Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-07T04:25:06.222Z Has data issue: false hasContentIssue false
  • English
  • Français

THE BERNSTEIN COPULA AND ITS APPLICATIONS TO MODELING AND APPROXIMATIONS OF MULTIVARIATE DISTRIBUTIONS

Published online by Cambridge University Press:  08 June 2004

Alessio Sancetta  and
Stephen Satchell
Alessio Sancetta
Affiliation:
Trinity College, University of Cambridge
Stephen Satchell
Affiliation:
Trinity College, University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

We define the Bernstein copula and study its statistical properties in terms of both distributions and densities. We also develop a theory of approximation for multivariate distributions in terms of Bernstein copulas. Rates of consistency when the Bernstein copula density is estimated empirically are given. In order of magnitude, this estimator has variance equal to the square root of the variance of common nonparametric estimators, e.g., kernel smoothers, but it is biased as a histogram estimator.We would thank Mark Salmon for interesting us in the copula function and Peter Phillips, an associate editor, and the referees for many valuable comments. All remaining errors are our sole responsibility.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 3 , June 2004 , pp. 535 - 562
Copyright
© 2004 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abadir, K.M. (1999) An introduction to hypergeometric functions for economists. Econometric Reviews 18, 287–330.Google Scholar
Abadir, K.M. & J.R. Magnus (2002) Notation in econometrics: A proposal for a standard. Econometrics Journal 5, 76–90.Google Scholar
Berens, H. & R. DeVore (1980) A characterization of Bernstein polynomials. In E.W. Cheney (ed.), Approximation Theory, vol. 3, pp. 213–219. Academic Press.
Bouyé, E., N. Gaussel, & M. Salmon (2001) Investigating Dynamic Dependence Using Copulae. Working paper, Financial Econometrics Research Centre.
Butzer, P.L. (1953) Linear combinations of Bernstein polynomials. Canadian Journal of Mathematics 5, 559–567.Google Scholar
Darsow, W.F., B. Nguyen, & E.T. Olsen (1992) Copulas and Markov processes. Illinois Journal of Mathematics 36, 600–642.Google Scholar
Devore, R.A. & G.G. Lorentz (1993) Constructive Approximations. Springer.
Embrechts, P., A. McNeil, & D. Straumann (1999) Correlation: Pitfalls and alternatives. Risk 5, 69–71.Google Scholar
Fortin, I. & C.A. Kuzmics (2002) Tail-dependence in stock-return pairs. International Journal of Intelligent Systems in Accounting, Finance, and Management 11, 89–107.Google Scholar
Genest, C. & L.P. Rivest (1993) Statistical inference procedures for bivariate archimedean copulas. Journal of the American Statistical Association 88, 1034–1043.Google Scholar
Hu, L. (2002) Dependence Patterns across Financial Markets: Methods and Evidence. Preprint, Department of Economics, Yale University.
Jackson, D. (1930) The Theory of Approximation. American Mathematical Society.
Joe, H. (1997) Multivariate Models and Dependence Concepts. Chapman and Hall.
Knight, J., S. Satchell, & K. Tran (1995) Statistical modelling of asymmetric risk in asset returns. Applied Mathematical Finance 2, 155–172.Google Scholar
Kulpa, T. (1999) On approximation of copulas. International Journal of Mathematics and Mathematical Science 22, 259–269.Google Scholar
Laherrère, J. & D. Sornette (1998) Stretched exponential distributions in nature and economy: “Fat tails” with characteristic scales. European Physical Journal B 2, 525–539.Google Scholar
Li, H., M. Scarsini, & M. Shaked (1996) Linkages: A tool for the construction of multivariate distributions with given nonoverlapping multivariate marginals. Journal of Multivariate Analysis 56, 20–41.Google Scholar
Li, X., P. Mikusiński, & M.D. Taylor (1998) Strong approximation of copulas. Journal of Mathematical Analysis and Applications 225, 608–623.Google Scholar
Longin, F. & B. Solnik (2001) Extreme correlation of international equity markets. Journal of Finance 56, 651–678.Google Scholar
Lorentz, G.G. (1953) Bernstein Polynomials. University of Toronto Press.
Marichev, O.I. (1983) Handbook of Integral Transforms of Higher Transcendental Functions. Ellis Horwood.
Nelsen, R.B. (1998) An Introduction to Copulas. Lecture Notes in Statistics 139. Springer-Verlag.
Ossiander, M. (1987) A central limit theorem under metric entropy with L2 bracketing. Annals of Probability 15, 897–919.Google Scholar
Patton, A.J. (2001) Modelling Time Varying Exchange Rate Dependence Using the Conditional Copula. Discussion Paper, Department of Economics, University of California, San Diego.
Phillips, P.C.B. (1982) Best Uniform and Modified Padé Approximations of Probability Densities in Econometrics. In W. Hildenbrand (ed.), Advances in Econometrics, pp. 123–167. Cambridge University Press.
Phillips, P.C.B. (1983) ERA's: A new approach to small sample theory. Econometrica 51, 1505–1525.Google Scholar
Pollard, D. (2001) General Bracketing Inequalities. Preprint, Department of Statistics, Yale University.
Rockinger, M. & E. Jondeau (2001) Conditional Dependency of Financial Series: An Application of Copulas. Preprint.
Sancetta, A. (2003) Nonparametric Estimation of Multivariate Distributions with Given Marginals: L2 Theory. Cambridge Working Papers in Economics 0320.
Schweizer, B. & E.F. Wolff (1981) On nonparametric measures of dependence for random variables. Annals of Statistics 9, 879–885.Google Scholar
Scott, D.W. (1992) Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley.
Silvapulle, P. & C. Granger (2001) Large returns, conditional correlation, and portfolio diversification: A value at risk approach. Quantitative Finance 1, 542–551.Google Scholar
Sklar, A. (1973) Random variables, joint distribution functions, and copulas. Kybernetika 9, 449–460.Google Scholar
Spanier, J. & E.H. Maize (1994) Quasi-random methods for estimating integrals using relatively small samples. SIAM Review 36, 18–44.Google Scholar
Stuart, A., & J.K. Ord (1994) Kendall's Advanced Theory of Statistics. Edward Arnold.
Van der Vaart, A. & J.A. Wellner (2000) Weak Convergence of Empirical Processes. Springer Series in Statistics. Springer.
White, H. (1994) Estimation, Inference, and Specification Analysis. Cambridge University Press.