Large Quantile Estimation for Distributions in the Domain of - PowerPoint PPT Presentation

Large Quantile Estimation for Distributions in the Domain of Attraction of a Max-Semistable Law Luísa Canto e Castro 1 Sandra Dias 2 1 Department of Statistics University of Lisbon, Portugal 2 Department of Mathematics University of Trás-os-Montes e Alto Douro, Portugal 4th Conference on Extreme Value Analysis Gothenburg, August 15-19, 2005 Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 1 / 30

Introduction F is in the domain of attraction of a max-stable d.f. G , F ∈ MS ( G ) , if and only if there exist a n > 0 and b n such that n → + ∞ F n ( a n x + b n ) = G ( x ) . lim (1) F is in the domain of attraction of a max-semistable d.f. G , F ∈ MSS ( G ) , if and only if there exist a n > 0 and b n such that n → + ∞ F k n ( a n x + b n ) = G ( x ) , ∀ x ∈ C G lim (2) where k n verifies n → + ∞ k n +1 /k n = r ≥ 1 ( r < ∞ ) . lim (3) Remark: Max-stable laws are a particular case of the max-semistable laws when r = 1 . Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 2 / 30

Introduction (continuation) Pioneers in max-semistable laws: Grinevich (1992, 1993) Pancheva (1992) (Grinevich 1992) G is a max-semistable d.f. if and only if there exist r > 1 , a > 0 and b ∈ R such that G is solution of the functional equation G ( x ) = G r ( ax + b ) . (4) Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 3 / 30

First characterization (Grinevich 1992) Unifying standard expressions for max-semistable d.f.’s For γ � = 0 � � − (1 + γx ) − 1 /γ ν (log(1 + γx )) G γ,ν ( x ) = exp , x ∈ R , 1 + γx > 0 a � = 1 and p = log a = γ log r For γ = 0 G 0 ,ν ( x ) = exp {− e − x ν ( x ) } , x ∈ R a = 1 and p = b = log r where ν is a positive, bounded and periodic function with period p . Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 4 / 30

Some graphical features of max-semistable laws 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 1 2 3 4 5 6 7 8 9 10 11 0 5 10 15 20 25 30 Figure: D.f.’s G 0 . 5 ( x ) = exp( − x − 2 ) and G 0 . 5 ,ν ( x ) = exp( − x − 2 (8 + cos(4 π log x ))) 11 10 9 8 7 6 5 4 3 2 1 0 0 5 10 15 20 25 30 Figure: QQ-Plot of G 0 . 5 ,ν against G 0 . 5 Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 5 / 30

Second characterization (Canto e Castro et al. 2000) General to location and scale, for any d.f. G max-semistable we have: − log( − log G ( s m + a m x )) = m log r + y ( x ) , ∀ x ∈ [0 , 1] , m ∈ Z (5) where y : [0 , 1] → [0 , log r ] is non decreasing, right continuous and continuous at x = 1 s m = m if a = 1 s m = ( a m − 1) / ( a − 1) if a � = 1 and a > 0 Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 6 / 30

Graph of the function − log( − log( G )) log r y 0 0 1 Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 7 / 30

Graph of the function − log( − log( G )) 2log r log r y 0 0 1 1+a Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 7 / 30

Graph of the function − log( − log( G )) 3log r 2log r log r y 0 1+a+a 2 0 1 1+a Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 7 / 30

Graph of the function − log( − log( G )) 3log r 2log r log r y 0 −log r 1+a+a 2 −a −1 0 1 1+a Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 7 / 30

Graph of the function − log( − log( G )) 4log r 3log r 2log r log r y 0 −log r −2log r 1+a+a 2 −a −2 −a −1 −a −1 1+a+a 2 +a 3 0 1 1+a Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 7 / 30

Statistical inference in max-semistable models Temido (2000) proposed that, in the estimation of the parameters r , p and γ , appropriated functions of the following sequence of statistics should be used Z s ( m ) := X ( m/s 2 ) − X ( m/s ) X ( m/s ) − X ( m ) where X ( m ) := X N − [ m ]+1: N are order statistics of a sample of size N from the random variable X m := m N is an intermediate sequence (that is, m N is an integer sequence verifying m N → + ∞ and m N /N → 0 ) Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 8 / 30

Behaviour of the sequence of statistics Z s 8 8 s = 1.9 ≠ r 0.5 = r s = e 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Figure: Sample trajectories of Z s ( m ) for s = 1 . 9 and s = e 0 . 5 ≈ 1 . 65 F ( x ) = 1 − x − 1 (14 + cos(4 π log x )) , r = e 0 . 5 Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 9 / 30

