Necessary and sufficient condition for the existence of a limit distribution of the nearest-neighbour density estimator

Abstract

Many convergence results in density estimation can be stated without any restriction on the function to be estimated. Unlike these universal properties, the asymptotic normality of estimators often requires hypotheses on the derivatives of the underlying density and additional conditions on the smoothing parameter. Yet, despite the possible bad local behaviour of the density (it is not continuous or has infinite derivative), the convergence in law of the nearest-neighbour estimator still may occur and provide confidence bands for the estimated density. Therefore, a natural question arises: Is it possible to get a necessary and sufficient condition for the existence of a limit distribution of the nearest-neighbour estimator? We answer this question by using the regularity index recently introduced by Beirlant, Berlinet and Biau [(2008), ‘Higher Order Estimation at Lebesgue Points’, Annals of the Institute of Statistical Mathematics, 60, 651–677]. As expected, when it does exist, the limit distribution is Gaussian. Its mean and variance are explicitly given as functions of the regularity index. The second-order term in the expansion of the small ball probability is shown to be the crucial parameter. In contrast to the former results on sufficiency of conditions for asymptotic normality, no continuity hypothesis is required for the underlying density.

Keywords:

• asymptotic normality
• nearest-neighbour estimator
• regularity index
• small ball probability

AMS Subject Classifications:

• 62G05
• 60E05

Acknowledgements

We would like to thank the referee for positive comments and sharp questions.