Bernstein estimators attracted considerable attention as smooth nonparametric estimators for distribution functions, densities, copulas and copula densities. The present paper adds a parallel result for the first-order derivative of a copula function. This result then leads to Bernstein estimators for a conditional distribution function and its important functionals such as the regression and quantile functions. Results of independent interest have been derived such as an almost sure oscillation behavior of the empirical copula process and a Bahadur-type almost sure asymptotic representation for the Bernstein estimator of a regression quantile function. Simulations demonstrate the good performance of the proposed estimators.
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The authors thank Mr. Charl Pretorius for his important help with the simulation section. They also thank the editor, associate editor and two referees for their valuable remarks and suggestions. The work was supported by the IAP Research Network P7/13 of the Belgian State (Belgian Science Policy). J. Swanepoel thanks the National Research Foundation of South Africa for financial support. N. Veraverbeke is also extraordinary professor at the North-West University, Potchefstroom, South Africa.
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Janssen, P., Swanepoel, J. & Veraverbeke, N. Bernstein estimation for a copula derivative with application to conditional distribution and regression functionals. TEST 25, 351–374 (2016). https://doi.org/10.1007/s11749-015-0459-x
- Asymptotic normality
- Asymptotic representation
- Bernstein estimation
- Copula density
- Oscillation of empirical copula process
- Quantile function
Mathematics Subject Classification