- Email: [email protected]
Large sample behavior of the Bernstein copula estimator
Journal of Statistical Planning and Inference 142 (2012) 1189–1197
Contents lists available at SciVerse ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Large sample behavior of the Bernstein copula estimator ¨ Veraverbeke a,n Paul Janssen a, Jan Swanepoel b, Noel a b
Hasselt University, Center for Statistics, Agoralaan, Gebouw D, 3590 Diepenbeek, Belgium North-West University, Potchefstroom Campus, Potchefstroom, South Africa
a r t i c l e i n f o
abstract
Article history: Received 24 February 2011 Received in revised form 4 October 2011 Accepted 29 November 2011 Available online 6 December 2011
Bernstein polynomial estimators have been used as smooth estimators for density functions and distribution functions. The idea of using them for copula estimation has been given in Sancetta and Satchell (2004). In the present paper we study the asymptotic properties of this estimator: almost sure consistency rates and asymptotic normality. We also obtain explicit expressions for the asymptotic bias and asymptotic variance and show the improvement of the asymptotic mean squared error compared to that of the classical empirical copula estimator. A small simulation study illustrates this superior behavior in small samples. & 2011 Elsevier B.V. All rights reserved.
Keywords: Asymptotic properties Bernstein estimator Copula estimator Mean squared error
1. The Bernstein copula estimator Copulas are functions that couple multivariate distribution functions to their one-dimensional marginal distribution functions. Copulas provide a very nice tool to model multivariate data and are therefore very useful in, for example, financial economics. See e.g. Sancetta and Satchell (2004) and Sancetta (2007). The analysis of multivariate survival data provides another example where copulas are instrumental. See e.g. Wienke (2011). Consider a random vector X ¼ ðX 1 , . . . ,X d ÞT with joint cumulative distribution function H and marginal distribution functions F 1 , . . . ,F d . From Sklar’s theorem (Sklar, 1959) there exists a d-variate function C such that Hðx1 , . . . ,xd Þ ¼ CðF 1 ðx1 Þ, . . . ,F d ðxd ÞÞ: For a detailed account on copulas we refer to the excellent book by Nelsen (2006), which includes many examples. A variety of parametric models for copulas and marginal distributions has been proposed and the corresponding parametric estimation methods are available. The study of nonparametric estimators of copulas (kernel smoothed empirical copulas) goes back to Deheuvels (1979) and Gaenssler and Stute (1987). In more recent papers kernel based smooth versions of the empirical copula have been studied. Omelka et al. (2009) study weak convergence of improved kernel estimators and include an excellent overview. Nonparametric estimation of copulas is also the focus of this paper. For simplicity we restrict the presentation to the case of bivariate data ðd ¼ 2Þ. We consider a bivariate random vector (X,Y) with joint distribution function H and marginal distribution functions F and G, i.e., Hðx,yÞ ¼ PðX rx,Y ryÞ; FðxÞ ¼ PðX rxÞ and GðyÞ ¼ PðY ryÞ. Sklar’s theorem says that there exists a copula C on ½0; 12 such that Hðx,yÞ ¼ CðFðxÞ,GðyÞÞ:
n
Corresponding author. E-mail addresses: [email protected] (P. Janssen), [email protected] (J. Swanepoel), [email protected] (N. Veraverbeke).
0378-3758/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2011.11.020
1190
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
We will assume throughout that F and G are continuous, which implies that C is unique and Cðu,vÞ ¼ HðF 1 ðuÞ,G1 ðvÞÞ: Given a sample ðX 1 ,Y 1 Þ, . . . ,ðX n ,Y n Þ from the bivariate distribution function H, the empirical copula estimator for Cðu,vÞ is given by 1 C n ðu,vÞ ¼ Hn ðF 1 n ðuÞ,Gn ðvÞÞ
with Hn ðx,yÞ ¼ n1
n X
IðX i r x,Y i ryÞ
i¼1
having marginals F n ðxÞ ¼ Hn ðx, þ 1Þ ¼ n1
n X
IðX i rxÞ,
i¼1
Gn ðyÞ ¼ Hn ð þ1,yÞ ¼ n1
n X
IðY i r yÞ:
i¼1
In this paper we consider Bernstein copula estimators, which are defined in terms of Bernstein polynomials. Before we give the precise definition of Bernstein copula estimators, we review some results on smooth estimators for the distribution function of univariate data, since the results obtained in this paper provide extensions of these results to multivariate (bivariate) data. For univariate data, Bernstein estimators have been used by Babu et al. (2002) to estimate the univariate distribution and density function. Leblanc (2008) proves the asymptotic normality of the Bernstein estimator of the univariate distribution function and shows that the estimator outperforms the classical empirical distribution function estimator by deriving an explicit formula for the first order improvement for the asymptotic variance. In Leblanc (2009) it is shown that the Chung–Smirnov property holds for the Bernstein estimator. In the present paper we extend both results to the Bernstein empirical copula. Towards the definition of the Bernstein copula estimator, recall that the Bernstein polynomial of order m 4 0 corresponding to the copula C is defined as m X m X k ‘ Bm ðu,vÞ ¼ , P ðuÞP l,m ðvÞ C m m k,m k¼0l¼0 with P k,m ðuÞ ¼
m uk ð1uÞmk k
the binomial probabilities. We have, uniformly in ðu,vÞ 2 ½0; 12 , lim Bm ðu,vÞ ¼ Cðu,vÞ
m-1
since C is continuous on ½0; 12 . Note that Bm is a copula by itself, since it satisfies the sufficient conditions given in Theorem 1 of Sancetta and Satchell (2004). They propose the following Bernstein estimator of order m 40 of the copula function C: m X m X k ‘ C m,n ðu,vÞ ¼ , P ðuÞPl,m ðvÞ, Cn m m k,m k¼0l¼0 where Cn is the empirical copula estimator. The order of m will depend on n and we will have that m-1 if n-1. In Section 2 we obtain Chung–Smirnov consistency rates for the Bernstein copula estimator. In Section 3 we use a stochastic approximation for the Bernstein copula estimator to study the asymptotic bias and variance. Based on these results we show that the Bernstein copula estimator outperforms the empirical copula estimator. The asymptotic normality of the smooth estimator follows as an easy consequence. Section 4 provides a small simulation study in which for a number of examples it is shown that the Bernstein copula estimator C m,n outperforms the empirical copula estimator Cn. Finally, some of the more technical proofs are collected in Appendix. 2. Chung–Smirnov consistency rates For any function g on ½0; 12 we define JgJ ¼ sup0 r u,v r 1 9gðu,vÞ9, the supremum norm. In this section we give uniform strong consistency rates for the Bernstein copula estimator.
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
1191
Theorem 1. If m mðnÞ-1 and n=ðm log log nÞ-c Z 0, then JC m,n CJ ¼ Oðn1=2 ðlog log nÞ1=2 Þ a.s., n-1. To prove this theorem we rely on the following lemma (see Appendix for the proof). Lemma 1. sup
Q n ¼ JC n CJ ¼
0 r u,v r 1
1 1 9Hn ðF 1 ðuÞ,G1 ðvÞÞ9 ¼ Oðn1=2 ðlog log nÞ1=2 Þ a:s:, n-1: n ðuÞ,Gn ðvÞÞHðF
Proof of Theorem 1. JC m,n CJ r JC m,n Bm J þ JBm CJ:
ð1Þ
Lemma 1 and the fact that P k,m ðuÞ and P‘,m ðvÞ, k,‘ ¼ 0, . . . ,m, are binomial probabilities, imply m X m X k ‘ k ‘ , C Pk,m ðuÞP‘,m ðvÞ Cn JC m,n Bm J ¼ sup m m mm 0 r u,v r 1k ¼ 0 ‘ ¼ 0 k ‘ k ‘ , C , r JC n CJ ¼ Oðn1=2 ðlog log nÞ1=2 Þ a:s:, n-1: r max C n m m m m 0 r k,‘ r m
ð2Þ
To obtain an order bound for the second term in the rhs of (1), we use the Lipschitz property of copulas and we write Bu and Bv to denote binomial random variables with parameters m and u, and m and v respectively. We then have X m X m k ‘ , Cðu,vÞ Pk,m ðuÞP‘,m ðvÞ C JBm CJ ¼ sup m m 0 r u,v r 1k ¼ 0 ‘ ¼ 0 2 !1=2 2 !1=2 m X m X k Bu Bv u þ ‘ v P k,m ðuÞP ‘,m ðvÞ r sup u v E þ sup E r sup m m m 0 r u,v r 1 k ¼ 0 ‘ ¼ 0 m 0rur1 0rvr1 1=2 1=2 uð1uÞ vð1vÞ 1 1 ¼ sup þ sup r þ ¼ m1=2 ¼ Oðn1=2 ðlog log nÞ1=2 Þ: ð3Þ m m 2m1=2 2m1=2 0rur1 0rvr1 The proof follows from (1)–(3).
