Asymptotics of Joint Maxima of Discrete Random Variables Anne Feidt - PowerPoint PPT Presentation

Asymptotics of Joint Maxima of Discrete Random Variables Anne Feidt University of Zurich with Christian Genest and Johanna Neˇ slehov´ a Disentis, 21 st July 2008

Introduction Notation ◮ X 1 , . . . , X n i.i.d. with cdf F ◮ Maximum X ( n ) := max 1 ≤ i ≤ n X i ◮ x F = sup x ∈ R { F ( x ) < 1 } right endpoint of F Question Under which conditions on F do there exist a n , b n ∈ R , a n > 0, and a non-degenerate df F ∗ such that � X ( n ) − b n � = F ∗ ( x ) , lim ≤ x n →∞ P a n i.e. such that F is in the maximum domain of attraction of F ∗ , F ∈ MDA ( F ∗ )?

Answer F has to satisfy (Leadbetter et al., 1983) 1 − F ( x ) lim 1 − F ( x − ) = 1 (1) x → x F Fisher-Tippett Theorem If (1) is fulfilled, there exist only 3 possible limit laws for the normalized maximum ( X ( n ) − a n ) / b n : � 0 , x ≤ 0 ◮ Fr´ echet: Φ α ( x ) = , x > 0 , α > 0 exp {− x − α } � exp {− ( − x ) − α } , x ≤ 0 ◮ Weibull: Ψ α ( x ) = , x > 0 , α > 0 1 ◮ Gumbel: Λ( x ) = exp {− e − x } , x ∈ R . (the extreme-value distributions F ∗ )

Univariate discrete random variables Problem (1) is not satisfied for discrete distributions such as the Binomial, Poisson, Geometric, Negative Binomial ⇒ no limit law for maxima!

Univariate discrete random variables Problem (1) is not satisfied for discrete distributions such as the Binomial, Poisson, Geometric, Negative Binomial ⇒ no limit law for maxima! Remedy Let a distribution parameter vary with the sample size n at a suitable rate. Then ◮ Poisson in MDA(Gumbel) (Anderson et al., 1997) ◮ Binomial, Geometric, Negative Binomial in MDA(Gumbel) (Nadarajah and Mitov, 2002)

Example: Geometric ◮ X 1 , . . . , X n i.i.d. ∼ Geo ( p ), 0 < p < 1, q=1-p ◮ W ( x ) := � n i =1 ✶ { X i ≥ x } = # exceedances of level x ◮ � � W ( x ) = 0 = { max 1 ≤ i ≤ n X i < ⌊ x ⌋} Approximate W ( x ) by a Poi ( nq ⌊ x ⌋ ) distribution: � − e − nq ⌊ x ⌋ � � � � � � � � ≤ q ⌊ x ⌋ � P 1 ≤ i ≤ n X i < ⌊ x ⌋ max (Stein-Chen method) n →∞ Choose p = p n − → 0 and a n = 1 / p n , b n = log n / p n . Then � � � � � 1 � � � � − exp {− e − x } � � ≤ q a n x + b n � P 1 ≤ i ≤ n X i ≤ a n x + b n max = O . n n

But, there exist discrete distributions such that (1) holds! Example Let ◮ X ≥ 0 absolutely continuous rv ◮ x F = ∞ ◮ hazard rate f ( x ) / (1 − F ( x )) → 0 as x → ∞ . ◮ e.g. Pareto distribution ◮ ⌈ x ⌉ := min { n ∈ N : n ≥ x } Then we discretize X to obtain ⌈ X ⌉ with df ⌈ F ⌉ ( x ) = P ( ⌈ X ⌉ ≤ x ) = P ( ⌈ X ⌉ ≤ ⌊ x ⌋ ) = P ( X ≤ ⌊ x ⌋ ) = F ( ⌊ x ⌋ ) → Can show that (1) holds for ⌈ X ⌉ and ⌈ F ⌉ ∈ MDA ( F ∗ ) ⇔ F ∈ MDA ( F ∗ )

In higher dimensions? Notation (d=2) ◮ ( X 1 , Y 1 ) , . . . , ( X n , Y n ) i.i.d. with joint df H and margins F , G ◮ componentwise maxima X ( n ) , Y ( n ) Question When do there exist a n , b n , c n and d n ∈ R , b n , d n > 0, and a non-degenerate df H ∗ such that � X ( n ) − a n � ≤ x , Y ( n ) − c n = H ∗ ( x , y ) , n →∞ P lim ≤ y b n d n i.e. when is H ∈ MDA ( H ∗ )?

