Basic Definitions and Notations The Bivariate Case The General Multivariate Case Estimation of the Angular Density in Multivariate Generalized Pareto Models René Michel michel@mathematik.uni-wuerzburg.de Institute of Applied Mathematics and Statistics University of Würzburg, Germany 18.08.2005 / EVA 2005 René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case The Multivariate Generalized Pareto Distribution Let X = ( X 1 , . . . , X d ) ∈ ( −∞ , 0 ) d , d ∈ N , be a random vector, which has a distribution function W with the representation � d � � � x 1 x d − 1 � W ( x ) = 1 + x i · D , . . . , � d � d i = 1 x i i = 1 x i i = 1 close to 0. Then X follows a generalized Pareto distribution (GPD) with uniform margins with the Pickands dependence function D : R d − 1 → [ 0 , 1 ] , R d − 1 being the unit simplex in R d − 1 . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case The Pickands Dependence Function D has the well known representation D ( t 1 , . . . , t d − 1 ) = � � d µ ( u ) max u 1 t 1 , . . . , u d − 1 t d − 1 , u d 1 − t i R d − 1 i ≤ d − 1 where u d = 1 − � i ≤ d − 1 u i with a measure µ on R d − 1 . We have µ ( R d − 1 ) = d . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case The Angular Density The distribution function L ( z 1 , . . . , z d − 1 ) = µ ([ 0 , z 1 ] × · · · × [ 0 , z d − 1 ]) of the measure µ is called angular distribution . If this measure possesses a density it is called the angular density l . Let ( X 1 , . . . , X d ) be a random vector following a GPD. Then D ( t 1 , . . . , t d − 1 ) = 1 ⇐ ⇒ µ ( { 0 } ) = 1 = µ { e i } �� 1 � d − 1 � d , . . . , 1 �� � D ( t ) = max t 1 , . . . , t d − 1 , 1 − t i ⇐ ⇒ µ = d d i = 1 with e i being the standard unit vectors in R d − 1 , the so called cases of independence and complete dependence . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Pickands Coordinates Let ( X 1 , X 2 ) be a bivariate random vector with X 1 < 0 , X 2 < 0 which is distributed by a GPD. The random variables Z := X 2 / ( X 1 + X 2 ) and C := X 1 + X 2 are the Pickands coordinates of ( X 1 , X 2 ) . Z is the angular component and C is the radial component . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Conditional Density of Z For any threshold c 0 close enough to 0 we have that the density of Z , conditional on C > c 0 , does not depend on c 0 . We denote this density by f ( z ) . One can show that f ( z ) g ( z ) := z ( 1 − z ) = constant · l ( z ) , l ( z ) being the angular density. René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Estimation of l Suppose now that we have n independent copies ( X 1 i , X 2 i ) of ( X 1 , X 2 ) and denote by Z i := X 2 i / ( X 1 i + X 2 i ) , C i := X 1 i + X 2 i the corresponding Pickands coordinates. Fix a threshold c 0 close to 0 and consider only those observations ( X 1 i , X 2 i ) with C i > c 0 . Denote these by (˜ X 11 , ˜ X 21 ) , . . . , (˜ X 1 m , ˜ X 2 m ) , where m is the random number of observations C i > c 0 . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Estimation of l A natural estimator of f is, therefore, the kernel density estimator with kernel function k and bandwidth h > 0 m � z − ˜ � Z j 1 ˆ � f m ( z ) := k . mh h j = 1 We can thus estimate a constant multiple of the angular density l by ˆ f m ( z ) ˆ g m ( z ) := z ( 1 − z ) . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Simulation Study Applying this estimator to 50 data points generated by a simulation algorithm for the logistic distribution (Michel 2004), whereby taking k to be the normal kernel and using automatic bandwidth selection we get quite good results. René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case The Case d > 2 The obvious generalization to d = 3 would be to compute threedimensional Pickands coordinates Z 1 := X 2 / ( X 1 + X 2 + X 3 ) , Z 2 := X 3 / ( X 1 + X 2 + X 3 ) and C := X 1 + X 2 + X 3 , take the density f ( z 1 , z 2 ) of ( Z 1 , Z 2 ) and set f ( z 1 , z 2 ) g ( z 1 , z 2 ) := z 1 z 2 ( 1 − z 1 − z 2 ) . But then we do NOT have g ( z 1 , z 2 ) = constant · l ( z 1 , z 2 ) . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Modified Pickands Coordinates Define instead the transformation T M ( x ) := 1 � 1 � , 1 + . . . + 1 x d − 1 x 1 , . . . , =: ( z M , c M ) . x 1 + . . . + 1 1 x 1 + . . . + 1 1 x 1 x d x d x d T M is called transformation to modified Pickands coordinates z M = ( z 1 , . . . , z d − 1 ) , c M . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case The Threshold Set A r , s We consider those coordinates under the condition that X ∈ A r , s , i.e., || X || ∞ < s and C M < − r . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Q r , s and χ ( r , s ) Set in addition � � d − 1 � � z i > 1 z i < 1 − 1 � � Q r , s := z ∈ R d − 1 rs , i = 1 , . . . , d − 1 , � � rs i = 1 and � χ ( r , s ) := l ( z ) dz . Q r , s Then Q r , s → r →∞ R d − 1 and χ ( r , s ) → r →∞ d . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Main Result Conditional on X ∈ A r , s the modified Pickands coordinate Z M has the density l ( z ) l ( z ) + O ( d − χ ( r , s )) → r →∞ d . d René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Convergence of χ The convergence of χ to d can be very different, see for example the logistic case 1 /λ � λ d − 1 d − 1 � � � t λ D λ ( t 1 , . . . , t d − 1 ) = i + 1 − t i , λ ∈ [ 1 , ∞ ) i = 1 i = 1 with λ = 6 and λ = 1 . 2. René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Estimation of l We can now estimate l ( z ) by a multivariate kernel density estimator with data sphering � � S − 1 / 2 z − Z ( i ) m 1 m M ˆ � , l m , r ( z ) = d · k ( det S m ) 1 / 2 mh d − 1 h i = 1 taking thereby only observations with X i ∈ A r , s and S m being the sample covariance matrix of the Z ( i ) M . Under suitable regularity conditions we have asymptotic normality of ˆ l m , r ( z ) for m → ∞ , r → ∞ . René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Simulation Study We show simulations of the estimator for various r for the logistic case with λ = 6 and λ = 1 . 2 in two and three dimensions. We begin with d = 2 and λ = 6, getting good and stable results. For d = 2 and λ = 1 . 2 we see varying results and slow convergence. René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Simulation Study For d = 3 and λ = 6 we get again good and stable results. But for d = 3 and λ = 1 . 2 we see again varying results and slow convergence. René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
Basic Definitions and Notations The Bivariate Case The General Multivariate Case Results If we are close to the case of complete dependence the estimator works fine, but close to the case of independence the estimator is only converging slowly to the desired result. This estimator was also used in Coles and Tawn (1991, 1994) and Coles, Heffernan & Tawn (1999) for the extreme value case. We see here that it is also applicable in the Generalized Pareto case. In the bivariate case a modified version of this estimator is able to deliver good results, independent of the threshold. René Michel michel@mathematik.uni-wuerzburg.de Estimation of the Angular Density
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