&
Remark 1. It is possible to modify the proof of Theorem 1 slightly and to obtain the same result under stronger conditions on C and weaker conditions on m. Indeed, if C has first order partial derivatives Cu and Cv that are Lipschitz continuous of order a ð0 o a r 1Þ and if n=ðm1 þ a log log nÞ-c Z 0 then JC m,n CJ ¼ Oðn1=2 ðlog log nÞ1=2 Þ a.s., n-1. 3. Asymptotic bias and variance of the Bernstein copula estimator Towards an asymptotic expression for the bias and the variance of the Bernstein copula estimator, the following decomposition is useful: n1=2 ðC m,n ðu,vÞCðu,vÞÞ ¼ n1=2 ðC m,n ðu,vÞBm ðu,vÞÞ þ n1=2 ðBm ðu,vÞCðu,vÞÞ: 2
ð4Þ
Assuming bounded third order partial derivatives on ð0; 1Þ for C, we have the following convergence rate of the Bernstein approximation: " 2 2 m X m X 1 k 1 ‘ u C uu ðu,vÞ þ v C vv ðu,vÞ Bm,n ðu,vÞCðu,vÞ ¼ 2 m 2 m k¼0‘¼0 k ‘ u v C uv ðu,vÞ P k,m ðuÞP ‘,m ðvÞ þ oðm1 Þ ¼ m1 bðu,vÞ þ oðm1 Þ, ð5Þ þ m m where bðu,vÞ ¼ 12½uð1uÞC uu ðu,vÞ þvð1vÞC vv ðu,vÞ: To handle the first term in (4) we rely on the following two lemmas. Lemma 2. For 0 ou o1, R1,m ðuÞ :¼ m1
m X
m X
k ¼ 0 ‘ ¼ kþ1
where
c1 ðuÞ ¼
uð1uÞ 1=2 : 4p
ðkmuÞP k,m ðuÞP ‘,m ðuÞ ¼ m1=2 fc1 ðuÞ þ oð1Þg,
1192
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
Proof. R1,m ðuÞ ¼ m1
m X
m X
ðkmuÞP k,m ðuÞP ‘,m ðuÞ ¼ m1
k ¼ 0 ‘ ¼ kþ1
¼ m1
m X
"
P ‘,m ðuÞ
‘¼1
m X
#
m X ‘¼1
ðkmuÞP k,m ðuÞ ¼ m1
k¼‘
P ‘,m ðuÞ
‘1 X
ðkmuÞP k,m ðuÞ
k¼0
h m i P ‘,m ðuÞ ‘ u‘ ð1uÞm‘ þ 1 , ‘ ‘¼1 m X
where the last equality is an identity in Johnson (1957). Then, m m m m X X X X ‘ ‘ u þu P 2‘,m ðuÞ ¼ uð1uÞ u P 2‘,m ðuÞ: ‘P2‘,m ðuÞ ¼ ð1uÞ P2‘,m ðuÞð1uÞ R1,m ðuÞ ¼ m1 ð1uÞ m m ‘¼1 ‘¼1 ‘¼1 ‘¼1 Now, by Lemma 3.1 in Babu et al. (2002), we have m X
P 2‘,m ðuÞ ¼ m1=2 fð4puð1uÞÞ1=2 þ oð1Þg
ð6Þ
‘¼1
and, using the Cauchy–Schwarz inequality and the fact that 0 rP ‘,m ðuÞ r1, we also have m X ‘ u P 2‘,m ðuÞ ¼ Oðm3=4 Þ: m ‘¼1
ð7Þ
Hence R1,m ðuÞ ¼ m1=2 fc1 ðuÞ þoð1Þg:
&
Lemma 3. Assume bounded second order partial derivatives for C on ð0; 1Þ2 . Then P (i) n1=2 ðC m,n ðu,vÞBm ðu,vÞÞ ¼ n1=2 ni¼ 1 Y mi þ oP ð1Þ where m X m X k ‘ k ‘ k ‘ k k C , C u , I Ui r Y mi ¼ I U i r ,V i r m m m m m m m m k¼0‘¼0
k ‘ ‘ ‘ C v , I Vi r P k,m ðuÞP ‘,m ðvÞ m m m m and where ðU 1 ,V 1 Þ, . . . ,ðU n ,V n Þ are independent random vectors with distribution function C and with uniform marginals on ½0; 1. (ii) The Ymi’s are bounded and have mean zero. (iii) VarðY mi Þ ¼ s2 ðu,vÞm1=2 Vðu,vÞ, where
s2 ðu,vÞ ¼ VarfIðU ru,V rvÞCðu,vÞC u ðu,vÞ½IðU r uÞuC v ðu,vÞ½IðV r vÞvg ¼ Cðu,vÞð1Cðu,vÞÞ þ uð1uÞC 2u ðu,vÞ þ vð1vÞC 2v ðu,vÞ2ð1uÞCðu,vÞC u ðu,vÞ2ð1vÞCðu,vÞC v ðu,vÞ þ 2C u ðu,vÞC v ðu,vÞ½Cðu,vÞuv and
"
# " # uð1uÞ 1=2 vð1vÞ 1=2 þ C v ðu,vÞð1C v ðu,vÞÞ : Vðu,vÞ ¼ C u ðu,vÞð1C u ðu,vÞÞ
p
p
Proof of Lemma 3. The proof of (i) is immediate since we have, uniformly in (u,v), the following asymptotic representation (see Fermanian et al., 2004, p. 857): C n ðu,vÞCðu,vÞ ¼ n1
n X
fIðU i ru,V i rvÞCðu,vÞC u ðu,vÞ½IðU i ruÞuC v ðu,vÞ½IðV i r vÞvgþ oP ðn1=2 Þ:
i¼1
The proof of (ii) is immediate and the technical proof of (iii) is available from Appendix. Theorem 2. Assume bounded third order partial derivatives for C on ð0; 1Þ2 . If n1=2 m1 -0, then for ðu,vÞ 2 ð0; 1Þ2 D
n1=2 ðC m,n ðu,vÞCðu,vÞÞ-N ð0, s2 ðu,vÞÞ:
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
1193
If n1=2 m1 -d, 0 o d o 1, then for ðu,vÞ 2 ð0; 1Þ2 D
n1=2 ðC m,n ðu,vÞCðu,vÞÞ-N ðdbðu,vÞ, s2 ðu,vÞÞ: Remark 2. Note that the asymptotic variance s2 ðu,vÞ in Theorem 2 coincides with the asymptotic variance in the limit distribution of n1=2 ðC n ðu,vÞCðu,vÞÞ. However, from Lemma 3(iii) it is clear that the first-order term in the expansion of the variance of n1=2 ðC m,n ðu,vÞCðu,vÞÞ is smaller than s2 ðu,vÞ (at the price of some bias in case n1=2 m1 -d 40). Remark 3. The asymptotic normality result in Theorem 2 can be strengthened to weak convergence of the process n1=2 ðC m,n ðu,vÞCðu,vÞÞ in the space ‘1 ðð0; 1Þ2 Þ of bounded functions. The limiting process is Gaussian with mean function dbðu,vÞ and with covariance function E½fIðU r u,V r vÞCðu,vÞC u ðu,vÞ½IðU ruÞuC v ðu,vÞ½IðV r vÞvgg fIðU r u0 ,V rv0 ÞCðu0 ,v0 ÞC u ðu0 ,v0 Þ½IðU ru0 Þu0 C v ðu0 ,v0 Þ½IðV r v0 Þv0 g for 0 ou,v,u0 ,v0 o 1. P Because of the representation in Lemma 3(i), it suffices to prove the asymptotic tightness of the process fn1=2 ni¼ 1 Y m,i g. 2 The Ymi as functions of u and v are uniformly bounded and continuous on ð0; 1Þ and have non-zero derivatives up to order m. By Theorem 2.7.1 in van der Vaart and Wellner (2000) such class of functions has finite e bracketing number of order Oðexpðe2=m ÞÞ. This result and the boundedness of the Ymi as functions of u and v guarantee, for m 41, the validity of the integrability condition on the square root of the logarithm of the bracketing number. An interesting further question, beyond the scope of this paper, is whether the weak convergence on ð0; 1Þ2 can be extended to ½0; 12 as Fermanian et al. (2004) did for the empirical copula process. Remark 4. The variance expression in Lemma 3(iii) is useful to arrive at an optimal choice of the order m. For the asymptotic mean squared error we have 2
AMSEðC m,n ðu,vÞÞ ¼ AsVarðC m,n ðu,vÞÞ þðAsBiasðC m,n ðu,vÞÞÞ2 ¼ n1 s2 ðu,vÞm1=2 n1 Vðu,vÞ þm2 b ðu,vÞ: Minimizing with respect to m gives !2=3 2 4b ðu,vÞ n2=3 m0 m0 ðu,vÞ ¼ Vðu,vÞ and 1
AMSEðC m0 ,n ðu,vÞÞ ¼ n
V 2 ðu,vÞ s ðu,vÞ3 16bðu,vÞ
!2=3
2
n4=3 :
Since AsVarðC n ðu,vÞÞ ¼ n1 s2 ðu,vÞ and since Cn is biased, it is clear that AMSEðC m0 ,n ðu,vÞÞ is less than AMSEðC n ðu,vÞÞ. 4. A Monte–Carlo comparison We now report on some small sample experiments which compare the empirical copula estimator Cn with the Bernstein copula estimator C m0 ,n . The quality of the estimators is firstly evaluated locally by empirically calculating their mean L1-norm, defined by RðC^ ,C y Þ ¼ EðLðC^ ,C y ÞÞ, where LðC^ ,C y Þ ¼ 9C^ ðu,vÞC y ðu,vÞ9 with C^ ¼ C n or C^ ¼ C m0 ,n where m0 ¼ m0 ðu,vÞ, and C y a given family of copulas. Secondly they are evaluated globally by calculating their mean discrete L2-norm, given by 8 9 1=2 n X n < 1 X i j i j 2= ^ ^ , C y , LðC ,C y Þ ¼ C :ðn1Þ2 ; n n n n i¼1j¼1
with C^ ¼ C n or C^ ¼ C m0 ,n where m0 ¼ m0 ðði=nÞ,ðj=nÞÞ, i,j ¼ 1, . . . ,n1. Data were generated from the following copulas (see e.g. Nelsen, 2006): 1. The Farlie–Gumbel–Morgenstern copula (FGMC), C y ðu,vÞ ¼ uv þ yuvð1uÞð1vÞ,
1 r y r 1:
In this case, Spearman’s rho, rS , is given by rS ¼ y=3.
