When uniform weak convergence fails: Empirical processes for - PowerPoint PPT Presentation

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs Axel B¨ ucher, Johan Segers and Stanislav Volgushev Universit´ e catholique de Louvain and Ruhr-Universit¨ at Bochum Van Dantzig Seminar, Mathematical Institute, Leiden University, 11 Apr 2014 1/ 32

Motivation Uniform convergence of bounded functions Strong implications vs. Restricted applicability 2/ 32

Motivation Uniform convergence of bounded functions Strong implications vs. Restricted applicability ◮ Implies pointwise, continuous, L p -convergence . . . 2/ 32

Motivation Uniform convergence of bounded functions Strong implications vs. Restricted applicability ◮ Implies pointwise, continuous, L p -convergence . . . ◮ Well-developed weak convergence theory Great success story in mathematical statistics [Van der Vaart and Wellner (1996): Weak convergence and empirical processes] 2/ 32

Motivation Uniform convergence of bounded functions Strong implications vs. Restricted applicability ◮ Implies pointwise, continuous, L p -convergence . . . ◮ Well-developed weak convergence theory Great success story in mathematical statistics [Van der Vaart and Wellner (1996): Weak convergence and empirical processes] ◮ Many applications through the continuous mapping theorem and the functional delta method 2/ 32

Motivation Uniform convergence of bounded functions Strong implications vs. Restricted applicability ◮ Implies pointwise, continuous, ◮ Continuous functions cannot L p -convergence . . . converge to jump functions ◮ Well-developed weak convergence 1.2 theory 1.0 Great success story in 0.8 mathematical statistics 0.6 0.4 [Van der Vaart and Wellner (1996): Weak 0.2 convergence and empirical processes] 0.0 ◮ Many applications through the − 0.2 continuous mapping theorem and 0.0 0.2 0.4 0.6 0.8 1.0 the functional delta method 2/ 32

Motivation Uniform convergence of bounded functions Strong implications vs. Restricted applicability ◮ Implies pointwise, continuous, ◮ Continuous functions cannot L p -convergence . . . converge to jump functions ◮ Well-developed weak convergence 1.2 theory 1.0 Great success story in 0.8 mathematical statistics 0.6 0.4 [Van der Vaart and Wellner (1996): Weak 0.2 convergence and empirical processes] 0.0 ◮ Many applications through the − 0.2 continuous mapping theorem and 0.0 0.2 0.4 0.6 0.8 1.0 the functional delta method ◮ Questions: Weaker metric? Weak convergence theory? Applications? 2/ 32

Empirical processes via epi- and hypographs The empirical copula process Weak convergence with respect to the uniform metric Non-smooth copulas: when weak convergence fails The hypi-semimetric and weak convergence Applications 3/ 32

Empirical processes via epi- and hypographs The empirical copula process Weak convergence with respect to the uniform metric Non-smooth copulas: when weak convergence fails The hypi-semimetric and weak convergence Applications 4/ 32

Copulas ◮ A d -variate copula C is a d -variate distribution function with uniform (0 , 1) margins. 5/ 32

Copulas ◮ A d -variate copula C is a d -variate distribution function with uniform (0 , 1) margins. ◮ Sklar’s (1959) theorem: If F is a d -variate distribution function with margins F 1 , . . . , F d , then there exists a copula C such that � � F ( x 1 , . . . , x d ) = C F 1 ( x 1 ) , . . . , F d ( x d ) 5/ 32

Copulas ◮ A d -variate copula C is a d -variate distribution function with uniform (0 , 1) margins. ◮ Sklar’s (1959) theorem: If F is a d -variate distribution function with margins F 1 , . . . , F d , then there exists a copula C such that � � F ( x 1 , . . . , x d ) = C F 1 ( x 1 ) , . . . , F d ( x d ) ◮ Moreover, if the margins are continuous, then C is unique and is given by the distribution function of ( F 1 ( X 1 ) , . . . , F d ( X d )), with ( X 1 , . . . , X d ) ∼ F : C ( u 1 , . . . , u d ) = P [ F 1 ( X 1 ) ≤ u 1 , . . . , F d ( X d ) ≤ u d ] = P [ X 1 ≤ F − 1 ( u 1 ) , . . . , X d ≤ F − d ( u d )] F − 1 ( u 1 ) , . . . , F − � � = F d ( u d ) with F − j ( u ) = inf { x : F j ( x ) ≥ u } the generalized inverse (quantile function) 5/ 32

Copulas ◮ A d -variate copula C is a d -variate distribution function with uniform (0 , 1) margins. ◮ Sklar’s (1959) theorem: If F is a d -variate distribution function with margins F 1 , . . . , F d , then there exists a copula C such that � � F ( x 1 , . . . , x d ) = C F 1 ( x 1 ) , . . . , F d ( x d ) ◮ Moreover, if the margins are continuous, then C is unique and is given by the distribution function of ( F 1 ( X 1 ) , . . . , F d ( X d )), with ( X 1 , . . . , X d ) ∼ F : C ( u 1 , . . . , u d ) = P [ F 1 ( X 1 ) ≤ u 1 , . . . , F d ( X d ) ≤ u d ] = P [ X 1 ≤ F − 1 ( u 1 ) , . . . , X d ≤ F − d ( u d )] F − 1 ( u 1 ) , . . . , F − � � = F d ( u d ) with F − j ( u ) = inf { x : F j ( x ) ≥ u } the generalized inverse (quantile function) ◮ Usage: Modelling dependence between components X 1 , . . . , X d , irrespective of their marginal distributions 5/ 32

The empirical copula ◮ Situation: ( X i ) i =1 ,..., n i.i.d. rvs, X i ∼ F = C ( F 1 , . . . , F d ), continuous marginals F j . [hence C ( u ) = F { F − 1 ( u 1 ) , . . . , F − d ( u d ) } with the generalized inverse F − j ( u ) = inf { x : F j ( x ) ≥ u } ]

The empirical copula ◮ Situation: ( X i ) i =1 ,..., n i.i.d. rvs, X i ∼ F = C ( F 1 , . . . , F d ), continuous marginals F j . [hence C ( u ) = F { F − 1 ( u 1 ) , . . . , F − d ( u d ) } with the generalized inverse F − j ( u ) = inf { x : F j ( x ) ≥ u } ] ◮ Goal: Estimate C nonparametrically.

