Uniform Consistency of Some Estimates of a Density Function

Abstract

Let $X_1, \cdots, X_n$ be independent random variables identically distributed with absolutely continuous distribution function $F$ and density function $f$. Loftsgaarden and Quesenberry [3] propose a consistent nonparametric point estimator $\hat{f}_n(z)$ of $f(z)$ which is quite easy to compute in practice. In this note we introduce a step-function approximation $f_n^\ast$ to $\hat{f}_n$, and show that both $\hat{f}_n$ and $\hat{f}_n^\ast$ converge uniformly (in probability) to $f$, assuming that $f$ is positive and uniformly continuous in $(-\infty, \infty)$. For more general $f$, uniform convergence over any compact interval where $f$ is positive and continuous follows. Uniform convergence is useful for estimation of the mode of $f$, for it follows from our theorem (see [4], section 3) that a mode of either $\hat{f}_n$ or $f_n^\ast$ is a consistent estimator of the mode of $f$. The mode of $f_n^\ast$ is particularly tractable; it is applied in [2] to some problems in pattern recognition. From the point of view of mode estimation, we thus obtain two new estimates which are similar in conception to those proposed by some previous authors. Let $k(n)$ be an appropriate sequence of numbers in each case. Chernoff [1] estimates the mode as the center of the interval of length $2k(n)$ containing the most observations. Venter [5] estimates the mode as the center (or endpoint) of the shortest interval containing $k(n)$ observations. The estimate based on $\hat{f}_n$ is that $z$ such that the distance from $z$ to the $k(n)$th closest observation is least. Finally, the estimate from $f_n^\ast$ is that observation such that the distance from it to the $k(n)$th closest observation is least.