1194
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
2. The Cuadras–Auge´ copula (CAC), C y ðu,vÞ ¼ fminðu,vÞgy ðuvÞ1y ,
0 r y r1:
Now, rS ¼ 3y=ð4yÞ. 3. The Plackett copula (PC): with y 40 and ya1, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 þðy1Þðu þ vÞ ½1 þ ðy1Þðu þ vÞ2 4uvyðy1Þ : C y ðu,vÞ ¼ 2ðy1Þ We now have that rS ¼ ððy þ 1Þ=ðy1ÞÞðð2y log yÞ=ðy1Þ2 Þ. Each entry in Tables 1–4 is based on 20 000 independent trials for sample sizes n ¼20, 30, 40. The standard errors of the Monte-Carlo estimates were found to be negligibly small and are not reported in the table. All calculations were done using double precision arithmetic in FORTRAN and routines from the IMSL library. The results in Tables 1–3 illustrate the good local performance of the Bernstein copula estimator compared to the classical empirical copula estimator for small to moderate sample sizes. It is clear that RðC m0 ,n ,C y Þ oRðC n ,C y Þ everywhere except for 4 out of the 45 cases, viz. the CAC when n ¼ 30; 40 and ðu,vÞ 2 fð0:50,0:50Þ,ð0:75,0:75Þg. The superior behavior of the Bernstein copula estimator is even more evident from Table 4. As far as global performance is concerned, the results in the table show that the strict inequality above holds everywhere.
Table 1 Monte–Carlo estimates of expected L1-norms for the empirical copula and Bernstein copula estimators. RðC n ,C y Þ
RðC m0 ,n ,C y Þ
0.50) 0.25) 0.75) 0.25) 0.75)
0.044 0.036 0.031 0.031 0.036
0.027 0.015 0.016 0.016 0.016
(0.20, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.037 0.031 0.028 0.028 0.033
0.024 0.012 0.011 0.010 0.017
(0.50, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.031 0.026 0.020 0.020 0.026
0.018 0.012 0.011 0.011 0.015
Copula
(u,v)
FGMC ðy ¼ 0:9, rS ¼ 0:3, n ¼ 20Þ
(0.50, (0.25, (0.25, (0.75, (0.75,
FGMC ðy ¼ 0:9, rS ¼ 0:3, n ¼ 30Þ
FGMC ðy ¼ 0:9, rS ¼ 0:3, n ¼ 40Þ
Table 2 Monte–Carlo estimates of expected L1-norms for the empirical copula and Bernstein copula estimators. RðC n ,C y Þ
RðC m0 ,n ,C y Þ
0.50) 0.25) 0.75) 0.25) 0.75)
0.047 0.036 0.031 0.031 0.041
0.045 0.020 0.012 0.012 0.041
(0.50, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.038 0.030 0.030 0.030 0.031
0.046 0.020 0.007 0.007 0.049
(0.50, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.031 0.026 0.022 0.022 0.026
0.048 0.023 0.006 0.006 0.053
Copula
(u,v)
CAC ðy ¼ 0:364, rS ¼ 0:3, n ¼ 20Þ
(0.50, (0.25, (0.25, (0.75, (0.75,
CAC ðy ¼ 0:364, rS ¼ 0:3, n ¼ 30Þ
CAC ðy ¼ 0:364, rS ¼ 0:3, n ¼ 40Þ
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
1195
Table 3 Monte–Carlo estimates of expected L1-norms for the empirical copula and Bernstein copula estimators. RðC n ,C y Þ
RðC m0 ,n ,C y Þ
0.50) 0.25) 0.75) 0.25) 0.75)
0.043 0.034 0.031 0.031 0.034
0.027 0.017 0.009 0.009 0.019
(0.50, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.036 0.030 0.029 0.029 0.032
0.025 0.015 0.009 0.009 0.017
(0.50, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.031 0.025 0.021 0.021 0.025
0.018 0.014 0.008 0.008 0.017
Copula
(u,v)
PC ðy ¼ 2:524, rS ¼ 0:3, n ¼ 20Þ
(0.50, (0.25, (0.25, (0.75, (0.75,
PC ðy ¼ 2:524, rS ¼ 0:3, n ¼ 30Þ
PC ðy ¼ 2:524, rS ¼ 0:3, n ¼ 40Þ
Table 4 Monte–Carlo estimates of expected L2-norms for the empirical copula and Bernstein copula estimators. Copula
n
RðC n ,C y Þ
RðC m0 ,n ,C y Þ
FGMC ðy ¼ 0:9, rS ¼ 0:3Þ
20 30 40
0.036 0.029 0.025
0.022 0.016 0.014
CAC ðy ¼ 0:364, rS ¼ 0:3Þ
20 30 40
0.037 0.030 0.026
0.023 0.022 0.022
PC ðy ¼ 2:524, rS ¼ 0:3Þ
20 30 40
0.036 0.029 0.025
0.022 0.017 0.014
In light of the asymptotic results derived above and the outcome of the simulation study, we therefore recommend the Bernstein copula technique as an effective way for estimating copulas nonparametrically.
Acknowledgment The authors thank Dr. James Allison for his important help with the Monte–Carlo section. This work was supported by the IAP Research Network P6/03 of the Belgian State (Belgian Science Policy). The second author thanks the National Research Foundation of South Africa for financial support. The third author acknowledges support from research Grant MTM2008-03129 of the Spanish Ministerio de Ciencia e Innovacion. Appendix
Proof of Lemma 1. Let F n and G n denote the marginal empirical distribution functions of the uniform ½0; 1 random variables U i ¼ FðX i Þ and V i ¼ GðY i Þ, i ¼ 1, . . . ,n and let H n denote their bivariate empirical distribution function. Then using 1 1 1 the identities (see e.g. Swanepoel, 1986) F n ðuÞ ¼ FðF 1 n ðuÞÞ and G n ðvÞ ¼ GðGn ðvÞÞ, it is easy to show that Qn can be rewritten as Qn ¼
sup 0 r u,v r 1
1
1
1
1
9H n ðF n ðuÞ,G n ðvÞÞCðu,vÞ9:
Now, Qn r
sup
1
1
1
1
9H n ðF n ðuÞ,G n ðvÞÞCðF n ðuÞ,G n ðvÞÞ9 þ sup 9F n ðuÞu9 þ sup 9G n ðvÞv9 ¼ Q n1 þ Q n2 þ Q n3 ,
0 r u,v r 1
where we used the Lipschitz property of C.
