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Nonparametric estimation of distributions with given marginals via Bernstein-Kantorovich polynomials: L1 and pointwise convergence theory

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  • Sancetta, Alessio

Abstract

The copula density is estimated using Bernstein-Kantorovich polynomials. The estimator is the usual one based on the smoothed histogram. Strong consistency is obtained in L1 and pointwise almost everywhere, allowing for dependent data. For L1 convergence, no condition is imposed on the copula density, while for pointwise convergence, the condition imposed on the true copula density appears to be minimal.

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  • Sancetta, Alessio, 2007. "Nonparametric estimation of distributions with given marginals via Bernstein-Kantorovich polynomials: L1 and pointwise convergence theory," Journal of Multivariate Analysis, Elsevier, vol. 98(7), pages 1376-1390, August.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:7:p:1376-1390
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    1. Axel Tenbusch, 1994. "Two-dimensional Bernstein polynomial density estimators," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 41(1), pages 233-253, December.
    2. Sancetta, Alessio & Satchell, Stephen, 2004. "The Bernstein Copula And Its Applications To Modeling And Approximations Of Multivariate Distributions," Econometric Theory, Cambridge University Press, vol. 20(3), pages 535-562, June.
    3. Bouezmarni, Taoufik & Scaillet, Olivier, 2005. "Consistency Of Asymmetric Kernel Density Estimators And Smoothed Histograms With Application To Income Data," Econometric Theory, Cambridge University Press, vol. 21(2), pages 390-412, April.
    4. W. Gawronski & U. Stadtmüller, 1981. "Smoothing histograms by means of lattice-and continuous distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 28(1), pages 155-164, December.
    5. Rüschendorf, Ludger & de Valk, Vincent, 1993. "On regression representations of stochastic processes," Stochastic Processes and their Applications, Elsevier, vol. 46(2), pages 183-198, June.
    6. Doukhan, Paul & Louhichi, Sana, 1999. "A new weak dependence condition and applications to moment inequalities," Stochastic Processes and their Applications, Elsevier, vol. 84(2), pages 313-342, December.
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    3. Tavin, Bertrand, 2015. "Detection of arbitrage in a market with multi-asset derivatives and known risk-neutral marginals," Journal of Banking & Finance, Elsevier, vol. 53(C), pages 158-178.

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