Uniform convergence rates for a nearest neighbor density estimator under dependence assumptions

Abstract

In this paper the rates of strong uniform convergence over any compact set for an alternative nearest neighbor density estimator are obtained when the observations satisfy a ø-mixing or an a-mixing condition. In the ø-mixing case we obtain a quite better convergence rate than for a-mixing processes and we do not require a geometric condition on the mixing coefficients. For independent or m-dependent observations, as a special case of ømixing, the result gives us the optimal rate of strong uniform convergence for density estimators.

Keywords:

• Nonparameiric density estimation
• nearest neighbor methods
• strong uniform consistency
• rate of consistency
• ømixing