Uniform strong convergence rate of nearest neighbor density estimation (Q796205)

In this paper, the authors study the strong convergence rate of the \(k_ n-NN\) density estimate \(\hat f{}_ n(x)\) of the population density f(x), proposed by \textit{D. O. Loftsgarden} and \textit{C. P. Quesenberry}, A nonparametric estimate of a multivariate density function. Ann. Math. Stat. 36, 1049-1051 (1965). They prove that, if \(f(x)>0\) and f satisfies the so-called \(\lambda\)-condition at x \((0<\lambda \leq 2)\), then for properly chosen \(k_ n\), \[ \lim \sup_{n\to \infty}(n/\log n)\quad^{\lambda /(1+2\lambda)}| \hat f_ n(x)-f(x)| \leq C\quad a.s. \] If f satisfies the \(\lambda\)-condition, then for properly chosen \(k_ n\), \[ \lim \sup_{n\to \infty}(n/\log n)^{\lambda /(1+3\lambda)}\sup_{x}| \hat f_ n(x)-f(x)| \leq C\quad a.s., \] where C is a constant. An order which the convergence rate of \(| f_ n(x)-f(x)|\) and \(\sup_{x}| f_ n(x)-f(x)|\) cannot reach is also given.