I N S T I T U T D E S T A T I S T I Q U E

Transcription

1 I N S T I T U T D E S T A T I S T I Q U E UNIVERSITÉ CATHOLIQUE DE LOUVAIN D I S C U S S I O N P A P E R 0822 ASYMPTOTIC PROPERTIES OF THE BERNSTEIN DENSITY COPULA FOR DEPENDENT DATA BOUEZMARNI, T., ROMBOUTS, J. ad A. TAAMOUTI This file ca be dowloaded from

2 Asymptotic Properties of the Berstei Desity Copula for Depedet Data Taoufik Bouezmari Jeroe V.K. Rombouts Abderrahim Taamouti Jue 30, 2008 ABSTRACT Copulas are extesively used for depedece modelig. I may cases the data does ot reveal how the depedece ca be modeled usig a particular parametric copula. Noparametric copulas do ot share this problem sice they are etirely data based. This paper proposes oparametric estimatio of the desity copula for α mixig data usig Berstei polyomials. We study the asymptotic properties of the Berstei desity copula, i.e., we provide the exact asymptotic bias ad variace, we establish the uiform strog cosistecy ad the asymptotic ormality. Key words: Noparametric estimatio; copula; Berstei polyomial; α mixig; asymptotic properties; boudary bias. Départemet de mathématiques et de statistique, Uiversité de Motréal ad Istitute of Statistics, Uiversité catholique de Louvai. Address: Départemet de mathématiques et de statistique, Uiversité de Motréal, C.P. 6128, succursale Cetre-ville Motréal, Caada, H3C 3J7. Istitute of Applied Ecoomics at HEC Motréal, CIRANO, CIRPEE, CORE (Uiversité catholique de Louvai. Address: 3000 Cote Saite Catherie, Motréal (QC, Caada, H3T 2A7. TEL: ; FAX: ; jeroe.rombouts@hec.ca Ecoomics Departmet, Uiversidad Carlos III de Madrid. Address: Departameto de Ecoomía Uiversidad Carlos III de Madrid Calle Madrid, Getafe (Madrid España. TEL: ; FAX: ; ataamout@eco.uc3m.es. 1

3 1 Itroductio The correlatio coefficiet of Pearso, the Kedall s tau, ad Spearma s rho are widely used to measure the depedece betwee variables. Despite their simplicity to implemet ad iterpret, they are ot able to capture all forms of depedece. The copula fuctio has the advatage to model completely the depedece amog variables. Further, the above coefficiets of depedece ca be deduced from the copula. Thaks to Sklar (1959, the copula fuctio is directly liked to the distributio fuctio. Ideed, the distributio fuctio ca be cotrolled by margial distributios, which give the iformatio o each compoet, ad the copula that captures the depedece betwee compoets. There are several ways to estimate copulas. First, the parametric approach imposes a specific model for both the copula ad margial distributios. We estimate the parameters usig the maximum likelihood or iferece fuctio for margis. This approach is widely used i practice because of its simplicity, see Oakes (1982 ad Joe (2005 for details. Secod, the semiparametric approach assumes a parametric model for the copula ad a oparametric model for the margial distributios. This approach is studied by Oakes (1982, Geest ad Rivest (1993, ad Geest, Ghoudi, ad Rivest (1995. These methods are ot efficiet sice they ivolve two-step estimatio. To overcome this problem, Che, Fa, ad Tsyreikov (2006 use the sieve maximum likelihood estimatio. Che ad Fa (2006 ivestigate the semiparametric approach for the estimatio of copula i the cotext of depedet data. I order to reduce dimesioality ad to remove the problem of boudary bias whe the support of the variables is bouded, Bouezmari ad Rombouts (2008 propose a semiparametric estimatio procedure for a multivariate desity with a parametric copula ad asymmetric kerels desity estimators for the margial desities. I this paper, we are iterested i a third way of estimatig copula fuctios, which is oparametric estimatio. This approach cosiders oparametric models for both the copula ad margial distributios. Deheuvels (1979 suggests the multivariate empirical distributio to estimate the copula fuctio. Gijbels ad Mieliczuk (1990 estimate a bivariate copula usig smoothig kerel methods. Also, they suggest the reflectio method i order to solve the boudary bias problem of the kerel methods. Che ad Huag (2007 propose a bivariate estimator based o the local liear estimator, which is cosistet everywhere i the support of the copula fuctio. Rödel (1987 uses the orthogoal series method. For idepedet ad idetically distributed (i.i.d. data, Sacetta ad Satchell (2004 develop a Berstei polyomial estimator of the copula fuctio ad fid a upper boud of the asymptotic bias ad variace ad the asymptotic ormality of the Berstei desity copula estimator. I this paper, we cosider α mixig depedet data ad propose oparametric estimatio of 2