In higher dimensions? Notation (d=2) ◮ ( X 1 , Y 1 ) , . . . , ( X n , Y n ) i.i.d. with joint df H and margins F , G ◮ componentwise maxima X ( n ) , Y ( n ) Question When do there exist a n , b n , c n and d n ∈ R , b n , d n > 0, and a non-degenerate df H ∗ such that � X ( n ) − a n � ≤ x , Y ( n ) − c n = H ∗ ( x , y ) , n →∞ P lim ≤ y b n d n i.e. when is H ∈ MDA ( H ∗ )? Answer for continuous margins Galambos’ Thm (1978). Uses copulas for modelling joint dfs.

What is a copula?

What is a copula? Definition A bivariate copula C : [0 , 1] 2 → [0 , 1] is a joint distribution function with standard uniform margins.

What is a copula? Definition A bivariate copula C : [0 , 1] 2 → [0 , 1] is a joint distribution function with standard uniform margins. Idea H ( x , y ) = P ( X ≤ x , Y ≤ y ) = P [ F ( X ) ≤ F ( x ) , G ( Y ) ≤ G ( y )] = P [ U ≤ F ( x ) , V ≤ G ( y )] , with U , V ∼ U [0 , 1] = C ( F ( x ) , G ( y )) ( for F , G continuous)

Sklar’s Theorem (i) If H is a joint df with margins F and G , then ∃ a copula C s.t. H ( x , y ) = C ( F ( x ) , G ( y )) ∀ x , y ∈ [ −∞ , ∞ ] (2) If F , G are continuous, then C is unique. If F , G are discrete, then C is uniquely determined on Ran ( F ) × Ran ( G ).

Sklar’s Theorem (i) If H is a joint df with margins F and G , then ∃ a copula C s.t. H ( x , y ) = C ( F ( x ) , G ( y )) ∀ x , y ∈ [ −∞ , ∞ ] (2) If F , G are continuous, then C is unique. If F , G are discrete, then C is uniquely determined on Ran ( F ) × Ran ( G ). (ii) If C is a copula and F and G are dfs, then H defined by (2) is a joint df with margins F and G .

Sklar’s Theorem (i) If H is a joint df with margins F and G , then ∃ a copula C s.t. H ( x , y ) = C ( F ( x ) , G ( y )) ∀ x , y ∈ [ −∞ , ∞ ] (2) If F , G are continuous, then C is unique. If F , G are discrete, then C is uniquely determined on Ran ( F ) × Ran ( G ). (ii) If C is a copula and F and G are dfs, then H defined by (2) is a joint df with margins F and G . If (2) holds, say C ∈ C ( H ), the class of copulas compatible with H .

Galambos’ Theorem For continuous margins Let H and H ∗ be joint dfs such that H ( x , y ) = C ( F ( x ) , G ( y )) with F and G continuous, and H ∗ ( x , y ) = C ∗ ( F ∗ ( x ) , G ∗ ( y )). Then, with u , v ∈ [0 , 1], � (i) F ∈ MDA ( F ∗ ) and G ∈ MDA ( G ∗ ) H ∈ MDA ( H ∗ ) (ii) lim t →∞ C t � u 1 / t , v 1 / t � ⇔ , = C ∗ ( u , v ) i.e. the extremal behaviour of H is determined by the extremal behaviour of its margins and its underlying copula.

What if the margins are discrete? Problem C is not unique, |C ( H ) | = ∞ (Genest and Neˇ a, 2007) . slehov´

What if the margins are discrete? Problem C is not unique, |C ( H ) | = ∞ (Genest and Neˇ a, 2007) . slehov´ → Can apply the following weak convergence result to prove Galambos’ theorem for the discrete case.

