Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Implicit Extremes and Implicit Max–Stable Laws Stilian Stoev ( sstoev@umich.edu ) University of Michigan, Ann Arbor September 19, 2014 Joint work with Hans-Peter Scheffler (University of Siegen). 1/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Problem Formulation 1 Limit Theory 2 Implicit Extreme Value Laws 3 Implicit Max–Stable Laws and their Domains of Attraction 4 An Example 5 2/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Problem Formulation 3/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Setup Let X 1 , . . . , X n be iid vectors in R d . They are hidden, i.e., unobserved. Observed are f ( X 1 ) , . . . , f ( X n ) for some loss function f : R d → [0 , ∞ ) . We want to know what is the behavior of the scenario that maximizes the loss X k ( n ) , where k ( n ) = Argmax k =1 ,..., n f ( X k ) . We refer to X k ( n ) as to the implicit extreme relative to the loss f . 4/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example First observations If the law of f ( X i ) is continuous, with probability one, there are no ties among f ( X 1 ) , . . . , f ( X n ) In the case of ties (discontinuous L ( f ( X i ))), k ( n ) is taken as the smallest index maximizing the losses f ( X i ) , i = 1 , . . . , n . The motivation stems from applications: We are interested in the structure of the complex (multivariate) events modeled by X i ’s that lead to extreme losses. These implicit extremes, depending on the loss function f , may or may not be associated with extreme values of the X i ’s... General Perspective: We are interested in the structure of events leading to extreme losses! 5/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example The simple Lemma that started it all Lemma Suppose the cdf G ( y ) := P ( f ( X 1 ) ≤ y ) is continuous. Then, for all measurable A ⊂ R , � G ( f ( x )) n − 1 P X ( d x ) P ( X k ( n ) ∈ A ) = n A Proof. There are no ties, a.s., and by symmetry and independence: P ( X k ( n ) ∈ A ) = nP ( X 1 ∈ A , f ( X i ) ≤ f ( X 1 ) , i = 2 , . . . , n ) � P ( f ( X 2 ) ≤ f ( x )) n − 1 P X ( d x ) . = n A Note: We can handle the general case of discontinuous G . 6/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Limit Theory 7/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Assumptions Homogeneous losses: The loss is non-negative f : R d → [0 , ∞ ) and f ( cx ) = cf ( x ) , for all c > 0 . This is not a terrible constraint, since Argmax ( f ( X 1 ) , . . . , f ( X n )) = Argmax ( h ◦ f ( X 1 ) , . . . , h ◦ f ( X n )) , for any strictly increasing h : [0 , ∞ ) → [ −∞ , ∞ ). d Regular variation on a cone: P X ∈ RV ( { a n } , D , ν ), where D ⊂ R is a closed cone, playing the role of zero. That is, v nP ( a − 1 n X ∈ · ) − → ν, as n → ∞ , d \ D . d in the space R D := R d This is an important generalization of the usual RV on R { 0 } . Note RV ( { a n } , { 0 } , ν ) ⊂ RV ( { a n } , D , ν ). However, the generalized notion of RV allows us to handle cases that are asymptotically trivial in the classical sense. Similar (but not the same as) Sid Resnick’s hidden regular variation. 8/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example RV on cones and generalized polar coordinates d := [ −∞ , ∞ ] d . Consider the compact space R Let τ : R d → [0 , ∞ ] be a continuous and homogeneous function. Define D := { τ = 0 } (necessarily) a compact in R d . d \ D with the relative topology. d Equip R D := R d that are d The compacts in R D are closed subsets of R d bounded away from D = { τ = 0 } . That is, K ⊂ R D is compact if it is closed and K ⊂ { τ > ǫ } , for some ǫ > 0. Polar coordinates: Let θ ( x ) := x /τ ( x ). Then d ( τ, θ ) : R D → (0 , ∞ ] × S , is a homeomorphism (of topological spaces), where d : τ ( x ) = 1 } S = { τ = 1 } = { x ∈ R is equipped with the relative topology. 