4 the desity copula based o the Berstei polyomials. We study the asymptotic properties of the Berstei desity copula, i.e., we provide the exact asymptotic bias ad variace, ad we establish the uiform strog cosistecy ad the asymptotic ormality of Berstei estimator for the desity copula. Motivated by Weierstrass theorem, Berstei polyomials are cosidered by Loretz (1953 who proves that ay cotiuous fuctio ca be approximated by Berstei polyomials. For desity fuctios, estimatio usig the Berstei polyomial is suggested by Vitale (1975 ad with a slight modificatio by Grawroski ad Stadtmüller (1981. Tebusch (1994 ivestigates the Berstei estimator for bivariate desity fuctios ad Bouezmari ad Roli (2007 prove the cosistecy of Berstei estimator for ubouded desity fuctios. Kakizawa (2004 ad Kakizawa (2006 cosider the Berstei polyomial to estimate desity ad spectral desity fuctios, respectively. Tebusch (1997 ad Brow ad Che (1999 propose estimators of the regressio fuctios based o the Berstei polyomial. I the Bayesia cotext, Berstei polyomials are explored by Petroe (1999a, Petroe (1999b, Petroe ad Wasserma (2002, ad Ghosal (2001. This paper is orgaized as follows. The Berstei copula estimator is itroduced i Sectio 2. Sectio 3 provides the asymptotic properties of the Berstei desity copula estimator, that is the asymptotic bias ad variace, the uiform strog cosistecy, ad the asymptotic ormality for α mixig depedet data. Sectio 4 cocludes. 2 Berstei copula estimator Let X = {(X i1,..., X id, i = 1,.., } be a sample of observatios of α-mixig vectors i IR d, with distributio fuctio F ad desity fuctio f. A sequece is α-mixig of order h if the mixig coefficiet α(h goes to zero as the order h goes to ifiite, where α(h = sup P(A B P(AP(B, A F1 t(x, B F t+h (X F t 1 (X ad F t+h (X are the σ-field of evets geerated by {X l, l t} ad {X l, l t + h}, respectively. The cocept of α-mixig is omipreset i time series aalysis ad is less restrictive tha β ad ρ-mixig. The followig coditio requires a α-mixig coefficiet with expoetial decay. We assume that the process X is α-mixig such that for some costat 0 < ρ < 1. α(h ρ h, h 1, (1 Accordig to Sklar (1959, the distributio fuctio of X ca be expressed via a copula: F(x 1,..., x d = C(F 1 (x 1,..., F d (x d, (2 3

