Goodness-of-Fit Testing with Empirical Copulas

Transcription

1 with Empirical Copulas Sami Umut Can John Einmahl Roger Laeven Department of Econometrics and Operations Research Tilburg University January 19, 2011 with Empirical Copulas

2 Overview of Copulas with Empirical Copulas

3 Overview of Copulas A bivariate copula C is a bivariate cdf defined on [0, 1] 2 with uniform marginal distributions on [0, 1]. with Empirical Copulas

4 Overview of Copulas A bivariate copula C is a bivariate cdf defined on [0, 1] 2 with uniform marginal distributions on [0, 1]. More precisely, a function C : [0, 1] 2 [0, 1] is called a bivariate copula if C(x, 0) = C(0, y) = 0 for any x, y [0, 1] C(x, 1) = x, C(1, y) = y for any x, y [0, 1] C(x 2, y 2 ) C(x 1, y 2 ) C(x 2, y 1 ) + C(x 1, y 1 ) 0 for any x 1, x 2, y 1, y 2 [0, 1] with x 1 x 2 and y 1 y 2 with Empirical Copulas

5 Sklar s Theorem: Let H be a bivariate cdf with continuous marginal cdf s H(x, ) = F(x), H(, y) = G(y). Then there exists a unique copula C such that H(x, y) = C(F(x), G(y)). (1) Conversely, for any univariate cdf s F and G and any copula C, (1) defines a bivariate cdf H with marginals F and G. with Empirical Copulas

6 Sklar s Theorem: Let H be a bivariate cdf with continuous marginal cdf s H(x, ) = F(x), H(, y) = G(y). Then there exists a unique copula C such that H(x, y) = C(F(x), G(y)). (1) Conversely, for any univariate cdf s F and G and any copula C, (1) defines a bivariate cdf H with marginals F and G. C captures the dependence structure of two random variables. It is used for dependence modeling in finance and actuarial science. with Empirical Copulas

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8 Given a sample (X 1, Y 1 ),..., (X n, Y n ) from an unknown bivariate distribution H, with unknown continuous marginal distributions F and G, and a corresponding copula C, how can we decide if a given copula C 0 or a given parametric family of copulas {C θ, θ Θ} is a good fit for the sample? with Empirical Copulas

9 Given a sample (X 1, Y 1 ),..., (X n, Y n ) from an unknown bivariate distribution H, with unknown continuous marginal distributions F and G, and a corresponding copula C, how can we decide if a given copula C 0 or a given parametric family of copulas {C θ, θ Θ} is a good fit for the sample? In other words, we would like to perform a hypothesis test about C, with a null hypothesis of the form C = C 0 or C {C θ, θ Θ}. For now, we consider the simple hypothesis (C = C 0 ) only. with Empirical Copulas

10 Given a sample (X 1, Y 1 ),..., (X n, Y n ) from an unknown bivariate distribution H, with unknown continuous marginal distributions F and G, and a corresponding copula C, how can we decide if a given copula C 0 or a given parametric family of copulas {C θ, θ Θ} is a good fit for the sample? In other words, we would like to perform a hypothesis test about C, with a null hypothesis of the form C = C 0 or C {C θ, θ Θ}. For now, we consider the simple hypothesis (C = C 0 ) only. We would like to use the empirical copula to construct a goodness-of-fit test. with Empirical Copulas

11 Note that we can write C(x, y) = H(F 1 (x), G 1 (y)), (x, y) [0, 1] 2, with F 1 (x) = inf{t R : F(t) x}, and similarly for G 1. with Empirical Copulas

12 Note that we can write C(x, y) = H(F 1 (x), G 1 (y)), (x, y) [0, 1] 2, with F 1 (x) = inf{t R : F(t) x}, and similarly for G 1. So a natural way of estimating the copula C is using the empirical copula with C n (x, y) = H n (Fn 1 (x), Gn 1 (y)), (x, y) [0, 1] 2, H n (x, y) = 1 n F n (x) = 1 n n 1{X i x, Y i y}, i=1 n 1{X i x}, G n (y) = 1 n i=1 n 1{Y i y} i=1 with Empirical Copulas

13 It is known that the empirical copula process D n (x, y) = n(c n (x, y) C(x, y)), (x, y) [0, 1] 2 converges weakly in l ([0, 1] 2 ) to a C-Brownian pillow, under the assumption that C x (x, y) is continuous on {(x, y) [0, 1] 2 : 0 < x < 1}, C y (x, y) is continuous on {(x, y) [0, 1] 2 : 0 < y < 1}. with Empirical Copulas

14 It is known that the empirical copula process D n (x, y) = n(c n (x, y) C(x, y)), (x, y) [0, 1] 2 converges weakly in l ([0, 1] 2 ) to a C-Brownian pillow, under the assumption that C x (x, y) is continuous on {(x, y) [0, 1] 2 : 0 < x < 1}, C y (x, y) is continuous on {(x, y) [0, 1] 2 : 0 < y < 1}. A C-Brownian sheet W (x, y) is a mean zero Gaussian process with covariance function Cov[W (x, y), W (x, y )] = C(x x, y y ), x, x, y, y [0, 1]. with Empirical Copulas