0rur1
0rvr1
1196
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
From Kiefer (1961) it follows that Q n1 ¼ Oðn1=2 ðlog log nÞ1=2 Þ a.s., n-1, and for the marginal quantiles we have that Qn2 and Qn3 are both Oðn1=2 ðlog log nÞ1=2 Þ a.s., n-1 (see e.g. Swanepoel, 1986). This proves that Q n ¼ Oðn1=2 ðlog log nÞ1=2 Þ a.s., n-1. Proof of Lemma 3(iii). VarðY mi Þ ¼
9 X
V mj ðu,vÞ,
j¼1
where V mj ðu,vÞ ¼
m X m X m m X X
0
Emj ðk,‘,k ,‘0 ÞPk,m ðuÞP‘,m ðvÞP k0 ,m ðuÞP‘0 ,m ðvÞ
k ¼ 0 ‘ ¼ 0 k0 ¼ 0 ‘ 0 ¼ 0
and 0
Em1 ðk,‘,k ,‘0 Þ ¼ C
0
Em2 ðk,‘,k ,‘0 Þ ¼ C u 0
Em3 ðk,‘,k ,‘0 Þ ¼ C v
0 0 0 k4k ‘4‘0 k ‘ k ‘ , C , C , , m m m m m m
0 0 0 0
k ‘ k ‘ k4k k k , Cu , , m m mm m m m
0 0
k ‘ k ‘ ‘4‘0 ‘ ‘0 , Cv , , m m mm m m m 0
0 0 k ‘0 k4k ‘ k ‘ k C , , C , , m m m m m m m
0
Em4 ðk,‘,k ,‘0 Þ ¼ C u
0 k ‘0 k ‘4‘0 k ‘ ‘0 , , , C C , m m m m m m m
0
Em5 ðk,‘,k ,‘0 Þ ¼ C v
0 0
0 k ‘ k4k ‘0 k ‘ k , C , , C , m m m m m m m
0
Em6 ðk,‘,k ,‘0 Þ ¼ C u
0 0 0
k ‘ k ‘4‘0 k ‘ ‘ , C , , C , m m m m m m m
0
Em7 ðk,‘,k ,‘0 Þ ¼ C v 0
Em8 ðk,‘,k ,‘0 Þ ¼ C u 0
Em9 ðk,‘,k ,‘0 Þ ¼ C v
0 0
k ‘ k ‘ k ‘0 k ‘0 , Cv , , C , m m m m mm m m
0 0 0 0
k ‘ k ‘ k ‘ ‘ k , Cu , C , : m m mm m m m m
Direct calculations give V 1m ðu,vÞ ¼ Cðu,vÞð1Cðu,vÞÞ þ 2C u ðu,vÞR1,m ðuÞ þ2C v ðu,vÞR1,m ðvÞ þ oðm1=2 Þ, V 2m ðu,vÞ ¼ uð1uÞC 2u ðu,vÞ þ2C 2u ðu,vÞR1,m ðuÞ þ oðm1=2 Þ, V 3m ðu,vÞ ¼ vð1vÞC 2v ðu,vÞ þ2C 2v ðu,vÞR1,m ðvÞ þ oðm1=2 Þ, V 4m ðu,vÞ ¼ ð1uÞCðu,vÞC u ðu,vÞ2C 2u ðu,vÞR1,m ðuÞ þ oðm1=2 Þ, V 5m ðu,vÞ ¼ ð1vÞCðu,vÞC v ðu,vÞ2C 2v ðu,vÞR1,m ðvÞ þ oðm1=2 Þ, V 6m ðu,vÞ ¼ ð1uÞCðu,vÞC u ðu,vÞ2C 2u ðu,vÞR1,m ðuÞ þ oðm1=2 Þ, V 7m ðu,vÞ ¼ ð1vÞCðu,vÞC v ðu,vÞ2C 2v ðu,vÞR1,m ðvÞ þ oðm1=2 Þ, V 8m ðu,vÞ ¼ C u ðu,vÞC v ðu,vÞ½Cðu,vÞuv þ oðm1=2 Þ, V 9m ðu,vÞ ¼ C u ðu,vÞC v ðu,vÞ½Cðu,vÞuv þ oðm1=2 Þ: Here are the details for V 1m ðu,vÞ in (8). The others follow in a similar way. m X m X k ‘ , P 2 ðuÞP 2‘,m ðvÞ V 1m ðu,vÞ ¼ C m m k,m k¼0‘¼0
ð8Þ
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
!2 0 m X m m X m m m X X X X k ‘ k4k ‘4‘0 , P k,m ðuÞP ‘,m ðvÞ þ , Pk,m ðuÞP‘,m ðvÞP k0 ,m ðuÞP‘0 ,m ðvÞ C C m m m m k¼0‘¼0 k ¼ 0 ‘ ¼ 0 k0 ¼ 0,k0 ak ‘0 ¼ 0,‘0 a‘ 0 m X m m m X m m X X X X k4k ‘ k ‘4‘0 2 Pk,m ðuÞPk0 ,m ðuÞP 2‘,m ðvÞ þ , , P k,m ðuÞP ‘,m ðvÞP‘0 ,m ðvÞ: C C þ m m m m 0 k¼0‘¼0 0 k ¼ 0 ‘ ¼ 0 ‘0 ¼ 0,‘0 a‘
1197
ð9Þ
k ¼ 0,k ak
The second term in (9) equals B2m ðu,vÞ. After Taylor expansion of Cððk=mÞ,ð‘=mÞÞ around (u,v) the first term in (9) becomes Cðu,vÞSm ðuÞSm ðvÞ þ OðSm ðuÞIm ðvÞ þSm ðvÞIm ðuÞÞ, 2 P Pm 2 where Sm ðuÞ ¼ m k ¼ 0 P k,m ðuÞ and Im ðuÞ ¼ k ¼ 0 ðk=mÞu P k,m ðuÞ. An expansion of the third term in (9) gives that it is equal to Cðu,vÞð1Sm ðuÞÞð1Sm ðvÞÞ þ 2C u ðu,vÞð1Sm ðvÞÞR1,m ðuÞ þ 2C v ðu,vÞð1Sm ðuÞÞR1,m ðvÞ þ OðR2,m ðuÞ þR2,m ðvÞÞ, P Pm 2 1 uð1uÞ ¼ Oðm1 Þ. where R2,m ðuÞ ¼ m2 m k¼0 ‘ ¼ k þ 1 ðkmuÞ P k,m ðuÞP ‘,m ðuÞ rm Similarly we obtain that the fourth and fifth term in (9) are equal to respectively Cðu,vÞSm ðvÞð1Sm ðuÞÞ þ2C u ðu,vÞSm ðvÞR1,m ðuÞ þ C v ðu,vÞð1Sm ðuÞÞOðIm ðuÞÞ and Cðu,vÞSm ðuÞð1Sm ðvÞÞ þ2C v ðu,vÞSm ðuÞR1,m ðvÞ þ C u ðu,vÞð1Sm ðvÞÞOðIm ðvÞÞ: Using Lemma 2, together with (6) and (7), gives that (8) holds. Finally use Lemma 2 to arrive at the expression for VarðY mi Þ. References Babu, G.J., Canty, A.J., Chaubey, Y.P., 2002. Application of Bernstein polynomials for smooth estimation of a distribution and density function. Journal of Statistical Planning and Inference 105, 377–392. Deheuvels, P., 1979. La fonction de de´pendence empirique et ses proprie´te´s. Un test non parame´trique d’inde´pendence. Acade´mie Royale de Belgique. Bulletin de la Classe des Sciences 65, 274–292. Fermanian, J.-D., Radulovic´, D., Wegkamp, M., 2004. Weak convergence of empirical copula processes. Bernoulli 10, 847–860. ¨ Gaenssler, P., Stute, W., 1987. Seminar on Empirical Processes. DMV Sem. 9. Birkhauser, Basel. Johnson, N.L., 1957. A note on the mean deviation of the binomial distribution. Biometrika 44, 532–533. Kiefer, J., 1961. On large deviations of the empiric d.f. of vector chance variables and a law of iterated logarithm. Pacific Journal of Mathematics 11, 649–660. Leblanc, A., 2008. A Note on the Estimation of Distribution Functions Using Bernstein Polynomials. Technical Report. University of Manitoba, Canada. Leblanc, A., 2009. Chung–Smirnov property for Bernstein estimators of distribution functions. Journal of Nonparametric Statistics 21, 133–142. Nelsen, R., 2006. An Introduction to Copulas, second ed. Springer, New York. Omelka, M., Gijbels, I., Veraverbeke, N., 2009. Improved kernel estimation of copulas: weak convergence and goodness-of-fit testing. Annals of Statistics 37, 3023–3058. Sancetta, A., Satchell, S., 2004. The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory 20, 535–562. Sklar, A., 1959. Fonctions de re´partition a n dimensions et leurs marges. Public Institute of Statistics University of Paris 8, 229–231. Swanepoel, J.W.H., 1986. A note on proving that the (modified) bootstrap works. Communications in Statistics—Theory and Methods 15, 3193–3203. Sancetta, A., 2007. Nonparametric estimation of distributions with given marginals via Bernstein–Kantarovich polynomials: L1 and pointwise convergence. Journal of Multivariate Analysis 98, 1376–1390. van der Vaart, A., Wellner, J.A., 2000. Weak Convergence of Empirical Processes. Springer, New York. Wienke, A., 2011. Frailty Models in Survival Analysis. Chapman & Hall/CRC Biostatistics Series, Boca Raton.