5 where F j is the margial distributio fuctio of radom variable X j = {X 1j,..., X j } ad C is a copula fuctio which captures the depedece i X. Deheuvels (1979 uses a oparametric approach, based o the empirical distributio fuctio, to estimate the distributio copula. Usig Berstei polyomials, to smooth the empirical distributio, Sacetta ad Satchell (2004 propose the empirical Berstei copula which is defied as follows: for s = (s 1,..., s d [0, 1] d Ĉ(s 1,..., s d = k 1 υ 1 =0... k 1 υ d =0 ( v1 F k,..., v d k d p υj (s j, (3 where k is a iteger playig the role of the badwidth parameter, F is the empirical distributio fuctio of X, ad p υj (s j is the biomial distributio fuctio: ( k 1 p υj (s j = s υ j j (1 s j k υj 1. υ j If we derive (2 with respect to (x 1,..., x d, we obtai the desity fuctio, say f(x 1,..., x d, of X that ca be expressed as follows: f(x 1,..., x d = (f 1 (x 1... f d (x d c (F 1 (x 1,..., F d (x d, where f j is the margial desity of the radom variable X j ad c is the copula desity. Hece, the estimatio of the joit desity fuctio ca be doe by estimatig the uivariate margial desities ad the copula desity fuctio. I this paper, we estimate the copula desity fuctio usig Berstei polyomials. Ideed, if we derive (3 with respect to (s 1,..., s d we obtai the Berstei desity copula: where S i = (F 1 (X i1,..., F d (X id, ad ĉ(s 1,..., s d = 1 k 1 K k (s, S i = k d υ 1 =0 A Si,υ = 1 {Si B υ}, with B υ = I what follows, we deote the multiple sums... K k (s, S i (4 k 1 A Si,υ υ d =0 d p υj (s j, [ υ1 k, υ ] [ υd... k k, υ ] d + 1. k k 1 υ 1 =0... k 1 υ d =0 by υ. The Berstei estimator for the desity copula fuctio is simple to implemet, o-egative, ad itegrates to oe. Sacetta ad Satchell (2004 give the upper bouds of the bias ad variace of the Berstei copula desity estimator for i.i.d observatios. I this paper, we provide the asymptotic properties of the Berstei copula desity for α-mixig depedet data. For such data, we give the exact asymptotic bias ad variace, we prove the uiform almost sure (a.s. covergece of the Berstei desity copula, ad we establish its asymptotic ormality. 4

6 3 Mai results This sectio studies the asymptotic properties of the Berstei desity copula estimator. We first show that the asymptotic bias of the Berstei desity copula estimator has a uiform rate of covergece. Hece, asymptotically there is o boudary bias problem. Secod, we provide the exact asymptotic variace of the estimator i the iterior regio. Fially, we establish the uiform strog cosistecy ad the asymptotic ormality of the Berstei desity copula estimator. We start by studyig the bias of the Berstei desity copula estimator. The followig propositio gives the exact asymptotic bias. To stress that the badwidth depeds o, we replace k by k. Propositio 1 (Asymptotic Bias. Suppose that the desity copula fuctio c is twice differetiable. Let ĉ be the Berstei desity copula estimator of c as defied i (4. The, for s = (s 1,..., s d (0, 1 d, if k, we have where γ (s = 1 2 IE(ĉ(s = c(s + k 1 γ (s + o(k 1 { } d dc(s (1 2s j + d2 c(s s j (1 s j. ds j Proof. Usig a secod order Taylor expasio ad various sums we have { } d IE(ĉ(s c(s = k d (c(u c(sdu p υj (s j υ B υ { d } d = k d dc(s (u l s l du p υj (s j du υ l l=1 B υ + kd d d 2 c(s d (u 2 l s l (u m s m du p du υ l du m l m B υ υj (s j { + kd d d 2 } d c(s 2 du 2 (u l s l 2 du p υj (s j + o(k 1 υ l=1 l B υ = 1 d dc d d 2 c 2 k 1 (s(1 2s j + (ss j (1 s j du j + o(k 1. ds 2 j du 2 j The last equality is obtaied by usig the mea ad variace of the Biomial distributio ad the fact that k d 2 d υ l m d 2 c(s du l du m (u l s l (u m s m du B υ d p υj (s j = O(k d. 5

