preamble The generalized Pareto process (?); characterizations and - PowerPoint PPT Presentation

preamble

The generalized Pareto process (?); characterizations and properties Ana Ferreira, Instituto Superior de Agronomia, UTL and CEAUL Laurens de Haan, Erasmus University Rotterdam and CEAUL Workshop on spatial extreme value theory and properties of max-stable random fields University of Poitiers, November 9, 2012, France Poitiers 2012 theslide – 1 / 12

Motivation ⊲ Motivation A stochastic process X in C ( S ) = { f : S → R , f continuous } , S ⊂ R d compact, is Simple GP process in the maximum domain of attraction of a max-stable process if, Finite dimensions GP process ∃ a n ( s ) > 0 and b n ( s ) ∈ R , continuous functions on S, such that Regular variation References � X i ( s ) − b n ( s ) � max , (1) a n ( s ) 1 ≤ i ≤ n s ∈ S X, X 1 , . . . , X n i.i.d., converges weakly in C ( S ) ; the limiting proc. is max-stable. Take the normalized process � 1 /γ ( s ) � 1 + γ ( s ) X ( s ) − b t ( s ) TX t ( s ) := , s ∈ S. a t ( s ) + It is well known (de Haan and Lin (2001)) that (1) is equivalent to: (i) uniform convergence of the marginal distributions, � X ( s ) − b t ( s ) � = (1 + γ ( s ) x ) − 1 /γ ( s ) , t →∞ tP lim > x 1 + γ ( s ) x > 0 , a t ( s ) uniformly in s , Poitiers 2012 theslide – 2 / 12

Motivation ⊲ Motivation (ii) convergence of measures, Simple GP process Finite dimensions C + ( S ) � � t →∞ tP ( TX t ∈ A ) = ν ( A ) , lim A ∈ B , GP process Regular variation References C + ( S ) = { f ∈ C ( S ) : f ≥ 0 } , for all A such that ν ( ∂A ) = 0 and inf { sup s ∈ S f ( s ) : f ∈ A } > 0 . Then ν is homogeneous of order -1: ν ( tA ) = t − 1 ν ( A ) for all t > 0 . The homogeneity property is basically what characterizes Pareto distributions. Recall, for a standard Pareto r.v. P ( Y > yu | Y > u ) = 1 /y , y, u > 1 , i.e. P ( Y > yu ) = y − 1 P ( Y > u ) . The resulting exponent measure and its homogeneity are also well known for random vectors verifying the maximum domain of attraction condition. Poitiers 2012 theslide – 3 / 12

Simple GP process Theorem. Let W be a stoch. proc. in C + ( S ) . Equivalent are: Motivation ⊲ Simple GP process Finite dimensions 1. (Random functions) GP process Regular variation � � (a) P sup s ∈ S W ( s ) ≥ 1 = 1 , References � � (b) E W ( s ) / sup s ∈ S W ( s ) > 0 for all s ∈ S , � � C + (c) P ( W ∈ rA ) = r − 1 P ( W ∈ A ) , for all r > 1 , A ∈ B 1 ( S ) , rA ≡ { rf, f ∈ A } , and C + 1 ( S ) := { f ∈ C + ( S ) : sup s ∈ S f ( s ) ≥ 1 } . 2. (POT - peaks-over-threshold - stability) = x − 1 , for x > 1 (stand. Pareto distr.), � � (a) P sup s ∈ S W ( s ) > x � � (b) E W ( s ) / sup s ∈ S W ( s ) > 0 for all s ∈ S , � � � � � sup s ∈ S W ( s ) > r W W � (c) P sup s ∈ S W ( s ) ∈ B = P sup s ∈ S W ( s ) ∈ B , � � C + ¯ for all r > 1 , B ∈ B 1 ( S ) with C + ¯ 1 ( S ) := { f ∈ C + ( S ) : sup s ∈ S f ( s ) = 1 } . Poitiers 2012 theslide – 4 / 12

