Statistics of Extremes

Transcription

1 Statistics of Extremes Anthony Davison c Multivariate Extremes 19 Componentwise maxima Standardization Limit distribution Special cases Parametric models Dependence measures Multivariate case

2 Multivariate Extremes slide 19 UBS and Credit Suisse Statistics of extremes Autumn 211 slide 191 UBS and Credit Suisse Statistics of extremes Autumn 211 slide

3 Multivariate extremes Many extremal problems are essentially multivariate in nature. Lack of data means the precision of extreme value estimates is often poor, so we often want to incorporate additional information in fitting/estimation, suggesting the use of multivariate models. Questions include: What issues are important when contemplating multivariate extremes? What are appropriate ways to summarize dependence in extremes? What models are suggested by asymptotic theory? How can inference be performed? Statistics of extremes Autumn 211 slide 193 Componentwise maxima If (X 1,Y 1 ),(X 2,Y 2 ),... iid F(x,y), define M X,n = max j=1,...,n {X j}, M Y,n = max j=1,...,n {Y j}. The vector of componentwise maxima, M n = (M X,n,M Y,n ), may not correspond to an actual observation and might even be physically impossible! The asymptotic theory of multivariate extremes begins with an analysis of M n as n, but it is helpful also to consider the point process of rescaled points (X j,y j ) {X j } and {Y j } considered separately are sequences of independent, univariate random variables, to which the earlier theory may be applied. Statistics of extremes Autumn 211 slide 194 Models for multivariate extremes In extension of the univariate case, we ask: If non-degenerate limiting distributions exist for maxima of rescaled pairs (X 1,Y 1 ),...,(X n,y n ) as n, then what forms can they have? This presupposes that limiting distributions exist for the rescaled margins individually, because otherwise any limiting joint distribution will be degenerate. Strategy: rescale the margins to have a standard (unit Fréchet) form; show that if a non-degenerate joint limiting distribution exists, it must be max-stable; show that the max-stable distributions have a particular (nonparametric) form. We use both maximum and point process arguments, in order to unify the discussion. Statistics of extremes Autumn 211 slide

4 Marginal standardization Suppose that sequences {an } > and {bn } exist such that as n, D a 1 n {max(x1,..., Xn ) bn } X GEV(, 1, ξx ). Then 1/ξX Pr(X x) = exp{ (1 + ξx x)+ 1/ξX so Z1 = (1 + ξx X )+ }, ξx R, has a unit Fre chet distribution, F (z) = exp( 1/z), z >. A similar argument holds for max Yj, yielding a unit Fre chet variable Z2. These transformations also apply to the point process {a 1 n (Xj bn ) : j = 1,..., n} for the Xs and the corresponding point process for the Y s. Now suppose that (Z1, Z2 ) has a limiting distribution Pr(Z1 z1, Z2 z2 ) = H(z1, z2 ), z1, z2 > ; note that H(z, ) = H(, z) = exp( 1/z), for z >, and that for any t >, H t (tz1, tz2 ) = H(z1, z2 ), z1, z2 >, by a straightforward extension of the argument for univariate max-stability. In this case the multivariate extreme-value distribution is sometimes called simple. Statistics of extremes Autumn 211 slide 196 Bivariate normal data Left: n = 1, bivariate normal observations with correlation ρ =.9; centre: transformed to unit Fre chet scale; right: with log axes. Transformed to unit Frechet Transformed to unit Frechet 1e+3 z2 1e e+1 y z Bivariate normal, rho= x z1 Statistics of extremes 5 1e 1 1e+1 1e+3 z1 Autumn 211 slide

5 Limit distribution of componentwise maxima Theorem 31 Let (Z 1,Z 2 ) be the linearly rescaled componentwise maxima of independent vectors (X j,y j ), transformed to have limiting unit Fréchet marginal distributions. If Pr(Z 1 z 2,Z 2 z 2 ) where H is a non-degenerate distribution function, then D H(z 1,z 2 ), z 1,z 2 >, H(z 1,z 2 ) = exp{ V(z 1,z 2 )}, z 1,z 2 >, where the function V is called the exponent measure and we can write 1 ( w V(z 1,z 2 ) = 2 max, 1 w ) dq(w), z 1 z 2 where Q, a spectral distribution function on [, 1], satisfies the mean constraint 1 wdq(w) = 1/2. If Q is differentiable with spectral density function q, then 1 ( w V(z 1,z 2 ) = 2 max, 1 w ) q(w) dw. z 1 z 2 Statistics of extremes Autumn 211 slide 198 Special cases Two important special cases of the above results: Independence: When Q is a measure with masses 1/2 on w = and w = 1, then H(z 1,z 2 ) = exp{ (z 1 1 +z 1 2 )}, z 1,z 2 >. Perfect dependence: When Q is a measure that places unit mass on w = 1/2, then H(z 1,z 2 ) = exp{ max(z 1 1,z 1 2 )}, z 1,z 2 >. which is the distribution function of variables that are marginally standard Fréchet, but which are perfectly dependent: Z 1 = Z 2 with probability one. For most modelling purposes, we aim to specify a parametric family for Q or q that encompasses a wide range of dependence types but not too wide, or it becomes impossible to estimate. Statistics of extremes Autumn 211 slide