7 Now, we compute the variace of the Berstei desity copula estimator. This is give by the followig propositio. Propositio 2 (Asymptotic Variace. Let ĉ be the Berstei desity copula estimator of c as defied i (4. The, for s (0, 1 d, uder coditio (1 ad if 1 k d/2 where V (s = (4π d/2 V ar(ĉ(s = 1 k d/2 V (s + o( 1 k k/2 c(s Q d (s j(1 s j 1/2. 0, we have Note that the formula of the variace at s = 0 ad s = 1 is give by Sacetta ad Satchell (2004. We see that the variace icreases with dimesio d ad it icreases ear the boudary because of the term (s j (1 s j 1/2 i the deomiator of V (s. Proof. We have V ar(ĉ(s = 1 V ar(k k (s, S i (1 1 icov ((K k (s, S 1, K k (s, S i. First, uder coditio (1 ad usig Billigsley s iequality, Lemma (3.1 i Bosq (1996 ad that K k (s, S i = O(k d/2, we obtai Secod, 1 2 (1 1 icov ((K k (s, S 1, K k (s, S i = o( 1 k d/2. 1 V ar(k k (s, S i = k2d V ar(a Si,υ υ d (s j From Sacetta ad Satchell (2004 V ar(a Si,υ = c( υ 1 k,..., υ d k k d + o(k d. Hece 1 V ar(k k (s, S i = k2d = k2d υ ( c( υ 1 k,..., υ d k k d [ (. + ] (. υ I c υ I V 1 + V 2 d + o(k d (s j 6

8 where { } I = υ = (υ 1,..., υ d ; υ j s j k < k δ, j = 1,..., d 1/3 < δ < 1/2. Let s start with the secod term V 2. By cosiderig the otatios m = sup s (c(s, ad whe v I c this meas that there exists d 0, for 1 d 0 d, elemets of v j such that υ j k s j > k δ, we have V 2 = υ I c ( c( υ 1 k,..., υ d k mkd d 0 m ( kd/2 k d υj k s j >k δ k 7/2 d 0 d + o(k d (s j = o( 1 k d/2. For the last iequality, o the oe had, we use (s j υ j /k s j >k δ O the other had, from Laplace s formula we have where P υj (s j = (s j d υ j /k s j >k δ j=d 0 +1 υj k s j <k δ p υj (s j = O(k 4, from Loretz (1953. k 1 2 (s j 1, as k P υj (s j k 1/2 2πs j (1 s j j+1 k j k [ exp k s j (1 s j (t s j 2 2 (s j Let v j ad v j be the smallest ad the biggest itegers such that υ j /k s j k δ. Usig the Laplace s formula ad the chage of variables we get υ j /k s j k δ (s j = 1 2πs j (1 s j k 1/2 υ j k υ j k [ exp k ] dt. s j (1 s j (t s j υj2 exp ( y 2 /2 dy πs j (1 s j 2π ( where υ j1 = 2k υj ( s j (1 s j k s j ad υ j2 = 2k υj s j (1 s j k s j. Note that, whe k, the j 1 ad j 2 +, because s j v j /k k δ, v j /k s j k δ, ad δ < 1/2. Thus, υ j /k s j k δ υ j1 (s j = O(k 1/2, as k. 7 ] dt

9 Now, for V 1 V 1 = k2d = kd υ I ( c( υ 1 k,..., υ d k k d d + o(k d (s j d c(s (s j + o( 1 k d/2, because s j υ j υ I = 1 k d/2 (4π d/2 c(s d (s j(1 s j 1/2 + o( 1 k d/2. k For α mixig depedet data, the uiform almost sure covergece of the Berstei desity copula estimator is stated i the followig propositio. Propositio 3 (Uiform a.s. Covergece. Suppose that the desity copula fuctio c is twice differetiable ad that {S i } is a α mixig sequece with coefficiet α(h = O(ρ h, for some 0 < ρ < 1. Let ĉ be the Berstei desity copula estimator of c as defied i (4. The, if k such that 1/2 k d/4 l( 0, we have sup ĉ(s c(s = O(k 1 + 1/2 k d/4 l(, a.s. s Proof. From the bias term ad uder the assumptio that c is twice differetiable, we have If we deote the we ca show that sup IE(ĉ(s c(s = O(k 1. s Y,i = 1 K k (s, S i, R 2 ( = supie Y,i 2 1/2 = O( 1 k k/4 i ad Y,i 1 k d/2. Hece, uder the above coditios o the badwidth parameter ad applyig Theorem (3.2 from Liebscher (1996 to Y,i, we get sup s IE(ĉ(s c(s = O( 1/2 k d/4 l(. The ext propositio establishes the asymptotic ormality of the Berstei desity copula estimator for α mixig depedet data. This result ca be applied i may cotexts. We ca use it for example to build copula-based tests of goodess-fit ad coditioal idepedece. 8