9/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Regular variation Definition A probability law P X ∈ RV ( { a n } , D , ν ), if there exists a regularly varying d sequence { a n } and a non-trivial Radon measure ν on R D , such that nP X ( a − 1 n X ∈ A ) − → ν ( A ) , as n → ∞ , for all measurable A , bounded away from D , i.e., A ⊂ { τ > ǫ } , for some ǫ > 0, and such that ν ( ∂ A ) = 0. Fact (Prop 3.8 in Scheffler & Stoev (2014)) P X ∈ RV ( { a n } , D , ν ) , if and only if nP ( a − 1 n τ ( X ) > x ) → n →∞ Cx − α and w P ( θ ( X ) ∈ ·| τ ( X ) > u ) − → u →∞ σ 0 ( · ) , where σ 0 is a finite measure on S. 10/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Example Cone and coors: Let τ ( x ) = min { ( x 1 ) + , . . . , ( x d ) + } so that d D = (0 , ∞ ] d . R The unit “sphere”, is now: i =1 [1 , ∞ ] i − 1 × { 1 } × [1 , ∞ ] d − i S := { x : τ ( x ) = 1 } = ∪ d Distribution: Let X = ( X i ) d i =1 with independent and Pareto X i ’s P ( X i > x ) = x − α i , ( α i > 0) , i = 1 , . . . , d . 11/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Example (cont’d) The classic RV: In R d { 0 } , we have asymptotic independence and the heaviest tail dominates: nP ( n − 1 /α ∗ X ∈ A ) − → µ ( A ) , where α ∗ := i =1 ,..., d α i , min and d � µ ( A ) = I { α i = α ∗ } ν i ,α ∗ ( A ) , i =1 where ν i ,α is concentrated on the positive part of the i -th axis and ν i ,α ( R i − 1 × [ x , ∞ ) × R d − i ) = x − α , x ≥ 0 . That is, the limit measure µ lives on the axes. 12/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Example (cont’d) The cone R d D := (0 , ∞ ] d : Then, X ∈ RV ( { n − 1 /α } , D , ν ), with α = α 1 + · · · + α d and where now ν lives on (0 , ∞ ) d and now has a density! Indeed, for A = ( x 1 , ∞ ] × · · · × ( x d , ∞ ] ⊂ (0 , ∞ ] d , d � P ( a − 1 n X ∈ A ) ∼ P ( X i > a n x i ) i =1 d � ( a n x i ) − α i =: a − α = ν ( A ) . n i =1 n X ∈ · ) → v ν ( · ), where By picking a n := n − 1 /α , we obtain nP ( a − 1 d d ν � α i x − α i − 1 x = ( x i ) d i =1 ∈ (0 , ∞ ] d . d x ( x ) = , i i =1 13/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Implicit Extreme Value Laws Assumptions: (RV α ) X ∈ RV α ( { a n } , D , ν ) d → [0 , ∞ ] is Borel, 1-homogeneous, f (0) = 0. (H) f : R (F) For all ǫ > 0, the set { f > ǫ } is bounded away from D and d x ∈ K f ( x ) > 0 , inf for all compact K ⊂ R D . (C) ν ( disc ( f )) = 0. Theorem (3.13 in Scheffler & Stoev (2014)) Under the above assumptions, we have 1 d X k ( n ) − → Y , as n → ∞ , a n where P Y ( dx ) = e − Cf ( x ) − α ν ( d x ) and C := ν { f > 1 } . 14/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Sketch of the proof By the above lemma, we have � P ( f ( X ) ≤ f ( x )) n − 1 P X ( d x ) P ( X k ( n ) ∈ a n A ) = n a n A � P ( f ( X ) ≤ f ( a n z ) n − 1 nP a − 1 = X ( d z ) (change of vars) n A � P ( f ( a − 1 n X ) ≤ f ( z )) n − 1 ν n ( dz ) = (homogeneity of f ) A where X ( d z ) ≡ nP ( a − 1 ν n ( d z ) := nP a − 1 n X ∈ d z ) . n Continuing... � 1 − nP ( f ( a − 1 � n X ) > f ( z ) � n − 1 P ( X k ( n ) ∈ a n A ) = ν n ( d z ) . n A 15/34
Problem Formulation Limit Theory Implicit Extreme Value Laws Domains of Attraction An Example Sketch of the proof (cont’d) � 1 − nP ( f ( a − 1 � � n − 1 n X ) > f ( z ) P ( X k ( n ) ∈ a n A ) = ν n ( d z ) n A � 1 − nP ( a − 1 � � n − 1 n X ∈ { f > f ( z ) } = ν n ( d z ) . n A The set B z := { f > f ( z ) } is bounded away from D . If it is a continuity set of ν , by the (RV α ) and ( H ) assumptions: nP ( a − 1 nP ( a − 1 n X ∈ B z ) = n X ∈ { f > f ( z ) } − → ν ( { f > f ( z ) } ) ν ( f ( z ) · { f > 1 } ) = f ( z ) − α ν { f > 1 } . = Since by (RV α ), we also have ν n → v ν , it can be shown that � e − ν { f > 1 } f ( z ) − α ν ( d z ) , P ( X k ( n ) ∈ a n A ) − → A for all ν -continuity sets A . ✷ 16/34
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