Contents lists available at SciVerse ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Large sample behavior of the Bernstein copula estimator ¨ Veraverbeke a,n Paul Janssen a, Jan Swanepoel b, Noel a b
Hasselt University, Center for Statistics, Agoralaan, Gebouw D, 3590 Diepenbeek, Belgium North-West University, Potchefstroom Campus, Potchefstroom, South Africa
a r t i c l e i n f o
abstract
Article history: Received 24 February 2011 Received in revised form 4 October 2011 Accepted 29 November 2011 Available online 6 December 2011
Bernstein polynomial estimators have been used as smooth estimators for density functions and distribution functions. The idea of using them for copula estimation has been given in Sancetta and Satchell (2004). In the present paper we study the asymptotic properties of this estimator: almost sure consistency rates and asymptotic normality. We also obtain explicit expressions for the asymptotic bias and asymptotic variance and show the improvement of the asymptotic mean squared error compared to that of the classical empirical copula estimator. A small simulation study illustrates this superior behavior in small samples. & 2011 Elsevier B.V. All rights reserved.
Keywords: Asymptotic properties Bernstein estimator Copula estimator Mean squared error
1. The Bernstein copula estimator Copulas are functions that couple multivariate distribution functions to their one-dimensional marginal distribution functions. Copulas provide a very nice tool to model multivariate data and are therefore very useful in, for example, financial economics. See e.g. Sancetta and Satchell (2004) and Sancetta (2007). The analysis of multivariate survival data provides another example where copulas are instrumental. See e.g. Wienke (2011). Consider a random vector X ¼ ðX 1 , . . . ,X d ÞT with joint cumulative distribution function H and marginal distribution functions F 1 , . . . ,F d . From Sklar’s theorem (Sklar, 1959) there exists a d-variate function C such that Hðx1 , . . . ,xd Þ ¼ CðF 1 ðx1 Þ, . . . ,F d ðxd ÞÞ: For a detailed account on copulas we refer to the excellent book by Nelsen (2006), which includes many examples. A variety of parametric models for copulas and marginal distributions has been proposed and the corresponding parametric estimation methods are available. The study of nonparametric estimators of copulas (kernel smoothed empirical copulas) goes back to Deheuvels (1979) and Gaenssler and Stute (1987). In more recent papers kernel based smooth versions of the empirical copula have been studied. Omelka et al. (2009) study weak convergence of improved kernel estimators and include an excellent overview. Nonparametric estimation of copulas is also the focus of this paper. For simplicity we restrict the presentation to the case of bivariate data ðd ¼ 2Þ. We consider a bivariate random vector (X,Y) with joint distribution function H and marginal distribution functions F and G, i.e., Hðx,yÞ ¼ PðX rx,Y ryÞ; FðxÞ ¼ PðX rxÞ and GðyÞ ¼ PðY ryÞ. Sklar’s theorem says that there exists a copula C on ½0; 12 such that Hðx,yÞ ¼ CðFðxÞ,GðyÞÞ:
n
Corresponding author. E-mail addresses: [email protected] (P. Janssen), [email protected] (J. Swanepoel), [email protected] (N. Veraverbeke).
0378-3758/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2011.11.020
1190
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
We will assume throughout that F and G are continuous, which implies that C is unique and Cðu,vÞ ¼ HðF 1 ðuÞ,G1 ðvÞÞ: Given a sample ðX 1 ,Y 1 Þ, . . . ,ðX n ,Y n Þ from the bivariate distribution function H, the empirical copula estimator for Cðu,vÞ is given by 1 C n ðu,vÞ ¼ Hn ðF 1 n ðuÞ,Gn ðvÞÞ
with Hn ðx,yÞ ¼ n1
n X
IðX i r x,Y i ryÞ
i¼1
having marginals F n ðxÞ ¼ Hn ðx, þ 1Þ ¼ n1
n X
IðX i rxÞ,
i¼1
Gn ðyÞ ¼ Hn ð þ1,yÞ ¼ n1
n X
IðY i r yÞ:
i¼1
In this paper we consider Bernstein copula estimators, which are defined in terms of Bernstein polynomials. Before we give the precise definition of Bernstein copula estimators, we review some results on smooth estimators for the distribution function of univariate data, since the results obtained in this paper provide extensions of these results to multivariate (bivariate) data. For univariate data, Bernstein estimators have been used by Babu et al. (2002) to estimate the univariate distribution and density function. Leblanc (2008) proves the asymptotic normality of the Bernstein estimator of the univariate distribution function and shows that the estimator outperforms the classical empirical distribution function estimator by deriving an explicit formula for the first order improvement for the asymptotic variance. In Leblanc (2009) it is shown that the Chung–Smirnov property holds for the Bernstein estimator. In the present paper we extend both results to the Bernstein empirical copula. Towards the definition of the Bernstein copula estimator, recall that the Bernstein polynomial of order m 4 0 corresponding to the copula C is defined as m X m X k ‘ Bm ðu,vÞ ¼ , P ðuÞP l,m ðvÞ C m m k,m k¼0l¼0 with P k,m ðuÞ ¼
m uk ð1uÞmk k
the binomial probabilities. We have, uniformly in ðu,vÞ 2 ½0; 12 , lim Bm ðu,vÞ ¼ Cðu,vÞ
m-1
since C is continuous on ½0; 12 . Note that Bm is a copula by itself, since it satisfies the sufficient conditions given in Theorem 1 of Sancetta and Satchell (2004). They propose the following Bernstein estimator of order m 40 of the copula function C: m X m X k ‘ C m,n ðu,vÞ ¼ , P ðuÞPl,m ðvÞ, Cn m m k,m k¼0l¼0 where Cn is the empirical copula estimator. The order of m will depend on n and we will have that m-1 if n-1. In Section 2 we obtain Chung–Smirnov consistency rates for the Bernstein copula estimator. In Section 3 we use a stochastic approximation for the Bernstein copula estimator to study the asymptotic bias and variance. Based on these results we show that the Bernstein copula estimator outperforms the empirical copula estimator. The asymptotic normality of the smooth estimator follows as an easy consequence. Section 4 provides a small simulation study in which for a number of examples it is shown that the Bernstein copula estimator C m,n outperforms the empirical copula estimator Cn. Finally, some of the more technical proofs are collected in Appendix. 2. Chung–Smirnov consistency rates For any function g on ½0; 12 we define JgJ ¼ sup0 r u,v r 1 9gðu,vÞ9, the supremum norm. In this section we give uniform strong consistency rates for the Bernstein copula estimator.
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
1191
Theorem 1. If m mðnÞ-1 and n=ðm log log nÞ-c Z 0, then JC m,n CJ ¼ Oðn1=2 ðlog log nÞ1=2 Þ a.s., n-1. To prove this theorem we rely on the following lemma (see Appendix for the proof). Lemma 1. sup
Q n ¼ JC n CJ ¼
0 r u,v r 1
1 1 9Hn ðF 1 ðuÞ,G1 ðvÞÞ9 ¼ Oðn1=2 ðlog log nÞ1=2 Þ a:s:, n-1: n ðuÞ,Gn ðvÞÞHðF
Proof of Theorem 1. JC m,n CJ r JC m,n Bm J þ JBm CJ:
ð1Þ
Lemma 1 and the fact that P k,m ðuÞ and P‘,m ðvÞ, k,‘ ¼ 0, . . . ,m, are binomial probabilities, imply m X m X k ‘ k ‘ , C Pk,m ðuÞP‘,m ðvÞ Cn JC m,n Bm J ¼ sup m m mm 0 r u,v r 1k ¼ 0 ‘ ¼ 0 k ‘ k ‘ , C , r JC n CJ ¼ Oðn1=2 ðlog log nÞ1=2 Þ a:s:, n-1: r max C n m m m m 0 r k,‘ r m
ð2Þ
To obtain an order bound for the second term in the rhs of (1), we use the Lipschitz property of copulas and we write Bu and Bv to denote binomial random variables with parameters m and u, and m and v respectively. We then have X m X m k ‘ , Cðu,vÞ Pk,m ðuÞP‘,m ðvÞ C JBm CJ ¼ sup m m 0 r u,v r 1k ¼ 0 ‘ ¼ 0 2 !1=2 2 !1=2 m X m X k Bu Bv u þ ‘ v P k,m ðuÞP ‘,m ðvÞ r sup u v E þ sup E r sup m m m 0 r u,v r 1 k ¼ 0 ‘ ¼ 0 m 0rur1 0rvr1 1=2 1=2 uð1uÞ vð1vÞ 1 1 ¼ sup þ sup r þ ¼ m1=2 ¼ Oðn1=2 ðlog log nÞ1=2 Þ: ð3Þ m m 2m1=2 2m1=2 0rur1 0rvr1 The proof follows from (1)–(3).
