A recursive kernel estimate of the functional modal regression under ergodic dependence condition

ABSTRACT

In this article, we consider an alternative estimator of the conditional mode when the explanatory variable takes values in a semimetric space. This alternative estimate is based in a recursive kernel method. Under the ergodicity hypothesis, we quantify the asymptotic properties of this estimate, by giving the almost complete convergence rate. The asymptotic normality of this estimate is also given. Our approach is illustrated by a real data application.

KEYWORDS:

• Functional data
• conditional mode
• conditional distribution
• recursive estimation
• ergodic condition

AMS SUBJECT CLASSIFICATION:

• 60G42
• 62F12
• 62G20
• 62G05

Acknowledgments

The authors thank the associate editor and the anonymous reviewer for their valuable comments and suggestions which improved substantially the quality of an earlier version of this article.

Notes

¹ Let be a sequence of real r.v.’s; we say that converges almost completely (a.co.) to zero if, and only if, , . Moreover, we say that the rate of almost complete convergence of to zero is of order (with and we write if, and only if, , . This kind of convergence implies both almost sure convergence and convergence in probability.