Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution

Transcription

1 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. /2 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution Dabao Zhang Department of Statistics Purdue University Joint Work with Martin T. Wells and Liang Peng

2 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 2/2 Water Levels Around Lake Ontario Here we explore the dependence structures among the water levels observed in Cape Vincent, Niagara Intake, Oswego and Rochester; The annual maximum/minimum water levels were recorded from 963 to 25, respectively

3 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 3/2 Annual Maximum Water Levels 73 Annual Maximum Water Level Niagara Rochester Oswego Cape Vincent year Table : (Pearson s ρ, Kendall s τ, Spearman s ρ) Niagara Rochester Oswego Cape Vincent Niagara (.7,.4,.2) (.,.7,.8) (.2,.3,.2) Rochester (.98,.85,.95) (.95,.83,.94) Oswego (.95,.79,.93)

4 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 4/2 Annual Minimum Water Levels 7 Annual Minimum Water Level Niagara Rochester Oswego Cape Vincent year Table 2: (Pearson s ρ, Kendall s τ, Spearman s ρ) Niagara Rochester Oswego Cape Vincent Niagara (.43,.5,.2) (.39,.,.4) (.46,.7,.24) Rochester (.94,.73,.88) (.85,.52,.66) Oswego (.85,.56,.72)

5 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 5/2 Outline Extreme Values and Their Distributions Review on Bivariate Dependence Function Estimates Multivariate Dependence Function Estimates Simulation Study Application to the Water Level Data

6 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 6/2 Extreme Values and Their Distributions Let Z, Z 2,, Z n be a sequence of iid random variables, and Y n = max{z, Z 2,, Z n }. Gnedenko (948): If there exist a n (, ), b n R, and non-degenerate distribution function F(x) such that lim P n ( Yn b n a n ) y = F(y), then, there exist µ R, σ (, ) and ξ R such that { ( F(y) = exp + ξ y µ ) } /ξ, ξ y µ >. σ σ

7 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 7/2 Three types of extreme value distributions: The Gumbel family F(y) = exp { ( exp y µ )}, < y < ; σ The Fréchet family F(y) =, { ) } α y µ, exp, y > µ; ( y µ σ The Webull family F(y) = exp { ( ) } α y µ, y < µ, σ, y µ.

8 Example: Protein Sequence Alignment Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 8/2

9 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 9/2 EVD vs. Normal Distribution Noraml Distribution.2.5 EVD

10 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. /2 Multivariate EVD Assume that each X i follows a EVD F i (x), i =, 2,, p; The joint distribution of (X, X 2,, X p ) is C(F (x ),, F p (x p )), where C is called the copula; Pickands (98) & Tawn (99): the copula C depends on a (p )-dimentional function A(s, s 2,, s p ), C(u, u 2,, u p ) { p = exp log u k A ( k= log u p k= log u,, k ) } log u p p k= log u. k

11 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. /2 The Dependence Function A The dependence function A(s,, s p ) is defined on the (p )-dimensional unit simplex S p = {(s,, s p ) : s k, k p ; p k= max(s, s 2,, s p ) A(s, s 2,, s p ) ; A(s, s 2,, s p ) is convex; A(s, s 2,, s p ) implies that each component of (X, X 2,, X p ) is independent of all the others; s k }. A(s, s 2,, s p ) = max(s, s 2,, s p ) implies that F (X ) F 2 (X 2 ) F p (X p ).

12 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 2/2 Two Parametric Models for A Symmetric Logistic Model (Gumbel, 96) A(s,, s p ) = { p i= p s r i + ( i= s i ) r }/r, r ; Asymmetric Logistic Model (Town, 99) A(s,, s p ) = c S { /rc (θ i,c s i ) c} r ; i c where θ i,c [, ] and the set S is the class of all nonempty subsets of {,, p}. Note that this model has 2 p (p + 2) (2p + ) parameters. Here we are interested in nonparametric estimate of A!

13 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 3/2 Review: Estimate A When p = 2 Data: iid bivariate extreme values, (X i, X i2 ), i n. Transformation: Y ik = log{f k (X ik )}, k =, 2. Pickands Estimator (Pickands, 98) Â P (s) = n n i= min(y i, Y i2 ); s s HT Estimator (Hall and Tajvidi, 2) Â HT (s) = n n i= min(y i/ŷ s, Y i2/ŷ 2 s ); where Ŷ = n i= Y / i n, Ŷ 2 = n i= Y i2/ n.