The empirical copula ◮ Situation: ( X i ) i =1 ,..., n i.i.d. rvs, X i ∼ F = C ( F 1 , . . . , F d ), continuous marginals F j . [hence C ( u ) = F { F − 1 ( u 1 ) , . . . , F − d ( u d ) } with the generalized inverse F − j ( u ) = inf { x : F j ( x ) ≥ u } ] ◮ Goal: Estimate C nonparametrically. ◮ Simple plug-in estimation: empirical cdfs n n F n ( x ) := 1 F nj ( x j ) := 1 � � I ( X i 1 ≤ x 1 , . . . , X id ≤ x d ) , I ( X ij ≤ x j ) . n n i =1 i =1 yield the empirical copula n C n ( u ) = F n { F − n 1 ( u 1 ) , . . . , F − nd ( u d ) } = n − 1 � I { X i 1 ≤ F − n 1 ( u 1 ) , . . . , X id ≤ F − nd ( u d ) } i =1 6/ 32

The empirical copula ◮ Situation: ( X i ) i =1 ,..., n i.i.d. rvs, X i ∼ F = C ( F 1 , . . . , F d ), continuous marginals F j . [hence C ( u ) = F { F − 1 ( u 1 ) , . . . , F − d ( u d ) } with the generalized inverse F − j ( u ) = inf { x : F j ( x ) ≥ u } ] ◮ Goal: Estimate C nonparametrically. ◮ Simple plug-in estimation: empirical cdfs n n F n ( x ) := 1 F nj ( x j ) := 1 � � I ( X i 1 ≤ x 1 , . . . , X id ≤ x d ) , I ( X ij ≤ x j ) . n n i =1 i =1 yield the empirical copula n C n ( u ) = F n { F − n 1 ( u 1 ) , . . . , F − nd ( u d ) } = n − 1 � I { X i 1 ≤ F − n 1 ( u 1 ) , . . . , X id ≤ F − nd ( u d ) } i =1 n � � ˆ U i 1 ≤ u 1 , . . . , ˆ = n − 1 + O ( n − 1 ) � I U id ≤ u d i =1 [where ˆ U ij = rank( X ij ) / n are ‘pseudo-observations’ of C (rescaled ranks)] 6/ 32

The empirical copula process u �→ C n ( u ) = √ n { C n ( u ) − C ( u ) } ∈ ℓ ∞ ([0 , 1] d ) is called empirical copula process. [ ℓ ∞ ([0 , 1] d ) the space of bounded functions on [0 , 1] d .] 7/ 32

The empirical copula process u �→ C n ( u ) = √ n { C n ( u ) − C ( u ) } ∈ ℓ ∞ ([0 , 1] d ) is called empirical copula process. [ ℓ ∞ ([0 , 1] d ) the space of bounded functions on [0 , 1] d .] Many applications. ◮ Testing for structural assumptions. Example: symmetry [Genest, Neˇ slehov´ a, Quessy (2012)]. Null hypothesis: C ( u , v ) = C ( v , u ) for all u , v . � � { C n ( u , v ) − C n ( v , u ) } 2 du dv { C n ( u , v ) − C n ( v , u ) } 2 du dv H 0 T n = n = 7/ 32

The empirical copula process u �→ C n ( u ) = √ n { C n ( u ) − C ( u ) } ∈ ℓ ∞ ([0 , 1] d ) is called empirical copula process. [ ℓ ∞ ([0 , 1] d ) the space of bounded functions on [0 , 1] d .] Many applications. ◮ Testing for structural assumptions. Example: symmetry [Genest, Neˇ slehov´ a, Quessy (2012)]. Null hypothesis: C ( u , v ) = C ( v , u ) for all u , v . � � { C n ( u , v ) − C n ( v , u ) } 2 du dv { C n ( u , v ) − C n ( v , u ) } 2 du dv H 0 T n = n = ◮ Minimum-distance estimators of parametric copulas [Tsukahara (2005)]. { C θ | θ ∈ Θ } class of parametric candidate models. Estimator: � { C θ ( u , v ) − C n ( u , v ) } 2 du dv . ˆ θ := argmin θ 7/ 32

The empirical copula process u �→ C n ( u ) = √ n { C n ( u ) − C ( u ) } ∈ ℓ ∞ ([0 , 1] d ) is called empirical copula process. [ ℓ ∞ ([0 , 1] d ) the space of bounded functions on [0 , 1] d .] Many applications. ◮ Testing for structural assumptions. Example: symmetry [Genest, Neˇ slehov´ a, Quessy (2012)]. Null hypothesis: C ( u , v ) = C ( v , u ) for all u , v . � � { C n ( u , v ) − C n ( v , u ) } 2 du dv { C n ( u , v ) − C n ( v , u ) } 2 du dv H 0 T n = n = ◮ Minimum-distance estimators of parametric copulas [Tsukahara (2005)]. { C θ | θ ∈ Θ } class of parametric candidate models. Estimator: � { C θ ( u , v ) − C n ( u , v ) } 2 du dv . ˆ θ := argmin θ ◮ Goodness-of fit tests, Asymptotics of estimators for Pickands dep. fct. ... 7/ 32