Kernel Conditional Density Estimation When the Regressor is Valued in a Semi-Metric Space

Abstract

This article deals with the conditional density estimation when the explanatory variable is functional. In fact, nonparametric kernel type estimator of the conditional density has been recently introduced when the regressor is valued in a semi-metric space. This estimator depends on a smoothing parameter which controls its behavior. Thus, we aim to construct and study the asymptotic properties of a data-driven criterion for choosing automatically and optimally this smoothing parameter. This criterion can be formulated in terms of a functional version of cross-validation ideas. Under mild assumptions on the unknown conditional density, it is proved that this rule is asymptotically optimal. A simulation study and an application on real data are carried out to illustrate, for finite samples, the behavior of our method. Finally, we mention that our results can also be considered as novel in the finite dimensional setting and several other open questions are raised in this article.

Keywords:

• Bandwidth selection
• Conditional density
• Conditional model
• Cross-validation
• Functional data
• Kernel estimator
• Small balls probability

Mathematics Subject Classification:

• Primary 62G05
• Secondary 62G07, 62G08, 62G35, 62G20

Acknowledgments

The authors would like to thank the Editor, an Associate Editor, and two anonymous reviewers for their valuable comments and suggestions which improved substantially the quality of an earlier version of this article.

Notes

fttp address: ftp://ftp.ncdc.noaa.gov/pub/data/ushcn/v2/monthly

Available at the website: “www.lsp.ups-tlse.fr/staph/npfda”

In Ferraty et al. (2010), the main aim is to state the rate of the uniform almost-complete convergence of the functional component. Such a result can be easily extended here (without precision of the convergence rate) to sup  _{a∈H n} by using the second part of assumption (15).