15 It is known that the empirical copula process D n (x, y) = n(c n (x, y) C(x, y)), (x, y) [0, 1] 2 converges weakly in l ([0, 1] 2 ) to a C-Brownian pillow, under the assumption that C x (x, y) is continuous on {(x, y) [0, 1] 2 : 0 < x < 1}, C y (x, y) is continuous on {(x, y) [0, 1] 2 : 0 < y < 1}. A C-Brownian sheet W (x, y) is a mean zero Gaussian process with covariance function Cov[W (x, y), W (x, y )] = C(x x, y y ), x, x, y, y [0, 1]. A C-Brownian pillow D(x, y) is a mean zero Gaussian process that is equal in distribution to the C-Brownian sheet W, conditioned on W (x, y) = 0 for any (x, y) [0, 1] 2 \ (0, 1) 2. with Empirical Copulas

16 We have D(x, y) = W (x, y) C x (x, y)w (x, 1) C y (x, y)w (1, y) (C(x, y) xc x (x, y) yc y (x, y))w (1, 1). with Empirical Copulas

17 We have D(x, y) = W (x, y) C x (x, y)w (x, 1) C y (x, y)w (1, y) (C(x, y) xc x (x, y) yc y (x, y))w (1, 1). So we know the asymptotic distribution of the empirical copula process D n (x, y) = n(c n (x, y) C(x, y)), and we can take a functional of D n (such as the sup over [0, 1] 2 or an appropriate integral) as a test statistic for a goodness-of-fit test. with Empirical Copulas

18 We have D(x, y) = W (x, y) C x (x, y)w (x, 1) C y (x, y)w (1, y) (C(x, y) xc x (x, y) yc y (x, y))w (1, 1). So we know the asymptotic distribution of the empirical copula process D n (x, y) = n(c n (x, y) C(x, y)), and we can take a functional of D n (such as the sup over [0, 1] 2 or an appropriate integral) as a test statistic for a goodness-of-fit test. Problem: The asymptotic distribution of D n, and that of the test statistic, depends on C. We would like to have a distribution-free goodness-of-fit test. with Empirical Copulas

19 Overview of Copulas with Empirical Copulas

20 Overview of Copulas Idea: Transform D n into another process, say Z n, whose asymptotic distribution is independent of C. Use an appropriate functional of the new process Z n as a test statistic for goodness-of-fit tests. with Empirical Copulas

21 Overview of Copulas Idea: Transform D n into another process, say Z n, whose asymptotic distribution is independent of C. Use an appropriate functional of the new process Z n as a test statistic for goodness-of-fit tests. We use E. Khmaladze s scanning idea to transform D into a standard two-parameter Wiener process Z defined on [0, 1] 2. The same transformation applied to D n will then produce a process Z n that will, hopefully, converge to Z. with Empirical Copulas

22 Overview of Copulas Idea: Transform D n into another process, say Z n, whose asymptotic distribution is independent of C. Use an appropriate functional of the new process Z n as a test statistic for goodness-of-fit tests. We use E. Khmaladze s scanning idea to transform D into a standard two-parameter Wiener process Z defined on [0, 1] 2. The same transformation applied to D n will then produce a process Z n that will, hopefully, converge to Z. Assumptions on C: Continuous first-order partial derivatives on [0, 1] 2 \ {(0, 0), (0, 1), (1, 0), (1, 1)}, continuous second-order partial derivatives on (0, 1) 2, strictly positive mixed partial C xy on (0, 1) 2, and more (to be determined). with Empirical Copulas

23 Define a grid {(x i, y j ) : 0 i, j N} on [0, 1] 2 such that 0 = x 0 < x 1 <... < x N = 1 0 = y 0 < y 1 <... < y N = 1 and define filtrations F x (x i ) = σ{d(x h, y k ) : 0 h i, 0 k N}, F y (y j ) = σ{d(x h, y k ) : 0 h N, 0 k j}, 0 i N 0 j N with Empirical Copulas

24 Define a grid {(x i, y j ) : 0 i, j N} on [0, 1] 2 such that and define filtrations 0 = x 0 < x 1 <... < x N = 1 0 = y 0 < y 1 <... < y N = 1 F x (x i ) = σ{d(x h, y k ) : 0 h i, 0 k N}, F y (y j ) = σ{d(x h, y k ) : 0 h N, 0 k j}, 0 i N 0 j N Scan the process D with respect to the filtration {F x }: K (N) i 1 ( 1 (x i, y j ) = D(x h+1, y j ) D(x h, y j ) h=0 ) E[D(x h+1, y j ) D(x h, y j ) F x (x h )] with Empirical Copulas

