On a Resampling Scheme for Empirical Copula Hideatsu Tsukahara - PowerPoint PPT Presentation

On a Resampling Scheme for Empirical Copula Hideatsu Tsukahara (tsukahar@seijo.ac.jp) Dept of Economics, Seijo University September 4, 2013 Asymptotic Statistics and Related Topics: Theories and Methodologies

Contents 1. Introduction to Copula Models 2. Empirical Copula 3. Bootstrap Approximations for Empirical Copula 4. A New Scheme by Prof. Sibuya

1. Introduction to Copula Models Copula : a df C on [ 0 , 1 ] d with uniform marginals ✓ ✏ Sklar’s Theorem For any d -dim df F with 1-dim marginals F 1 ,..., F d , there exists a copula C s. t. � � F ( x 1 ,..., x d ) = C F 1 ( x 1 ) ,..., F d ( x d ) . ✒ ✑ C is called a copula associated with F . For continuous F , C is unique and is given by � � F − 1 1 ( u 1 ) ,..., F − 1 C ( u 1 ,..., u d ) = F d ( u d ) .

Examples of Bivariate Copulas 1. Clayton family � � − 1 / θ , u − θ + v − θ − 1 θ > 1 C θ ( u , v ) = 2. Gumbel-Hougaard family � ( − log u ) θ +( − log v ) θ � 1 / θ � � θ ≥ 1 C θ ( u , v ) = exp − , 3. Frank family � � 1 + ( e θ u − 1 )( e θ v − 1 ) C θ ( u , v ) = 1 θ ∈ R , θ log e θ − 1

4. Plackett family  � { 1 +( θ − 1 )( u + v ) } 2 − 4 uv θ ( θ − 1 ) 1 +( θ − 1 )( u + v ) −   θ > 0  ,  2 ( θ − 1 ) θ � = 1 C θ ( u , v ) =     θ = 1 uv , 5. Gaussian family � � C θ ( u , v ) = Φ θ Φ − 1 ( u ) , Φ − 1 ( v ) − 1 ≤ θ ≤ 1 , where �� 0 � � 1 θ �� Φ θ : N , df θ 1 0 and Φ : N ( 0 , 1 ) df

Advantages of Copula Modeling • Better understanding of (scale-free) dependence • Separate modeling for marginals and dependence structure in non- Gaussian multivariate distributions • Easy simulation of multivariate random samples Books on copulas • R. B. Nelsen, An Introduction to Copulas , 2nd ed., Springer, 2006. • H. Joe, Multivariate Models and Dependence Concepts , Chapman & Hall, 1997. • D. Drouet Mari and S. Kotz, Correlation and Dependence , Imperial College Press, 2001.

Semiparametric Estimation Problem X k = ( X k 1 ,..., X k d ) , k = 1 ,..., n iid with continuous df F = C θ ( F 1 ,..., F d ) • { C θ } θ ∈ Θ ⊂ R m : given parametric family of copulas • Marginals F 1 ,..., F d : unknown (nonparametric part) ◮ Semiparametric estimators of θ have asymptotic variances which depend on the unknown C θ 0 .

Goodness-of-fit Tests X k = ( X k 1 ,..., X k d ) , k = 1 ,..., n iid with continuous df F = C ( F 1 ,..., F d ) ◮ For a given C 0 , test H 0 : C = C 0 vs. H 1 : C � = C 0 One can utilize � er-von Mises distance: ρ CvM ( C , D ) = [ 0 , 1 ] d [ C ( u ) − D ( u )] 2 du • Cram´ • Kolmogorov-Smirnov distance: ρ KS ( C , D ) = sup u ∈ [ 0 , 1 ] d | C ( u ) − D ( u ) | to devise test statistics

2. Empirical Copula X k = ( X k 1 ,..., X k d ) , k = 1 ,..., n iid with continuous df F = C ( F 1 ,..., F d ) � � F − 1 1 ( u 1 ) ,..., F − 1 Recall C ( u 1 ,..., u d ) = F d ( u d ) ✓ ✏ Definition � � F − 1 n 1 ( u 1 ) ,..., F − 1 C n ( u ) : = F n nd ( u d ) where n n F n ( x ) : = 1 F ni ( x i ) : = 1 ∑ ∑ d ≤ x d } , 1 { X k 1 { X k 1 ≤ x 1 ,..., X k i ≤ x i } n n k = 1 k = 1 ✒ ✑