14 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 4/2 CFG Estimator (Caperaa, Fougeres and Genest, 997) Ĥ n (z): the empirical distribution of Z i = Y i Y i +Y i2, i =, 2,, n; Two different estimators: { t log  (s) = exp log  (s) = exp CFG Estimator: { } Ĥ n (z) z z( z) dz t Ĥ n (z) z z( z) dz log Â(s) = λ(s) log  (s) + { λ(s)} log  (s); One Choice: λ(s) = s., } ;

15 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 5/2 x x Comparison of results using HT Estimatior, CFG Estimator and the New Estimator

16 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 6/2 Straightforward Extensions For any (s, s 2,, s p ) S p, let s p = p k= s k. Denote Y ik = log{f k (X ik )}; Extension of Pickands Estimator: Â P (s, s 2,, s p ) = n i= n p j= Y ij s j. Extension of HT Estimator: Â HT (s, s 2,, s p ) = n i= n p j=. Y ij /Ŷ j s j New Estimator: Â ZW (s, s 2,, s p ) = p k= ( n i= Y ik) s k n i= p k= Y ik/s k.

17 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 7/2 Extension of CFG Estimator Proposition: Let (Y, Y 2,, Y p ) be marginally distributed with standard exponential distributions and their dependence function be A(s,, s p ). Then Z k = follows the distribution function H k (z) = z + z( z) l k Y l s l Y k s k + l k z log A ( zs s k,, zs k s k, z, zs k+ s k,, zs p s k Y l s l ). Therefore, log A(s,, s p ) = sk H k (z) z z( z) dz.

18 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 8/2 Ĥ k : the empirical distribution function of H k ; p different nonparametric estimators for the dependence function A, log{â k (s,, s p )} = sk Ĥ k (z) z z( z) dz; Extension of CFG Estimator: p Â(s,, s p ) = Â k (s,, s p ) λ k(s,,s p ), k= where p k= λ k(s,, s p ) = ; One Choice: λ k (s,, s p ) = s k.

19 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 9/2 Let B(u,, u p ) denote a Gaussian process on [, ] p with E{B(u,, u p )} = and, E{B(u,, u p )B(v,, v p )} = C(u v,, u p v p ) C(u,, u p )C(v,, v p ). Let B j (u) denote B(u,, u p ) with u j = u and u i = for all i j. Theorem: For any fixed (s,, s p ) S p, we have and sup Â(s,, s p ) A(s,, s p ) (s,,s p ) S p n{log Â(s,, s p ) log A(s,, s p )} p, d p λ j (s,, s p ) j= sj B j (z) z( z) dz.

20 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 2/2 Simulation Study Mean Integrated Square Errors (MISE) and Maximum Mean Square Errors (MMSE) in Simulation Study: the results are for a symmetric logistic dependence function (SLDF) and an asymmetric logistic dependence function (ALDF). n = 25 n = 5 n = n = 2 MISE 5 SLDF ALDF SLDF ALDF SLDF ALDF SLDF ALDF Â P Â D Â HT Â MMSE 5 Â P Â D Â HT Â

21 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 2/2 Application to the Water Level Data Here we explore the dependence structures among the water levels observed in Cape Vincent, Niagara Intake, Oswego and Rochester; The annual maximum/minimum water levels were recorded from 963 to 25, respectively

22 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 22/2 Annual Maximum Values Annual Minimum Values.9.9 A(s,s 2 ).8.7 A(s,s 2 ) s (Oswego, Niagara, Rochester).2.4 s s (Oswego, Niagara, Rochester).2.4 s A(s,s 2 ) A(s,s 2 ) s2.2.5 s s s.6.8 (Oswego, Rochester, Cape Vincent) (Oswego, Rochester, Cape Vincent)

23 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 23/2 Future Work? Taking block maxima is a wasteful approach! Alternative approach: Using the values that exceed some high threshold. Assume X F with F in the extreme value domain of attraction. Then, for large enough u, P(Y = X u < y X > u) H(y) = ( + ξy σ ) /ξ How to choose the threshold u? Other problems: parametric models to describe tail behaviors of multivariate variables! Risk Analysis: Environmental Risk (e.g., extreme rainfall extreme, Southern California wildfire, global warming); Financial Risk (e.g., insurance claims, returns on stock prices); Bioinformatics?

24 Thank You! Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. 24/2