&
Remark 1. It is possible to modify the proof of Theorem 1 slightly and to obtain the same result under stronger conditions on C and weaker conditions on m. Indeed, if C has first order partial derivatives Cu and Cv that are Lipschitz continuous of order a ð0 o a r 1Þ and if n=ðm1 þ a log log nÞ-c Z 0 then JC m,n CJ ¼ Oðn1=2 ðlog log nÞ1=2 Þ a.s., n-1. 3. Asymptotic bias and variance of the Bernstein copula estimator Towards an asymptotic expression for the bias and the variance of the Bernstein copula estimator, the following decomposition is useful: n1=2 ðC m,n ðu,vÞCðu,vÞÞ ¼ n1=2 ðC m,n ðu,vÞBm ðu,vÞÞ þ n1=2 ðBm ðu,vÞCðu,vÞÞ: 2
ð4Þ
Assuming bounded third order partial derivatives on ð0; 1Þ for C, we have the following convergence rate of the Bernstein approximation: " 2 2 m X m X 1 k 1 ‘ u C uu ðu,vÞ þ v C vv ðu,vÞ Bm,n ðu,vÞCðu,vÞ ¼ 2 m 2 m k¼0‘¼0 k ‘ u v C uv ðu,vÞ P k,m ðuÞP ‘,m ðvÞ þ oðm1 Þ ¼ m1 bðu,vÞ þ oðm1 Þ, ð5Þ þ m m where bðu,vÞ ¼ 12½uð1uÞC uu ðu,vÞ þvð1vÞC vv ðu,vÞ: To handle the first term in (4) we rely on the following two lemmas. Lemma 2. For 0 ou o1, R1,m ðuÞ :¼ m1
m X
m X
k ¼ 0 ‘ ¼ kþ1
where
c1 ðuÞ ¼
uð1uÞ 1=2 : 4p
ðkmuÞP k,m ðuÞP ‘,m ðuÞ ¼ m1=2 fc1 ðuÞ þ oð1Þg,
1192
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
Proof. R1,m ðuÞ ¼ m1
m X
m X
ðkmuÞP k,m ðuÞP ‘,m ðuÞ ¼ m1
k ¼ 0 ‘ ¼ kþ1
¼ m1
m X
"
P ‘,m ðuÞ
‘¼1
m X
#
m X ‘¼1
ðkmuÞP k,m ðuÞ ¼ m1
k¼‘
P ‘,m ðuÞ
‘1 X
ðkmuÞP k,m ðuÞ
k¼0
h m i P ‘,m ðuÞ ‘ u‘ ð1uÞm‘ þ 1 , ‘ ‘¼1 m X
where the last equality is an identity in Johnson (1957). Then, m m m m X X X X ‘ ‘ u þu P 2‘,m ðuÞ ¼ uð1uÞ u P 2‘,m ðuÞ: ‘P2‘,m ðuÞ ¼ ð1uÞ P2‘,m ðuÞð1uÞ R1,m ðuÞ ¼ m1 ð1uÞ m m ‘¼1 ‘¼1 ‘¼1 ‘¼1 Now, by Lemma 3.1 in Babu et al. (2002), we have m X
P 2‘,m ðuÞ ¼ m1=2 fð4puð1uÞÞ1=2 þ oð1Þg
ð6Þ
‘¼1
and, using the Cauchy–Schwarz inequality and the fact that 0 rP ‘,m ðuÞ r1, we also have m X ‘ u P 2‘,m ðuÞ ¼ Oðm3=4 Þ: m ‘¼1
ð7Þ
Hence R1,m ðuÞ ¼ m1=2 fc1 ðuÞ þoð1Þg:
&
Lemma 3. Assume bounded second order partial derivatives for C on ð0; 1Þ2 . Then P (i) n1=2 ðC m,n ðu,vÞBm ðu,vÞÞ ¼ n1=2 ni¼ 1 Y mi þ oP ð1Þ where m X m X k ‘ k ‘ k ‘ k k C , C u , I Ui r Y mi ¼ I U i r ,V i r m m m m m m m m k¼0‘¼0
k ‘ ‘ ‘ C v , I Vi r P k,m ðuÞP ‘,m ðvÞ m m m m and where ðU 1 ,V 1 Þ, . . . ,ðU n ,V n Þ are independent random vectors with distribution function C and with uniform marginals on ½0; 1. (ii) The Ymi’s are bounded and have mean zero. (iii) VarðY mi Þ ¼ s2 ðu,vÞm1=2 Vðu,vÞ, where
s2 ðu,vÞ ¼ VarfIðU ru,V rvÞCðu,vÞC u ðu,vÞ½IðU r uÞuC v ðu,vÞ½IðV r vÞvg ¼ Cðu,vÞð1Cðu,vÞÞ þ uð1uÞC 2u ðu,vÞ þ vð1vÞC 2v ðu,vÞ2ð1uÞCðu,vÞC u ðu,vÞ2ð1vÞCðu,vÞC v ðu,vÞ þ 2C u ðu,vÞC v ðu,vÞ½Cðu,vÞuv and
"
# " # uð1uÞ 1=2 vð1vÞ 1=2 þ C v ðu,vÞð1C v ðu,vÞÞ : Vðu,vÞ ¼ C u ðu,vÞð1C u ðu,vÞÞ
p
p
Proof of Lemma 3. The proof of (i) is immediate since we have, uniformly in (u,v), the following asymptotic representation (see Fermanian et al., 2004, p. 857): C n ðu,vÞCðu,vÞ ¼ n1
n X
fIðU i ru,V i rvÞCðu,vÞC u ðu,vÞ½IðU i ruÞuC v ðu,vÞ½IðV i r vÞvgþ oP ðn1=2 Þ:
i¼1
The proof of (ii) is immediate and the technical proof of (iii) is available from Appendix. Theorem 2. Assume bounded third order partial derivatives for C on ð0; 1Þ2 . If n1=2 m1 -0, then for ðu,vÞ 2 ð0; 1Þ2 D
n1=2 ðC m,n ðu,vÞCðu,vÞÞ-N ð0, s2 ðu,vÞÞ:
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
1193
If n1=2 m1 -d, 0 o d o 1, then for ðu,vÞ 2 ð0; 1Þ2 D
n1=2 ðC m,n ðu,vÞCðu,vÞÞ-N ðdbðu,vÞ, s2 ðu,vÞÞ: Remark 2. Note that the asymptotic variance s2 ðu,vÞ in Theorem 2 coincides with the asymptotic variance in the limit distribution of n1=2 ðC n ðu,vÞCðu,vÞÞ. However, from Lemma 3(iii) it is clear that the first-order term in the expansion of the variance of n1=2 ðC m,n ðu,vÞCðu,vÞÞ is smaller than s2 ðu,vÞ (at the price of some bias in case n1=2 m1 -d 40). Remark 3. The asymptotic normality result in Theorem 2 can be strengthened to weak convergence of the process n1=2 ðC m,n ðu,vÞCðu,vÞÞ in the space ‘1 ðð0; 1Þ2 Þ of bounded functions. The limiting process is Gaussian with mean function dbðu,vÞ and with covariance function E½fIðU r u,V r vÞCðu,vÞC u ðu,vÞ½IðU ruÞuC v ðu,vÞ½IðV r vÞvgg fIðU r u0 ,V rv0 ÞCðu0 ,v0 ÞC u ðu0 ,v0 Þ½IðU ru0 Þu0 C v ðu0 ,v0 Þ½IðV r v0 Þv0 g for 0 ou,v,u0 ,v0 o 1. P Because of the representation in Lemma 3(i), it suffices to prove the asymptotic tightness of the process fn1=2 ni¼ 1 Y m,i g. 2 The Ymi as functions of u and v are uniformly bounded and continuous on ð0; 1Þ and have non-zero derivatives up to order m. By Theorem 2.7.1 in van der Vaart and Wellner (2000) such class of functions has finite e bracketing number of order Oðexpðe2=m ÞÞ. This result and the boundedness of the Ymi as functions of u and v guarantee, for m 41, the validity of the integrability condition on the square root of the logarithm of the bracketing number. An interesting further question, beyond the scope of this paper, is whether the weak convergence on ð0; 1Þ2 can be extended to ½0; 12 as Fermanian et al. (2004) did for the empirical copula process. Remark 4. The variance expression in Lemma 3(iii) is useful to arrive at an optimal choice of the order m. For the asymptotic mean squared error we have 2
AMSEðC m,n ðu,vÞÞ ¼ AsVarðC m,n ðu,vÞÞ þðAsBiasðC m,n ðu,vÞÞÞ2 ¼ n1 s2 ðu,vÞm1=2 n1 Vðu,vÞ þm2 b ðu,vÞ: Minimizing with respect to m gives !2=3 2 4b ðu,vÞ n2=3 m0 m0 ðu,vÞ ¼ Vðu,vÞ and 1
AMSEðC m0 ,n ðu,vÞÞ ¼ n
V 2 ðu,vÞ s ðu,vÞ3 16bðu,vÞ
!2=3
2
n4=3 :
Since AsVarðC n ðu,vÞÞ ¼ n1 s2 ðu,vÞ and since Cn is biased, it is clear that AMSEðC m0 ,n ðu,vÞÞ is less than AMSEðC n ðu,vÞÞ. 4. A Monte–Carlo comparison We now report on some small sample experiments which compare the empirical copula estimator Cn with the Bernstein copula estimator C m0 ,n . The quality of the estimators is firstly evaluated locally by empirically calculating their mean L1-norm, defined by RðC^ ,C y Þ ¼ EðLðC^ ,C y ÞÞ, where LðC^ ,C y Þ ¼ 9C^ ðu,vÞC y ðu,vÞ9 with C^ ¼ C n or C^ ¼ C m0 ,n where m0 ¼ m0 ðu,vÞ, and C y a given family of copulas. Secondly they are evaluated globally by calculating their mean discrete L2-norm, given by 8 9 1=2 n X n < 1 X i j i j 2= ^ ^ , C y , LðC ,C y Þ ¼ C :ðn1Þ2 ; n n n n i¼1j¼1
with C^ ¼ C n or C^ ¼ C m0 ,n where m0 ¼ m0 ðði=nÞ,ðj=nÞÞ, i,j ¼ 1, . . . ,n1. Data were generated from the following copulas (see e.g. Nelsen, 2006): 1. The Farlie–Gumbel–Morgenstern copula (FGMC), C y ðu,vÞ ¼ uv þ yuvð1uÞð1vÞ,
1 r y r 1:
In this case, Spearman’s rho, rS , is given by rS ¼ y=3.