◮ L ( C n ) is the same for all F whose copula is C ⇒ Enough to consider ξ k = ( ξ k 1 ,..., ξ k d ) : iid with df C ( k = 1 ,..., n ) Put n n G n ( u ) : = 1 G ni ( u i ) : = 1 ∑ ∑ d ≤ u d } , 1 { ξ k 1 { ξ k 1 ≤ u 1 ,..., ξ k i ≤ u i } n n k = 1 k = 1 � � n ( u ) : = √ n • U C G n ( u ) − C ( u ) : Multivariate empirical process � � n ( u ) : = √ n • D C C n ( u ) − C ( u ) : Empirical copula process

Asymptotic representation theorem ✓ ✏ Assume C is differentiable with continuous i th partial derivatives ∂ i C ( u ) : = ∂ C ( u ) / ∂ u i , i = 1 ,..., d . Then we have d ∑ ∂ i C ( u ) U C D C n ( u ) = U C n ( u ) − n ( 1 , u i , 1 )+ R n ( u ) , i = 1 where sup u | R n ( u ) | = o P ( 1 ) as n → ∞ . ✒ ✑ ◮ With stronger conditions on C , one can show � n − 1 / 4 ( log n ) 1 / 2 ( loglog n ) 1 / 4 � | R n ( u ) | = O , a.s. sup u [Tsukahara (2005), with Erratum (2011)]

Proof : Write d ∑ ∂ i C ( u ) U C R n ( u ) = D C n − U C n + n ( 1 , u i , 1 ) i = 1 = : R 1 n ( u )+ R 2 n ( u ) where � � G − 1 n 1 ( u 1 ) ,..., G − 1 R 1 n ( u ) : = U C − U C nd ( u d ) n ( u ) n � R 2 n ( u ) : = √ n � � G − 1 n 1 ( u 1 ) ,..., G − 1 nd ( u d ) − C ( u ) C �� d � ∑ ∂ i C + G ni ( u i ) − u i i = 1

a . s . ◮ sup u | R 1 n ( u ) | − → 0 : Use • Probability inequality for the oscillation of U C n [Einmahl (1987)] � n − 1 / 2 ( loglog n ) 1 / 2 � • Smirnov LIL: sup | G − 1 ni ( u ) − u | = O P ◮ sup u | R 2 n ( u ) | − → 0 : Use • Mean value theorem and 0 ≤ ∂ i C ≤ 1 (Lipschitz continuity of C) • Kiefer (1970): � � � √ n ( G − 1 � ni ( u i ) − u i + G ni ( u i ) − u i ) sup u i � n − 1 / 4 ( log n ) 1 / 2 ( loglog n ) 1 / 4 � = O a.s.

Weak convergence ✓ ✏ → D C in D ([ 0 , 1 ] d ) L n → ∞ D C − n where d ∑ ∂ i C ( u ) U C ( 1 , u i , 1 ) D C ( u ) : = U C ( u ) − i = 1 and U C is a centered Gaussian process with Cov ( U C ( u ) , U C ( v )) = C ( u ∧ v ) − C ( u ) C ( v ) ✒ ✑

3. Bootstrap Approximations for Empirical Copula Define n C n ( u ) : = 1 ∑ � 1 { F n 1 ( X k 1 ) ≤ u 1 ,..., F nd ( X k d ) ≤ u d } n k = 1 Noting that n C n ( x ) = 1 ∑ nd ( u d ) } , 1 { X k 1 ≤ F − 1 d ≤ F − 1 n 1 ( u 1 ) ,..., X k n k = 1 one can show C n ( u ) − C n ( u ) | ≤ d u ∈ [ 0 , 1 ] d | � sup n

(i) Traditional Bootstrap (Fermanian-Radulovi´ c-Wegkamp (2004)) Define � � F # − 1 n 1 ( u 1 ) ,..., F # − 1 C # n ( u ) : = F # nd ( u d ) n where n n n ( x ) : = 1 ni ( x i ) : = 1 ∑ ∑ F # F # d ≤ x d } , W ni 1 { X k W ni 1 { X k 1 ≤ x 1 ,..., X k i ≤ x i } n n k = 1 k = 1 ( W n 1 ,..., W nn ) ∼ Multinomial ( 1 / n ,..., 1 / n ) Then √ n ( C # P W D C n ( u ) − C n ) �

(ii) Multiplier with Derivative Estimates (R´ emillard-Scaillet (2009)) n n ( u ) : = 1 ∑ C ∗ d ) ≤ u d } , Z i 1 { F n 1 ( X k 1 ) ≤ u 1 ,..., F nd ( X k n k = 1 where Z 1 ,..., Z n : iid mean 0 and variance 1 ⇒ β n : = √ n ( � n − Z n C n ) � U C (unconditional) C ∗ = ∂ i C ( u ) : = C n ( u 1 ,..., u i + h ,..., u d ) − C n ( u 1 ,..., u i − h ,..., u d ) � 2 h with h : = n − 1 / 2 . Then n ∑ ∂ i C ( u ) β n ( 1 , u i , 1 ) � D C ( unconditionally ) � β n ( u ) − i = 1

(iii) Multiplier Bootstrap (B¨ ucher-Dette (2010)) Define � � C ♭ n ( u ) : = F ♭ F ♭ − 1 n 1 ( u 1 ) ,..., F ♭ − 1 nd ( u d ) n where ξ i ξ i n n n ( x ) : = 1 ni ( x i ) : = 1 ∑ ∑ F ♭ F ♭ d ≤ x d } , 1 { X k 1 { X k 1 ≤ x 1 ,..., X k i ≤ x i } ξ n ξ n n n k = 1 k = 1 ξ 1 ,..., ξ n : iid positive rv’s with E ( ξ i ) = µ , Var ( ξ i ) = τ 2 > 0 Then √ n µ P τ ( C ♭ ξ D C n ( u ) − C n ) �

4. A New Scheme by Prof. Sibuya Let d = 2 for simplicity ( X 1 , Y 1 ) ,..., ( X n , Y n ) : iid with continuous df F ( x , y ) = C ( F 1 ( x ) , F 2 ( y )) For each i = 1 ,..., n , R ni : = rank of X i among X 1 ,..., X n Q ni : = rank of Y i among Y 1 ,..., Y n The vectors of ranks ( R n 1 , Q n 1 ) ,..., ( R nn , Q nn ) are sufficient for C ⇒ Why don’t we resample based only on ( R n 1 , Q n 1 ) ,..., ( R nn , Q nn ) ?

Let U 1 ,..., U n , V 1 ,..., V n be independent U(0,1) random variables independent of ( X 1 , Y 1 ) ,..., ( X n , Y n ) , and • U 1: n < ··· < U n : n : order statistics for U 1 ,..., U n • V 1: n < ··· < V n : n : order statistics for V 1 ,..., V n For each i = 1 ,..., n , put � � U ni : = U R ni : n , V ni : = V Q ni : n One can easily see that 1. ( � U n 1 , � V n 1 ) ,..., ( � U nn , � V nn ) are NOT independent 2. ( � U n 1 , � V n 1 ) ,..., ( � U nn , � V nn ) are identically distributed with the distri- bution varying with n

• Marginal df: n P ( U r : n ≤ u ) · 1 ∑ P ( � U n 1 ≤ u ) = E [ P ( U R ni : n ≤ u | R ni )] = n r = 1 � n − 1 � � u n ∑ t r − 1 ( 1 − t ) n − r d t = r − 1 0 r = 1 � u n − 1 ∑ = p n − 1 , ν ( t ) d t = u 0 ν = 0 where � n � t k ( 1 − t ) n − k p n , k ( t ) = k � � = ⇒ U ni ∼ U ( 0 , 1 ) , V ni ∼ U ( 0 , 1 ) ( i = 1 ,..., n )

• Joint df: H n ( u , v ) : = P ( � U ni ≤ u , � V ni ≤ v ) H n ( u , v ) = E [ P ( U R ni : n ≤ u , V Q ni : n ≤ v ) | R ni , Q ni ] n ∑ = P ( U r : n ≤ u ) P ( V q : n ≤ v ) P ( R ni = r , Q ni = q ) r , q = 1 � u � v n n ! n ! ∑ = ( r − 1 ) ! ( n − r ) ! ( q − 1 ) ! ( n − q ) ! 0 0 r , q = 1 t r − 1 ( 1 − t ) n − r s q − 1 ( 1 − s ) n − q P ( R ni = r , Q ni = q ) d t d s � u � v = : 0 J ( s , t ) d t d s 0 Let K n ( u , v ) : = P ( R ni ≤ nu , Q ni ≤ nv ) . Then � r − 1 � n , q − 1 < R ni n ≤ r < Q ni n ≤ q P ( R ni = r , Q ni = q ) = P n n n