Simple GP process Motivation 3. (Constructive approach) W ( s ) = Y V ( s ) , for all s ∈ S , for some Y and ⊲ Simple GP process V = { V ( s ) } s ∈ S verifying: Finite dimensions GP process Regular variation (a) Y is a standard Pareto random variable, F Y ( y ) = 1 − 1 /y , y > 1 , References (b) V ∈ C + ( S ) is a stochastic process verifying sup s ∈ S V ( s ) = 1 a.s., and EV ( s ) > 0 for all s ∈ S , (c) Y and V are independent. Definition. W as characterized above is called simple Pareto process. Sketch of proof. ( 2 . ⇒ 3 . ) Note that sup s ∈ S W ( s ) < ∞ a.s. Take: W Y = sup W ( s ) and V = sup s ∈ S W ( s ) . s ∈ S Poitiers 2012 theslide – 5 / 12

Simple GP process Motivation � � C + ¯ ( 3 . ⇒ 1 . ) Let, for all r > 1 , B ∈ B 1 ( S ) ⊲ Simple GP process Finite dimensions � � f f ∈ C + ( S ) : sup s ∈ S f ( s ) > r, A r,B = sup s ∈ S f ( s ) ∈ B = r × A 1 ,B ; GP process Regular variation References � W � � � P W ∈ A r,B = P sup W ( s ) > r, sup s ∈ S W ( s ) ∈ B s ∈ S = P ( Y > r, V ∈ B ) = P ( Y > r ) P ( V ∈ B ) = 1 � W � = 1 � � r P sup W ( s ) > 1 , sup s ∈ S W ( s ) ∈ B r P W ∈ A 1 ,B . s ∈ S = 1 = 1 � � � � ( 1 . ⇒ 2 . ) For any r ≥ 1 , P sup s ∈ S W ( s ) ≥ r r P sup s ∈ S W ( s ) ≥ 1 r . � � C + ¯ Also for any B ∈ B 1 ( S ) , � W � P sup W ( s ) ≥ r, sup s ∈ S W ( s ) ∈ B s ∈ S = 1 � W � = 1 � W � r P sup W ( s ) ≥ 1 , sup s ∈ S W ( s ) ∈ B r P sup s ∈ S W ( s ) ∈ B . s ∈ S Poitiers 2012 theslide – 6 / 12

Simple GP process Motivation Formulas for distribution functions: ⊲ Simple GP process Finite dimensions Let w, W ∈ C + ( S ) , with w > 0 and W simple Pareto process. Then, GP process Regular variation References � V ( s ) � � V ( s ) � P ( W ≤ w ) = E sup − E sup . w ( s ) ∧ 1 w ( s ) s ∈ S s ∈ S Other simple but motivating formulas are: if E inf s ∈ S V ( s ) > 0 , (i) for x ∈ R , � 1 , x < 1 P ( W > x | W > 1) = 1 /x , x ≥ 1 , (ii) for s ∈ S , x ∈ R , � 1 , x < 1 P ( W ( s ) > x | W ( s ) > 1) = 1 /x , x ≥ 1 . Poitiers 2012 theslide – 7 / 12