Behaviour of the sequence of statistics Z s (continuation) Dias and Canto e Castro (2004) proved that n → + ∞ a c if and only if s = r c , c ∈ N . P Z s ( m ) − → Furthermore, if s � = r c then Z s ( m ) as an oscillatory behaviour. Using these result we have that if s = r c , c ∈ N , then Z s 2 ( m ) P R s ( m ) := − → 1 , n → + ∞ ( Z s ( m )) 2 P → log a c = cp, � P s ( m ) := log ( Z s ( m )) − n → + ∞ , γ � = 0 P γ s ( m ) := log( Z s ( m )) − → γ, n → + ∞ � log s Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 10 / 30

Behaviour of the sequence of statistics Z s (continuation) Dias and Canto e Castro (2004) proved that n → + ∞ a c if and only if s = r c , c ∈ N . P Z s ( m ) − → Furthermore, if s � = r c then Z s ( m ) as an oscillatory behaviour. Using these result we have that if s = r c , c ∈ N , then Z s 2 ( m ) P R s ( m ) := − → 1 , n → + ∞ ( Z s ( m )) 2 P → log a c = cp, � P s ( m ) := log ( Z s ( m )) − n → + ∞ , γ � = 0 P γ s ( m ) := log( Z s ( m )) − → γ, n → + ∞ � log s Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 10 / 30

Proposed estimators for the parameters r , p and γ 35 2 s = 1.5 s = 1.5 s = 1.7 1.8 s = 1.7 30 s = 1.9 s = 1.9 1.6 25 1.4 1.2 20 1 15 0.8 0.6 10 0.4 5 0.2 0 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Figure: Left: Sample trajectories of R s ( m ) for s = 1 . 5 , s = 1 . 7 and s = 1 . 9 Right: Magnified version r = e 0 . 5 ≈ 1 . 65 F ( x ) = 1 − x − 1 (14 + cos(4 π log x )) , Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 11 / 30

Proposed estimators for the parameters r , p and γ 35 2 s = 1.5 s = 1.5 s = 1.7 1.8 s = 1.7 30 s = 1.9 s = 1.9 1.6 25 1.4 1.2 20 1 15 0.8 0.6 10 0.4 5 0.2 0 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Figure: Left: Sample trajectories of R s ( m ) for s = 1 . 5 , s = 1 . 7 and s = 1 . 9 Right: Magnified version r = e 0 . 5 ≈ 1 . 65 F ( x ) = 1 − x − 1 (14 + cos(4 π log x )) , Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 11 / 30

Proposed estimators for the parameters r , p and γ (continuation) An estimate of r � � � r = mode arg max B s ( ǫ ) , ǫ = 0 . 01 , (0 . 01) , 0 . 1 s =1 . 1 , (0 . 1) , 3 . 0 where P k ◮ B s ( ǫ ) := 1 I { m ( i ) ∈ A n : | R s ( m ( i ) ) − 1 | <ǫ } ( m ( i ) ) (percentage of time that the 1 k i =1 sequence of statistics R s spends in a ǫ -neighborhood of 1) A n = { m (1) , m (2) , ..., m ( k ) } ◮ A n is a set of suitable values of m ◮ k = # A n Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 12 / 30

Proposed estimators for the parameters r , p and γ (continuation) An estimate of p ◮ if γ � = 0 X X p = 1 r ( m ) = 1 b b P b log( Z b r ( m )) k k m ∈ A n m ∈ A n ◮ if γ = 0 b p = log b r An estimate of γ γ = � p/ log � r ( γ � = 0) � Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 13 / 30

Simulation study Selected d.f.’s 1 − F ( x ) = (1 − F 0 ( x )) θ ( x ) where F 0 ∈ MS ( G ) for some max-stable d.f. G . In particular ◮ Generalized Pareto d.f. ( 1 − (1 + γx ) − 1 /γ x ∈ R , 1 + γx > 0 and γ � = 0 F 0 ( x ) = 1 − e − x x ∈ R and γ = 0 ◮ Burr d.f. F 0 ( x ) = 1 − (1 + x − ρ/γ ) − 1 /ρ , x ≥ 0 , ρ < 0 and γ > 0 θ positive and periodic function Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 14 / 30

Simulation study (continuation) 70 70 N = 1000 N = 5000 N = 2000 N = 10000 60 60 50 50 40 40 30 30 20 20 10 10 0 0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Figure: Empirical distribution of the estimates of r F ( x ) = 1 − x − 2 (8 + cos(4 π log x )) , r = e ≈ 2 . 71 Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 15 / 30

Simulation study (continuation) 90 90 N = 1000 N = 5000 N = 2000 N = 10000 80 80 70 70 60 60 50 50 40 40 30 30 20 20 10 10 0 0 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 Figure: Empirical distribution of the estimates of r F ( x ) = 1 − x − 1 (27 + cos(8 π log x )) , r = e 0 . 25 ≈ 1 . 28 , r 2 ≈ 1 . 65 , r 3 ≈ 2 . 12 and r 4 ≈ 2 . 72 Canto e Castro, Dias Quantile Estimation in MSS EVA 2005 16 / 30