10 Propositio 4 (Asymptotic Normality. Suppose that the desity copula fuctio c is twice differetiable ad that {S i } is a α mixig sequece with coefficiet α(h = O(ρ h, for some 0 < ρ < 1. Let ĉ be the Berstei desity copula estimator of c as defied i (4. The, if k such that k = O( 2/(4+d, we have 1/2 k d/4 ĉ(s c(c k 1 γ (s N(0, 1. V (s Remark that if we choose k = O( 2/(4+d, the the bias term disappears. Proof. Based o Propositio (1, we eed to show that 1/2 k d/4 ( ĉ(s IE(ĉ(s V (s N(0, 1, for s (0, 1 d. (5 If we deote Y i = K k (s, S i IE(K k (s, S i V (s. the ( 1/2 k d/4 ĉ(s IE(ĉ(s V (s = 1/2 k d/4 Y i 1/2 I. We follow Doob s method to show the asymptotic ormality for depedet radom vectors, see Doob (1953. We cosider the variables V i = k d/4 (Y (i 1(p+q Y ip+(i 1q ad Vi = k d/4 (Y ip+(i 1q Y i(p+q. For r(p + q r(p + q + 1, I = r V i + r Vi + k d/4 i=r(p+q Y i. (6 We ca show that 1/2 r Vi + k d/4 i=r(p+q Y i P 0. Ideed, if we choose r a, p 1 a, adq c, where 0 < a < 1 ad 0 < c < 1 a, we get ( r 1 V ar Vi = O( a+c 1, ad 1 k d/2 V ar Y i = O( a 1. i=r(p+q The two last terms i the right side of (6 are asymptotically egligible. Now, we show that V i are asymptotically mutually idepedet. Let U i = exp(itv i which is F j i - measurable, where i = (i 1(p + q + 1 ad j = ip + (i 1q, hece from Volkoski ad Razaov 9

11 (1959 ( ( IE exp it r V i r IE(exp(itV i 16(r 1α(q + 1. Lastly, we employ the Lyapouov s theorem for the asymptotic ormality of 1/2 V i. If we choose a > d+2 d+4, we obtai, r IE( V i 3 (r var(v 1 3/2 V i (r var(v 1 1/2 p k d/4 K k (s, t (r var(v 1 1/2 = O( d+2 d+4 a = o(1 because K k (s, X i = O(k d/2. 4 Coclusio A oparametric Berstei polyomial-based estimator of desity copula for depedet data is provided. The proposed estimator ca be applied i several cotexts, ad we ca use it to build copula-based tests of, for example, goodess-fit ad coditioal idepedece, see Bouezmari, Rombouts, ad Taamouti (2008. We provide the exact asymptotic bias ad variace of the Berstei copula desity estimator ad we establish its uiform strog cosistecy, ad the asymptotic ormality. Our results ca be exteded to the right cesored data usig smoothed Kapla-Meier estimator istead of the empirical distributio fuctio. A badwidth choice i practice remais a ope questio ad existig methods like cross-validatio ca be ivestigated. Refereces Bosq, D. (1996: Noparametric Statistics for Stochastic Processes, Estimatio ad Predictio. Spriger-Verlag, New York. Bouezmari, T., ad J. Roli (2007: Berstei Estimator for Ubouded Desity Fuctio, Joural of Noparametric Statistics, 19, Bouezmari, T., ad J. Rombouts (2008: Semiparametric Multivariate Desity Estimatio for Positive Data Usig Copulas, Computatioal Statistics ad Data Aalysis, forthcomig. 10

