x X 1 > x) = 0 where (X 1, X 2 ) are identically distributed with x F = sup x R {x : P(X 1 x) < 1}. If λ > 0, then (X 1, X 2 ) is called asymptotic dependent