Finite dimensions Motivation Finite dimensional distributions: Simple GP process For x i > 0 , i = 1 , . . . , d , for all integer d , ⊲ Finite dimensions GP process � V ( s i ) � � V ( s i ) � Regular variation P ( W ( s 1 ) ≤ x 1 , . . . , W ( s d ) ≤ x d ) = E max − E max References x i ∧ 1 x i 1 ≤ i ≤ d 1 ≤ i ≤ d which matches to known formulas for Pareto random vectors. E.g., in Rootz´ en and Tajvidi (2006) the definition for a d − dimensional d.f. of a Pareto r.v. is, (log G ( x 1 ∧ 0 , . . . , x d ∧ 0) − log G ( x 1 , . . . , x d )) / log G (1 , . . . , 1) . Recall the fin-dim distributions of simple max-stable η ∈ C + ( S ) (de Haan (1984), for s i ∈ S , x i > 0 , i = 1 , . . . , d , all d , � V ( s i ) � G ( x 1 , . . . , x d ) = P ( η ( s 1 ) ≤ x 1 , . . . , η ( s d ) ≤ x d ) = exp − E max . x i 1 ≤ i ≤ d Recall the representation for η (Penrose, 1992) η = d � Z i V i . i =1 , 2 ,... (0 , ∞ ] , r − 2 dr where { Z i } ∞ and V i ∈ C + ( S ) i.i.d. with � � i =1 are from a PPP EV i ( s ) = 1 ∀ s and E sup s ∈ S V i ( s ) < ∞ . Poitiers 2012 theslide – 8 / 12

GP process Motivation Let Simple GP process Finite dimensions γ = { γ ( s ) } s ∈ S extreme value index function ⊲ GP process Regular variation µ = { µ ( s ) } s ∈ S location function References σ = { σ ( s ) } s ∈ S > 0 scalling function all in C ( S ) . The generalized Pareto process in C ( S ) is given by W µ,σ,γ = µ + σ W γ − 1 . γ � � C + E.g. it verifies the homogeneity property, for all r > 1 and A ∈ B 1 ( S ) , �� � 1 /γ � �� � 1 /γ � 1 + γ W µ,σ,γ − µ 1 + γ W µ,σ,γ − µ = r − 1 P P ∈ rA ∈ A . σ σ Poitiers 2012 theslide – 9 / 12

Regular variation Motivation (de Haan and Lin 2001, Hult and Lindskog 2005) X in C ( S ) is regularly varying if Simple GP process ∃ α > 0 and a probability measure ρ , Finite dimensions GP process � � ⊲ Regular variation X P sup s ∈ S X ( s ) > tx, sup s ∈ S X ( s ) ∈ · → W x − α ρ ( · ) , References x > 0 , t → ∞ , � � P sup s ∈ S X ( s ) > t on { f ∈ C ( S ) : sup s ∈ S f ( s ) = 1 } . In a ‘standard’ form α = 1 . On the other hand, the condition for the convergence of measures in the max. dom. of attr. cond. for X ∈ C ( S ) is equivalent to (de Haan and Lin 2001), � � P sup s ∈ S TX t ( s ) > x � = 1 lim x , x > 1 , � P sup s ∈ S TX t ( s ) > 1 t →∞ and � TX t � � sup � t →∞ P lim sup s ∈ S TX t ( s ) ∈ B TX t ( s ) > 1 = ρ ( B ) , s ∈ S � � C + ¯ , ρ ( ∂B ) = 0 , ρ some probability measure on ¯ C + with B ∈ B 1 ( S ) 1 ( S ) . Poitiers 2012 theslide – 10 / 12

Regular variation Motivation Hence, Simple GP process Finite dimensions � � T X t P sup s ∈ S TX t ( s ) > 1 , sup s ∈ S T X t ( s ) ∈ B GP process ⊲ Regular variation � � P sup s ∈ S TX t ( s ) > 1 References � � T X t P sup s ∈ S TX t ( s ) > tx, sup s ∈ S T X t ( s ) ∈ B � � P sup s ∈ S TX t ( s ) > t ∼ � � � � P sup s ∈ S TX t ( s ) > t P sup s ∈ S TX t ( s ) > tx i.e. � � T X t P sup s ∈ S TX t ( s ) > tx, sup s ∈ S T X t ( s ) ∈ B ∼ x − 1 ρ ( B ) . � � P sup s ∈ S TX t ( s ) > t That is, X (or its probab. distribution) verifying regularly varying is basically the condition on the convergence of measures in the max. dom. of attr. or, the convergence of the ‘dependence part’ to a Pareto process. Poitiers 2012 theslide – 11 / 12