12 Bouezmari, T., J. Rombouts, ad T. Taamouti (2008: A Noparametric Test for Coditioal Idepedece Usig Berstei Desity Copulas, Mimeo. Brow, B., ad S. Che (1999: Beta-Berstei Smoothig for Regressio Curves with Compact Supports, Scadiavia Joural of Statistics, 26, Che, S. X., ad T. Huag (2007: Noparametric Estimatio of Copula Fuctios for Depedet Modelig, Caadia Joural of Statistics, 35, Che, X., ad Y. Fa (2006: Estimatio of Copula-based Semiparametric Time Series Models, Joural of Ecoometrics, 130, Che, X., Y. Fa, ad V. Tsyreikov (2006: Efficiet Estimatio of Semiparametric Multivariate Copula Models, Joural of the America Statistical Associatio, 101, Deheuvels, P. (1979: La Foctio de Dépedace Empirique et ses Propriétés. U Test o Paramétrique D idepedace, Bulletti de l académie Royal de Belgique, Classe des Scieces, pp Doob, J. (1953: Stochastic Processes. Joh Wiley ad Sos, New York. Geest, C., K. Ghoudi, ad L. Rivest (1995: Semiparametric Estimatio Procedure of Depedace Parameters i Multivariate Families of Distributios, Biometrika, 82, Geest, C., ad L. Rivest (1993: Statistical Iferece Procedure for Bivariate Archimedea Copulas, Joural of the America Statistical Associatio, 88, Ghosal, S. (2001: Covergece Rates for Desity Estimatio with Berstei Polyomials, Aals of Statistics, 28, Gijbels, I., ad J. Mieliczuk (1990: Estimatig The Desity of a Copula Fuctio, Commuicatios i Statistics - Theory ad Methods, 19, Grawroski, N., ad U. Stadtmüller (1981: Smoothig Histograms by Lattice ad Cotiuous Distributios, Metrika, 28, Joe, H. (2005: Asymptotic Efficiecy of the Two-Stage Estimatio Method for Copula-Based Models, Joural of Multivariate Aalysis, 94, Kakizawa, Y. (2004: Berstei Polyomial Probability Desity Estimatio, Joural of Noparametric Statistics, 16,

13 (2006: Berstei Polyomial Estimatio of a Spectral Desity, Joural of Time Series Aalalysis, 27, Liebscher, E. (1996: Strog Covergece of Sums of α-mixig Radom Variables with Applicatios to Desity Estimatio, Stochastic Processes ad their Applicatios, 65, Loretz, G. (1953: Berstei Polyomials. Uiversity of Toroto Press. Oakes, D. (1982: A Model for Associatio i Bivariate Survival Data, Joural of the Royal Statistical Society, Series B, 44, Petroe, S. (1999a: Bayesia Desity Estimatio Usig Berstei Polyomials, Caadia Joural of Statistics, 27, (1999b: Radom Berstei Polyomials, Scadiavia Joural of Statistics, 26, 373 Petroe, S., ad L. Wasserma (2002: Cosistecy of Berstei Polyomial Desity Estimators, Joural of the Royal Statistical Society, Ser. B, 64, Rödel, E. (1987: R-Estimatio of Normed Bivariate Desity Fuctios, Statistics, 18, Sacetta, A., ad S. Satchell (2004: The Berstei Copula ad its Applicatios to Modelig ad Approximatios of Multivariate Distributios, Ecoometric Theory, 20, Sklar, A. (1959: Foctio de Répartitio à Dimesios et leurs Marges, Publicatios de l Istitut de Statistique de l Uiversité de Paris, 8, Tebusch, A. (1994: Two-dimesioal Berstei Polyomial Desity Estimatio, Metrika, 41, (1997: Noparametric Curve Estimatio with Berstei Estimates, Metrika, 45, Vitale, R. (1975: A Berstei Polyomial Approach to Desity Estimatio, Puri, M.L. (Ed., Statistical Iferece ad Related Topics, vol. 2. Academic Press, New York,, pp Volkoski, V. A., ad Y. A. Razaov (1959: Some Limit Theorems for Radom Fuctios. I, Theory of Probability ad Its Applicatios, 4,