Statistics of Extremes

Transcription

1 Statistics of Extremes Anthony Davison c 29 Multivariate Extremes 184 Europe Componentwise maxima Standardization Limit distribution Special cases Parametric models Asymmetric models Dependence measures Multivariate case

2 Multivariate Extremes slide 184 European heatwave 23 The 23 European heat wave was one of the hottest summers on record in Europe, especially in France. Temperatures exceeded 4 in many places in northern Europe, and reached almost 5 C in Portugal. It led to health crises in several countries and combined with drought to create a crop shortfall in Southern Europe. It has been linked to forest fires in Portugal, to dried-up rivers, to buckling of train track in the UK, and to flash floods in the Alps. Over 35, more people than usual are thought to have died in Europe that summer. Some scenarios of climate change predict increases of up to 7 C in summer temperatures in northern Europe over the next century. How can we predict their likely consequences? Statistics of extremes Autumn 29 slide 185 Europe 23 Statistics of extremes Autumn 29 slide

3 Sites 4 altitude in meters Basel Binningen (316) Neuchatel (485) Chateau d'oex (985) Montreux Clarens (45) Gd St Bernard (2472) Oeschberg Koppigen (483) Bern Liebefeld (565) Montana (158) 3 Jungfraujoch (358) 3 3 Zurich MeteoSchweiz (556) Engelberg (135) Locarno Monti (366) Lugano (273) Santis (249) Bad Ragaz (496) Davos Dorf (159) Arosa (184) 3 Statistics of extremes Autumn 29 slide 187 Swiss summer temperatures Maximum temperature: June, July, August, 2125 Jungfraujoch (358 m) Santis (249 m) GdStBernard (2472 m) Arosa (184 m) DavosDorf (159 m) Montana (158 m) Temperature anomaly (degrees Celsius) Engelberg (135 m) Chateau d'oex (985 m) BernLiebefeld (565 m) ZurichMeteoSchweiz (556 m) Bad Ragaz (496 m) Neuchatel (485 m) OeschbergKoppigen (483 m) MontreuxClarens (45 m) LocarnoMonti (366 m) BaselBinningen (316 m) Lugano (273 m) Statistics of extremes Autumn 29 slide

4 Zurich, 28 The Basel Committee rules regulate the behaviour of banks and other financial institutions Part of this involves estimating a quantity known as the Value at Risk, which can be viewed mathematically as an extreme quantile of a distribution (though this is far from being the entire story!) How should this be estimated from a series of daily percent returns, defined as where P t is the closing price on day t? Y t = 1 log(p t /P t 1 ), How to measure the risk to a financial system of big changes in the values of several companies simultaneously? Statistics of extremes Autumn 29 slide 189 UBS and Credit Suisse Statistics of extremes Autumn 29 slide

5 UBS and Credit Suisse Statistics of extremes Autumn 29 slide 191 UBS and Credit Suisse Statistics of extremes Autumn 29 slide

6 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, 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 29 slide 193 Componentwise maxima If (X 1,Y 1 ),(X 2,Y 2 ),... iid F(x,y), define The vector of componentwise maxima is M X,n = max j=1,...,n {X j} and M Y,n = max j=1,...,n {Y j}. M n = (M X,n,M Y,n ); this 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 29 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 29 slide

7 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 ) = G(z1, z2 ), z1, z2 > ; note that G(z, ) = G(, z) = exp( 1/z), for z >, and that for any t >, Gt (tz1, tz2 ) = G(z1, z2 ), z1, z2 >, by a straightforward extension of the argument for univariate max-stability. Statistics of extremes Autumn 29 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 29 slide

8 Limit distribution of componentwise maxima Theorem 28 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 G is a non-degenerate distribution function, then where we can write D G(z 1,z 2 ), z 1,z 2 >, G(z 1,z 2 ) = exp{ V (z 1,z 2 )}, z 1,z 2 >, V (z 1,z 2 ) = 2 1 ( w max, 1 w ) dh(w), z 1 z 2 and H is a distribution function on [,1] satisfying the mean constraint If H is differentiable with density h, then V (z 1,z 2 ) = wdh(w) = 1/2. ( w max, 1 w ) h(w)dw. z 1 z 2 Statistics of extremes Autumn 29 slide 198 Special cases Two important special cases of the above results: Independence: When H is a measure with masses 1/2 on w = and w = 1, then G(z 1,z 2 ) = exp{ (z z 1 2 )}, z 1,z 2 >. Perfect dependence: When H is a measure that places unit mass on w = 1/2, then G(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, try to specify a parametric family for H or h that can encompasses a wide range of dependence types but not too wide, or it becomes impossible to estimate. Statistics of extremes Autumn 29 slide

9 Parametric models It is not easy to formulate parametric models that satisfy the mean constraints, particularly in higher dimensions. Several different models exist for bivariate data, however. Example 29 Simple widely-used model is the logistic, for which h(w) = 1 2 (α 1 1){w(1 w)} 1 1/α {w 1/α + (1 w) 1/α } α 2, for < w < 1 and < α 1. In this case { ( ) G(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 29 slide 2 Asymmetric models Asymmetric alternatives to the logistic model include: the bilogistic model h(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 = ; h(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, Statistics of extremes Autumn 29 slide

10 Dependence measures A simple summary of dependence is the extremal coefficient: θ = zv (z,z) = 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 ). 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 is convex; 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; and may be written as A(t) = 1 t + t H([,w])dw, t 1. Statistics of extremes Autumn 29 slide 22 Limit distribution of componentwise maxima Theorem 3 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 G is a non-degenerate distribution function, then where we can write D G(z 1,...,z D ), z 1,...,z D >, G(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 and H is a distribution function on the D-dimensional simplex z d ) dh(w 1,...,w D ), S D = {(w 1,...,w D ) : w d = 1,w d } satisfying the mean constraints S D w d dh(w 1,...,w D ) = 1/D, d = 1,...,D. Statistics of extremes Autumn 29 slide

11 Summary As in the scalar case, max-stability imposes a strong constraint on 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 very 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 29 slide