25 We compute K (N) 1 (x i, y j ) i 1 = D(x i, y j ) h=0 ( ) E[D(xh, y j )D(x h+1, y j )] D(x h, y j ) E[D(x h, y j ) 2 1 ] with Empirical Copulas

26 We compute K (N) 1 (x i, y j ) i 1 = D(x i, y j ) h=0 ( ) E[D(xh, y j )D(x h+1, y j )] D(x h, y j ) E[D(x h, y j ) 2 1 ] Making the x-partitioning finer and finer, we obtain x K 1 (x, y j ) = D(x, y j ) D(s, y j )ξ 1 (ds, y j ) 0 as a limit in probability, where ξ 1 is an absolutely continuous measure whose density is determined by C and its first- and second-order derivatives. with Empirical Copulas

27 Next, we scan K 1 with respect to the filtration {F y } and take the limit as the y-partition gets finer and finer: x K (x, y) = D(x, y) + 0 x y 0 y D(s, y)ξ 1 (ds, y) 0 D(s, t)ξ 1 (ds, t)ξ 2 (s, dt), 0 D(x, t)ξ 2 (x, dt) where ξ 2 is another absolutely continuous measure whose density is determined by C and its first- and second-order derivatives. with Empirical Copulas

28 Theorem: K is a C-Brownian sheet. with Empirical Copulas

29 Theorem: K is a C-Brownian sheet. Proof: with Empirical Copulas

30 Theorem: K is a C-Brownian sheet. Proof: K is a mean zero Gaussian process since D is, so it remains to show that K has the covariance structure of a C-Brownian sheet. with Empirical Copulas

31 Theorem: K is a C-Brownian sheet. Proof: K is a mean zero Gaussian process since D is, so it remains to show that K has the covariance structure of a C-Brownian sheet. K has independent (rectangle) increments by construction, so it will suffice to show that Var[K (x, y)] = C(x, y) for all (x, y) [0, 1] 2. with Empirical Copulas

32 Theorem: K is a C-Brownian sheet. Proof: K is a mean zero Gaussian process since D is, so it remains to show that K has the covariance structure of a C-Brownian sheet. K has independent (rectangle) increments by construction, so it will suffice to show that Var[K (x, y)] = C(x, y) for all (x, y) [0, 1] 2. We need a technical lemma about the densities ψ 1 and ψ 2 of the measures ξ 1 and ξ 2, namely that sup x(1 x) ψ 1 (x, y) <, (x,y) [0,1] 2 sup y(1 y) ψ 2 (x, y) <. (x,y) [0,1] 2 with Empirical Copulas

33 Corollary: The process Z (x, y) = x y dk (s, t), (x, y) [0, C xy 1]2 (s, t) is a standard two-parameter Wiener process. with Empirical Copulas

34 Corollary: The process Z (x, y) = x y dk (s, t), (x, y) [0, C xy 1]2 (s, t) is a standard two-parameter Wiener process. We have thus transformed the C-Brownian pillow D into a standard two-parameter Wiener process Z, through a two-step transformation: D K Z. with Empirical Copulas

35 We apply the same two-step transformation to D n, i.e. we define K n (x, y) = D n (x, y) Z n (x, y) = x y 0 for (x, y) [0, 1] x 0 y 0 x y 0 D n (s, y)ξ 1 (ds, y) D n (x, t)ξ 2 (x, dt) 0 1 C xy (s, t) dk n(s, t) D n (s, t)ξ 1 (ds, t)ξ 2 (s, dt), with Empirical Copulas

36 Theorem: Z n converges weakly to Z. with Empirical Copulas

37 Theorem: Z n converges weakly to Z. As an intermediate step, we need to show that K n converges weakly to K. with Empirical Copulas

38 Theorem: Z n converges weakly to Z. As an intermediate step, we need to show that K n converges weakly to K. Thus the asymptotical distribution of Z n is independent of C, and functionals of Z n can be used for goodness-of-fit tests. with Empirical Copulas

39 Theorem: Z n converges weakly to Z. As an intermediate step, we need to show that K n converges weakly to K. Thus the asymptotical distribution of Z n is independent of C, and functionals of Z n can be used for goodness-of-fit tests. Future: Construct actual test statistics and procedures for goodness-of-fit tests. Consider composite null hypotheses of the form C {C θ : θ Θ} and consider m-dimensional copulas with m > 2. with Empirical Copulas

40 Theorem: Z n converges weakly to Z. As an intermediate step, we need to show that K n converges weakly to K. Thus the asymptotical distribution of Z n is independent of C, and functionals of Z n can be used for goodness-of-fit tests. Future: Construct actual test statistics and procedures for goodness-of-fit tests. Consider composite null hypotheses of the form C {C θ : θ Θ} and consider m-dimensional copulas with m > 2. Thank you for listening! with Empirical Copulas