We continue the development of a method for the selection of a bandwidth or a number of design parameters in density estimation. We provide explicit non-asymptotic density-free inequalities that relate the $L_1$ error of the selected estimate with that of the best possible estimate, and study in particular the connection between the richness of the class of density estimates and the performance bound. For example, our method allows one to pick the bandwidth and kernel order in the kernel estimate simultaneously and still assure that for {\it all densities}, the $L_1$ error of the corresponding kernel estimate is not larger than about three times the error of the estimate with the optimal smoothing factor and kernel plus a constant times $\sqrt{\log n/n}$, where $n$ is the sample size, and the constant only depends on the complexity of the family of kernels used in the estimate. Further applications include multivariate kernel estimates, transformed kernel estimates, and variable kernel estimates.
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Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number
281.
Find related papers by JEL classification: C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Estimation C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Semiparametric and Nonparametric Methods
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