A multivariate Bahadur–Kiefer representation for the empirical copula process

We provide a multivariate extension of the Kiefer (1970) strong limit law for the uniform Bahadur–Kiefer reperesentation. This allows us to derive optimal rates for the strong approximation of empirical copula processes by sequences of Gaussian processes. We also provide a hill characterization of empirical copulas in a general framework. Bibliography: 30 titles.

[1]  Nathalie Castelle,et al.  Strong approximations of bivariate uniform empirical processes , 1998 .

[2]  Pascal Massart,et al.  STRONG APPROXIMATION FOR MULTIVARIATE EMPIRICAL AND RELATED PROCESSES, VIA KMT CONSTRUCTIONS , 1989 .

[3]  Winfried Stute,et al.  The Oscillation Behavior of Empirical Processes: The Multivariate Case , 1984 .

[4]  Paul Deheuvels,et al.  Bahadur-Kiefer-Type Processes , 1990 .

[5]  R. Nelsen.  An Introduction to Copulas , 1998 .

[6]  Approximations fortes pour des processus bivariés , 2002, Canadian Journal of Mathematics.

[7]  Galen R. Shorack,et al.  Kiefer's theorem via the Hungarian construction , 1982 .

[8]  Carles M. Cuadras,et al.  Distributions With Given Marginals and Statistical Modelling , 2002 .

[9]  K. Chung,et al.  An estimate concerning the Kolmogoroff limit distribution , 1949 .

[10]  L. Rüschendorf.  Construction of multivariate distributions with given marginals , 1985 .

[11]  Abe Sklar,et al.  Random variables, joint distribution functions, and copulas , 1973, Kybernetika.

[12]  M. Yor,et al.  On quadratic functionals of the Brownian sheet and related processes , 2006 .

[13]  M. Sklar.  Fonctions de repartition a n dimensions et leurs marges , 1959 .

[14]  Samuel Kotz,et al.  Advances in Probability Distributions with Given Marginals , 1991 .

[15]  P. Sen,et al.  Theory of rank tests , 1969 .

[16]  Weighted Multivariate Tests of Independence , 2007 .

[17]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[18]  S. A. van de Geer,et al.  Lectures on Empirical Processes: Theory and Statistical Applications , 2007 .

[19]  Ludger Rüschendorf,et al.  Asymptotic Distributions of Multivariate Rank Order Statistics , 1976 .

[20]  M. Fréchet.  Sur les tableaux de correlation dont les marges sont donnees , 1951 .

[21]  B. Schweizer,et al.  Thirty Years of Copulas , 1991 .

[22]  J. Kiefer,et al.  Deviations Between the Sample Quantile Process and the Sample DF , 1985 .

[23]  Helen Finkelstein.  The Law of the Iterated Logarithm for Empirical Distribution , 1971 .

[24]  T. Lai.  Reproducing kernel Hilbert spaces and the law of the iterated logarithm for Gaussian processes , 1974 .

[25]  C. Genest,et al.  The Joy of Copulas: Bivariate Distributions with Uniform Marginals , 1986 .