1194
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
2. The Cuadras–Auge´ copula (CAC), C y ðu,vÞ ¼ fminðu,vÞgy ðuvÞ1y ,
0 r y r1:
Now, rS ¼ 3y=ð4yÞ. 3. The Plackett copula (PC): with y 40 and ya1, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½1 þðy1Þðu þ vÞ ½1 þ ðy1Þðu þ vÞ2 4uvyðy1Þ : C y ðu,vÞ ¼ 2ðy1Þ We now have that rS ¼ ððy þ 1Þ=ðy1ÞÞðð2y log yÞ=ðy1Þ2 Þ. Each entry in Tables 1–4 is based on 20 000 independent trials for sample sizes n ¼20, 30, 40. The standard errors of the Monte-Carlo estimates were found to be negligibly small and are not reported in the table. All calculations were done using double precision arithmetic in FORTRAN and routines from the IMSL library. The results in Tables 1–3 illustrate the good local performance of the Bernstein copula estimator compared to the classical empirical copula estimator for small to moderate sample sizes. It is clear that RðC m0 ,n ,C y Þ oRðC n ,C y Þ everywhere except for 4 out of the 45 cases, viz. the CAC when n ¼ 30; 40 and ðu,vÞ 2 fð0:50,0:50Þ,ð0:75,0:75Þg. The superior behavior of the Bernstein copula estimator is even more evident from Table 4. As far as global performance is concerned, the results in the table show that the strict inequality above holds everywhere.
Table 1 Monte–Carlo estimates of expected L1-norms for the empirical copula and Bernstein copula estimators. RðC n ,C y Þ
RðC m0 ,n ,C y Þ
0.50) 0.25) 0.75) 0.25) 0.75)
0.044 0.036 0.031 0.031 0.036
0.027 0.015 0.016 0.016 0.016
(0.20, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.037 0.031 0.028 0.028 0.033
0.024 0.012 0.011 0.010 0.017
(0.50, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.031 0.026 0.020 0.020 0.026
0.018 0.012 0.011 0.011 0.015
Copula
(u,v)
FGMC ðy ¼ 0:9, rS ¼ 0:3, n ¼ 20Þ
(0.50, (0.25, (0.25, (0.75, (0.75,
FGMC ðy ¼ 0:9, rS ¼ 0:3, n ¼ 30Þ
FGMC ðy ¼ 0:9, rS ¼ 0:3, n ¼ 40Þ
Table 2 Monte–Carlo estimates of expected L1-norms for the empirical copula and Bernstein copula estimators. RðC n ,C y Þ
RðC m0 ,n ,C y Þ
0.50) 0.25) 0.75) 0.25) 0.75)
0.047 0.036 0.031 0.031 0.041
0.045 0.020 0.012 0.012 0.041
(0.50, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.038 0.030 0.030 0.030 0.031
0.046 0.020 0.007 0.007 0.049
(0.50, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.031 0.026 0.022 0.022 0.026
0.048 0.023 0.006 0.006 0.053
Copula
(u,v)
CAC ðy ¼ 0:364, rS ¼ 0:3, n ¼ 20Þ
(0.50, (0.25, (0.25, (0.75, (0.75,
CAC ðy ¼ 0:364, rS ¼ 0:3, n ¼ 30Þ
CAC ðy ¼ 0:364, rS ¼ 0:3, n ¼ 40Þ
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
1195
Table 3 Monte–Carlo estimates of expected L1-norms for the empirical copula and Bernstein copula estimators. RðC n ,C y Þ
RðC m0 ,n ,C y Þ
0.50) 0.25) 0.75) 0.25) 0.75)
0.043 0.034 0.031 0.031 0.034
0.027 0.017 0.009 0.009 0.019
(0.50, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.036 0.030 0.029 0.029 0.032
0.025 0.015 0.009 0.009 0.017
(0.50, (0.25, (0.25, (0.75, (0.75,
0.50) 0.25) 0.75) 0.25) 0.75)
0.031 0.025 0.021 0.021 0.025
0.018 0.014 0.008 0.008 0.017
Copula
(u,v)
PC ðy ¼ 2:524, rS ¼ 0:3, n ¼ 20Þ
(0.50, (0.25, (0.25, (0.75, (0.75,
PC ðy ¼ 2:524, rS ¼ 0:3, n ¼ 30Þ
PC ðy ¼ 2:524, rS ¼ 0:3, n ¼ 40Þ
Table 4 Monte–Carlo estimates of expected L2-norms for the empirical copula and Bernstein copula estimators. Copula
n
RðC n ,C y Þ
RðC m0 ,n ,C y Þ
FGMC ðy ¼ 0:9, rS ¼ 0:3Þ
20 30 40
0.036 0.029 0.025
0.022 0.016 0.014
CAC ðy ¼ 0:364, rS ¼ 0:3Þ
20 30 40
0.037 0.030 0.026
0.023 0.022 0.022
PC ðy ¼ 2:524, rS ¼ 0:3Þ
20 30 40
0.036 0.029 0.025
0.022 0.017 0.014
In light of the asymptotic results derived above and the outcome of the simulation study, we therefore recommend the Bernstein copula technique as an effective way for estimating copulas nonparametrically.
Acknowledgment The authors thank Dr. James Allison for his important help with the Monte–Carlo section. This work was supported by the IAP Research Network P6/03 of the Belgian State (Belgian Science Policy). The second author thanks the National Research Foundation of South Africa for financial support. The third author acknowledges support from research Grant MTM2008-03129 of the Spanish Ministerio de Ciencia e Innovacion. Appendix
Proof of Lemma 1. Let F n and G n denote the marginal empirical distribution functions of the uniform ½0; 1 random variables U i ¼ FðX i Þ and V i ¼ GðY i Þ, i ¼ 1, . . . ,n and let H n denote their bivariate empirical distribution function. Then using 1 1 1 the identities (see e.g. Swanepoel, 1986) F n ðuÞ ¼ FðF 1 n ðuÞÞ and G n ðvÞ ¼ GðGn ðvÞÞ, it is easy to show that Qn can be rewritten as Qn ¼
sup 0 r u,v r 1
1
1
1
1
9H n ðF n ðuÞ,G n ðvÞÞCðu,vÞ9:
Now, Qn r
sup
1
1
1
1
9H n ðF n ðuÞ,G n ðvÞÞCðF n ðuÞ,G n ðvÞÞ9 þ sup 9F n ðuÞu9 þ sup 9G n ðvÞv9 ¼ Q n1 þ Q n2 þ Q n3 ,
0 r u,v r 1
where we used the Lipschitz property of C.