Proposition 1 Let ( X 1 , Y 1 ), ( X 2 , Y 2 ) , . . . be mutually independent random pairs such that ( X n , Y n ) has joint df H n and margins F n , G n . Let ( X , Y ) be a random pair with joint df H and margins F , G . Then, the following are equivalent: (a) ( X n , Y n ) w → ( X , Y ), as n → ∞ .

Proposition 1 Let ( X 1 , Y 1 ), ( X 2 , Y 2 ) , . . . be mutually independent random pairs such that ( X n , Y n ) has joint df H n and margins F n , G n . Let ( X , Y ) be a random pair with joint df H and margins F , G . Then, the following are equivalent: (a) ( X n , Y n ) w → ( X , Y ), as n → ∞ . w w (b) X n → X and Y n → Y , as n → ∞ , and ∃ C ∈ C ( H ) and ∃ a sequence ( C n ) with C n ∈ C ( H n ) such that C n → C on Ran ( F ) × Ran ( G ).

Proposition 1 Let ( X 1 , Y 1 ), ( X 2 , Y 2 ) , . . . be mutually independent random pairs such that ( X n , Y n ) has joint df H n and margins F n , G n . Let ( X , Y ) be a random pair with joint df H and margins F , G . Then, the following are equivalent: (a) ( X n , Y n ) w → ( X , Y ), as n → ∞ . w w (b) X n → X and Y n → Y , as n → ∞ , and ∃ C ∈ C ( H ) and ∃ a sequence ( C n ) with C n ∈ C ( H n ) such that C n → C on Ran ( F ) × Ran ( G ). w w (c) X n → X and Y n → Y as n → ∞ , and ∀ C n ∈ C ( H n ) and ∀ C ∈ C ( H ), we have C n → C uniformly on Ran ( F ) × Ran ( G ). Proof: use triangle inequality, Lipschitz-property of copulas, Continuous Mapping thm, Arzel´ a-Ascoli thm

General Galambos (for i.i.d. pairs) Apply Proposition 1 to normalized maxima: Proposition 2 Let ( X 1 , Y 1 ), ( X 2 , Y 2 ) , . . . be mutually independent random pairs with common joint df H and margins F , G . Let H ∗ be a joint df with margins F ∗ , G ∗ and copula C ∗ . Then, the following are equivalent: ∈ MDA ( H ∗ ) (a) H ∈ MDA ( F ∗ ) and G ∈ MDA ( G ∗ ) and ∃ C (b) F ∈ C ( H ) such that lim t →∞ C t � u 1 / t , v 1 / t � = C ∗ ( u , v ) for all ( u , v ) ∈ [0 , 1] 2 . ∈ MDA ( F ∗ ) and G ∈ MDA ( G ∗ ) and ∀ C (c) F ∈ C ( H ), lim t →∞ C t � u 1 / t , v 1 / t � = C ∗ ( u , v ) holds uniformly on [0 , 1] 2 .

General Galambos (for triangular arrays) If margins are Bin, Poi, Geo, NB, . . . ⇒ let parameter vary with n Proposition 2 Let ( X 1 , Y 1 ), ( X 2 , Y 2 ) , . . . be mutually independent random pairs with common joint df H and margins F , G . Let H ∗ be a joint df with margins F ∗ , G ∗ and copula C ∗ . Then, the following are equivalent: ∈ MDA ( H ∗ ) (a) H ∈ MDA ( F ∗ ) and G ∈ MDA ( G ∗ ) and ∃ C (b) F ∈ C ( H ) such that lim t →∞ C t � u 1 / t , v 1 / t � = C ∗ ( u , v ) for all ( u , v ) ∈ [0 , 1] 2 . ∈ MDA ( F ∗ ) and G ∈ MDA ( G ∗ ) and ∀ C (c) F ∈ C ( H ), lim t →∞ C t � u 1 / t , v 1 / t � = C ∗ ( u , v ) holds uniformly on [0 , 1] 2 .