# Inequalities for a new data-based method for selecting nonparametric density estimates

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## Abstract

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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## Bibliographic Info

Paper provided by Department of Economics and Business, Universitat Pompeu Fabra in its series Economics Working Papers with number 281.

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Handle: RePEc:upf:upfgen:281

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Web page: http://www.econ.upf.edu/

## Related research

Keywords: Density estimation; Kernel estimate; convergence; smoothing factor; minimum distance estimate; asymptotic optimality;

Find related papers by JEL classification:

• C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
• C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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## References

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1. Devroye, Luc, 1982. "Bounds for the uniform deviation of empirical measures," Journal of Multivariate Analysis, Elsevier, vol. 12(1), pages 72-79, March.
2. Duc Devroye & J. Beirlant & R. Cao & R. Fraiman & P. Hall & M. Jones & Gábor Lugosi & E. Mammen & J. Marron & C. Sánchez-Sellero & J. Uña & F. Udina & L. Devroye, 1997. "Universal smoothing factor selection in density estimation: theory and practice," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 6(2), pages 223-320, December.
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## Citations

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Cited by:
1. Luc Devroye & Gábor Lugosi, 1999. "Almost sure testability of classes of densities," Economics Working Papers 375, Department of Economics and Business, Universitat Pompeu Fabra.
2. Luc Devroye & Gábor Lugosi, 1998. "Variable Kernel estimates: On the impossibility of tuning the parameters," Economics Working Papers 325, Department of Economics and Business, Universitat Pompeu Fabra.

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