0rur1
0rvr1
1196
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
From Kiefer (1961) it follows that Q n1 ¼ Oðn1=2 ðlog log nÞ1=2 Þ a.s., n-1, and for the marginal quantiles we have that Qn2 and Qn3 are both Oðn1=2 ðlog log nÞ1=2 Þ a.s., n-1 (see e.g. Swanepoel, 1986). This proves that Q n ¼ Oðn1=2 ðlog log nÞ1=2 Þ a.s., n-1. Proof of Lemma 3(iii). VarðY mi Þ ¼
9 X
V mj ðu,vÞ,
j¼1
where V mj ðu,vÞ ¼
m X m X m m X X
0
Emj ðk,‘,k ,‘0 ÞPk,m ðuÞP‘,m ðvÞP k0 ,m ðuÞP‘0 ,m ðvÞ
k ¼ 0 ‘ ¼ 0 k0 ¼ 0 ‘ 0 ¼ 0
and 0
Em1 ðk,‘,k ,‘0 Þ ¼ C
0
Em2 ðk,‘,k ,‘0 Þ ¼ C u 0
Em3 ðk,‘,k ,‘0 Þ ¼ C v
0 0 0 k4k ‘4‘0 k ‘ k ‘ , C , C , , m m m m m m
0 0 0 0
k ‘ k ‘ k4k k k , Cu , , m m mm m m m
0 0
k ‘ k ‘ ‘4‘0 ‘ ‘0 , Cv , , m m mm m m m 0
0 0 k ‘0 k4k ‘ k ‘ k C , , C , , m m m m m m m
0
Em4 ðk,‘,k ,‘0 Þ ¼ C u
0 k ‘0 k ‘4‘0 k ‘ ‘0 , , , C C , m m m m m m m
0
Em5 ðk,‘,k ,‘0 Þ ¼ C v
0 0
0 k ‘ k4k ‘0 k ‘ k , C , , C , m m m m m m m
0
Em6 ðk,‘,k ,‘0 Þ ¼ C u
0 0 0
k ‘ k ‘4‘0 k ‘ ‘ , C , , C , m m m m m m m
0
Em7 ðk,‘,k ,‘0 Þ ¼ C v 0
Em8 ðk,‘,k ,‘0 Þ ¼ C u 0
Em9 ðk,‘,k ,‘0 Þ ¼ C v
0 0
k ‘ k ‘ k ‘0 k ‘0 , Cv , , C , m m m m mm m m
0 0 0 0
k ‘ k ‘ k ‘ ‘ k , Cu , C , : m m mm m m m m
Direct calculations give V 1m ðu,vÞ ¼ Cðu,vÞð1Cðu,vÞÞ þ 2C u ðu,vÞR1,m ðuÞ þ2C v ðu,vÞR1,m ðvÞ þ oðm1=2 Þ, V 2m ðu,vÞ ¼ uð1uÞC 2u ðu,vÞ þ2C 2u ðu,vÞR1,m ðuÞ þ oðm1=2 Þ, V 3m ðu,vÞ ¼ vð1vÞC 2v ðu,vÞ þ2C 2v ðu,vÞR1,m ðvÞ þ oðm1=2 Þ, V 4m ðu,vÞ ¼ ð1uÞCðu,vÞC u ðu,vÞ2C 2u ðu,vÞR1,m ðuÞ þ oðm1=2 Þ, V 5m ðu,vÞ ¼ ð1vÞCðu,vÞC v ðu,vÞ2C 2v ðu,vÞR1,m ðvÞ þ oðm1=2 Þ, V 6m ðu,vÞ ¼ ð1uÞCðu,vÞC u ðu,vÞ2C 2u ðu,vÞR1,m ðuÞ þ oðm1=2 Þ, V 7m ðu,vÞ ¼ ð1vÞCðu,vÞC v ðu,vÞ2C 2v ðu,vÞR1,m ðvÞ þ oðm1=2 Þ, V 8m ðu,vÞ ¼ C u ðu,vÞC v ðu,vÞ½Cðu,vÞuv þ oðm1=2 Þ, V 9m ðu,vÞ ¼ C u ðu,vÞC v ðu,vÞ½Cðu,vÞuv þ oðm1=2 Þ: Here are the details for V 1m ðu,vÞ in (8). The others follow in a similar way. m X m X k ‘ , P 2 ðuÞP 2‘,m ðvÞ V 1m ðu,vÞ ¼ C m m k,m k¼0‘¼0
ð8Þ
P. Janssen et al. / Journal of Statistical Planning and Inference 142 (2012) 1189–1197
!2 0 m X m m X m m m X X X X k ‘ k4k ‘4‘0 , P k,m ðuÞP ‘,m ðvÞ þ , Pk,m ðuÞP‘,m ðvÞP k0 ,m ðuÞP‘0 ,m ðvÞ C C m m m m k¼0‘¼0 k ¼ 0 ‘ ¼ 0 k0 ¼ 0,k0 ak ‘0 ¼ 0,‘0 a‘ 0 m X m m m X m m X X X X k4k ‘ k ‘4‘0 2 Pk,m ðuÞPk0 ,m ðuÞP 2‘,m ðvÞ þ , , P k,m ðuÞP ‘,m ðvÞP‘0 ,m ðvÞ: C C þ m m m m 0 k¼0‘¼0 0 k ¼ 0 ‘ ¼ 0 ‘0 ¼ 0,‘0 a‘
1197
ð9Þ
k ¼ 0,k ak
The second term in (9) equals B2m ðu,vÞ. After Taylor expansion of Cððk=mÞ,ð‘=mÞÞ around (u,v) the first term in (9) becomes Cðu,vÞSm ðuÞSm ðvÞ þ OðSm ðuÞIm ðvÞ þSm ðvÞIm ðuÞÞ, 2 P Pm 2 where Sm ðuÞ ¼ m k ¼ 0 P k,m ðuÞ and Im ðuÞ ¼ k ¼ 0 ðk=mÞu P k,m ðuÞ. An expansion of the third term in (9) gives that it is equal to Cðu,vÞð1Sm ðuÞÞð1Sm ðvÞÞ þ 2C u ðu,vÞð1Sm ðvÞÞR1,m ðuÞ þ 2C v ðu,vÞð1Sm ðuÞÞR1,m ðvÞ þ OðR2,m ðuÞ þR2,m ðvÞÞ, P Pm 2 1 uð1uÞ ¼ Oðm1 Þ. where R2,m ðuÞ ¼ m2 m k¼0 ‘ ¼ k þ 1 ðkmuÞ P k,m ðuÞP ‘,m ðuÞ rm Similarly we obtain that the fourth and fifth term in (9) are equal to respectively Cðu,vÞSm ðvÞð1Sm ðuÞÞ þ2C u ðu,vÞSm ðvÞR1,m ðuÞ þ C v ðu,vÞð1Sm ðuÞÞOðIm ðuÞÞ and Cðu,vÞSm ðuÞð1Sm ðvÞÞ þ2C v ðu,vÞSm ðuÞR1,m ðvÞ þ C u ðu,vÞð1Sm ðvÞÞOðIm ðvÞÞ: Using Lemma 2, together with (6) and (7), gives that (8) holds. Finally use Lemma 2 to arrive at the expression for VarðY mi Þ. References Babu, G.J., Canty, A.J., Chaubey, Y.P., 2002. Application of Bernstein polynomials for smooth estimation of a distribution and density function. Journal of Statistical Planning and Inference 105, 377–392. Deheuvels, P., 1979. La fonction de de´pendence empirique et ses proprie´te´s. Un test non parame´trique d’inde´pendence. Acade´mie Royale de Belgique. Bulletin de la Classe des Sciences 65, 274–292. Fermanian, J.-D., Radulovic´, D., Wegkamp, M., 2004. Weak convergence of empirical copula processes. Bernoulli 10, 847–860. ¨ Gaenssler, P., Stute, W., 1987. Seminar on Empirical Processes. DMV Sem. 9. Birkhauser, Basel. Johnson, N.L., 1957. A note on the mean deviation of the binomial distribution. Biometrika 44, 532–533. Kiefer, J., 1961. On large deviations of the empiric d.f. of vector chance variables and a law of iterated logarithm. Pacific Journal of Mathematics 11, 649–660. Leblanc, A., 2008. A Note on the Estimation of Distribution Functions Using Bernstein Polynomials. Technical Report. University of Manitoba, Canada. Leblanc, A., 2009. Chung–Smirnov property for Bernstein estimators of distribution functions. Journal of Nonparametric Statistics 21, 133–142. Nelsen, R., 2006. An Introduction to Copulas, second ed. Springer, New York. Omelka, M., Gijbels, I., Veraverbeke, N., 2009. Improved kernel estimation of copulas: weak convergence and goodness-of-fit testing. Annals of Statistics 37, 3023–3058. Sancetta, A., Satchell, S., 2004. The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory 20, 535–562. Sklar, A., 1959. Fonctions de re´partition a n dimensions et leurs marges. Public Institute of Statistics University of Paris 8, 229–231. Swanepoel, J.W.H., 1986. A note on proving that the (modified) bootstrap works. Communications in Statistics—Theory and Methods 15, 3193–3203. Sancetta, A., 2007. Nonparametric estimation of distributions with given marginals via Bernstein–Kantarovich polynomials: L1 and pointwise convergence. Journal of Multivariate Analysis 98, 1376–1390. van der Vaart, A., Wellner, J.A., 2000. Weak Convergence of Empirical Processes. Springer, New York. Wienke, A., 2011. Frailty Models in Survival Analysis. Chapman & Hall/CRC Biostatistics Series, Boca Raton.