6 Parametric models It is not easy to formulate parametric models that satisfy the mean constraints, particularly in higher dimensions. A variety of models exist for bivariate data, however. Example 32 A simple widely-used model is the logistic, for which q(w) = 1 2 (α 1 1){w(1 w)} 1 1/α {w 1/α +(1 w) 1/α } α 2, < w < 1, < α 1. In this case { ( ) H(z 1,z 2 ) = exp z 1/α 1 +z 1/α α } 2, z 1,z 2 >. Independence and perfect dependence arise as limits as α 1 and α respectively. A limitation of this model is its symmetry. Statistics of extremes Autumn 211 slide 2 Logistic and Dirichlet densities Left: logistic density functions with α =.1 (black),.3 (red),.5 (blue),.9 (green). Right: Dirichlet densities with parameters (α, β) = (.5,.5) (black), (.5, 1) (red), (.5, 2) (blue) and (2,3) (green). Density Density w w Statistics of extremes Autumn 211 slide

7 Asymmetric parametric models Asymmetric alternatives to the logistic model include: the bilogistic model q(w) = 1 2 (1 α)(1 w) 1 w 2 (1 u)u 1 α {α(1 u)+βu} 1, < w < 1, where < α,β < 1, and u = u(w,α,β) satisfies and the Dirichlet model for parameters α,β >. (1 α)(1 w)(1 u) β (1 β)wu α = ; q(w) = αβγ(α+β +1)(αw)α 1 {β(1 w)} β 1 2Γ(α)Γ(β){αw +β(1 w)} α+β+1, < w < 1, The function fbvevd (see also dbvevd) in the R library evd allows fitting of the logistic, asymmetric logistic, Husler Reiss, negative logistic, asymmetric negative logistic, bilogistic, negative bilogistic, Coles Tawn and asymmetric mixed models, so there are plenty of parametric bivariate densities. Statistics of extremes Autumn 211 slide 22 Extremal coefficient A simple summary of dependence is the extremal coefficient: θ = V(1,1), which satisfies θ = 1 for perfectly dependent data, and θ = 2 for independent data, and is (loosely) interpreted as the number of independent maxima contributing to (Z 1,Z 2 ), because Pr{max(Z 1,Z 2 ) z} = Pr(Z 1 z,z 2 z) = exp{ V(z,z)} = exp {V(1,1)/z}, z >, which is a Fréchet distribution with parameter θ = V(1,1). θ also has an interpretation in terms of limiting conditional probabilities of rare events: lim Pr(Z 1 > z Z 2 > z) = 2 θ, θ 1. z Statistics of extremes Autumn 211 slide

8 Pickands dependence function A richer summary is Pickands dependence function A, determined by ( ) z1 V(z 1,z 2 ) = (1/z 1 +1/z 2 )A, z 1 +z 2 which satisfies max(t,1 t) A(t) 1 for t [,1]; satisfies A(t) = 1 for independent data, and A(t) = max(t,1 t) for perfectly dependent data; is convex; and may be written as A(t) = 1 t+2 t Q([,w])dw, t 1. This last formula enables one to compute Q from A, since { 1+A (w), w < 1, Q([,w]) = 2, w = 1, where A is the right-hand derivative of A. Further differentiation gives q, if it exists. Statistics of extremes Autumn 211 slide 24 Pickands dependence functions Left: for logistic density with α =.1 (black),.3 (red),.5 (blue),.9 (green). Right: for Dirichlet density with parameters (α, β) = (.5,.5) (black), (.5, 1) (red), (.5, 2) (blue) and (2,3) (green). A(t) A(t) t t Statistics of extremes Autumn 211 slide

9 Limit distribution of componentwise maxima Theorem 33 Let (Z 1,...,Z D ) be the linearly rescaled componentwise maxima of independent vectors transformed to have limiting unit Fréchet marginal distributions. If Pr(Z 1 z 2,...,Z D z D ) where H is a non-degenerate distribution function, then where we can write D H(z 1,...,z D ), z 1,...,z D >, H(z 1,...,z D ) = exp{ V(z 1,...,z D )}, z 1,...,z D >, V(z 1,...,z D ) = D max S D d ( wd z d ) dq(w 1,...,w D ), and Q, a spectral distribution function on the D-dimensional simplex { } D S D = (w 1,...,w D ) : w d = 1,w d, satisfies the mean constraints d=1 S D w d dq(w 1,...,w D ) = 1/D, d = 1,...,D. Statistics of extremes Autumn 211 slide 26 Summary As in the scalar case, max-stability constrains the possible limiting distributions for bi- and multi-variate maxima. After transformation of the margins to a standard form, the joint distributions have a nonparametric structure subject to mean constraints. There are numerous parametric models in the bivariate case, but few in higher dimensions. Dependence measures exist: in particular the extremal coefficient is a single-number summary of dependence that satisfies θ = 1 for fully dependent data and θ = D for independent data. Statistics of extremes Autumn 211 slide