The uniform convergence rate of kernel density estimate (Q1068485)

In this paper, we study the uniform convergence rate of the kernel density estimate \(\hat f_ n\) and obtain the optimal uniform rate of convergence without the assumption of compact support for the kernel function. It is proved that if the density function f satisfies the \(\lambda\)-condition and the kernel function K is \(\lambda\)-good, then we have \[ \limsup_{n\to \infty}(n/\log n)^{\lambda /(1+2\lambda)}\sup_{x\in R^ 1}| \hat f_ n(x)-f(x)| \leq const.\quad a.s. \]