On Strong Consistency of Density Estimates

Abstract

Let $X_1, X_2, \cdots, X_n$, be a sample of $n$ independent observations of a random variable $X$ with distribution $F(x) = F(x_1, \cdots, x_m)$ or $R^m$ and Lebesgue density $f(x) = f(x_1, \cdots, x_m)$. To estimate the density $f(x)$ consider estimates of the form \begin{equation*} \tag{1} f_n(x) = n^{-1} \sum^n_{j=1} K_n(x, X_j),\quad K_n(x, X_j) = h_n^{-m}K(h_n^{-1}(x - X_j));\end{equation*} where $K(u) = K(u_1, \cdots, u_m)$ is a real-valued Borel-measurable function on $R^m$ such that \begin{equation*} \tag{2} K(u) \text{is a density on}\quad R^m\end{equation*} \begin{equation*} \tag{3} \sup_{u\varepsilon R^m} K(u) < \infty\end{equation*} \begin{equation*} \tag{4} \|u\|^m K(u) \rightarrow 0 \text{as} \|u\|^2 = \sum^m_{i=1} u_i^2 \rightarrow \infty\end{equation*} and $\{h_n\}$ is a sequence of numbers such that \begin{equation*} \tag{5} h_n > 0,\quad n = 1, 2, \cdots;\quad \lim_{n\rightarrow\infty} h_n = 0 \text{and} \lim_{n\rightarrow\infty} nh_n^m = \infty.\end{equation*} Such density estimates have been shown to be weakly consistent (that is, $f_n(x) \rightarrow f(x)$ in probability as $n \rightarrow \infty$) on the continuity set, $C(f),$ of the density $f(x)$ by Parzen [4] for $m = 1$ and by Cacoullos [1] for $m > 1$. In Theorem 1, we state conditions under which strong consistency (that is, $f_n(x) \rightarrow f(x)$ with probability one as $n \rightarrow \infty$) of such estimates obtains. Theorem 2 gives conditions under which uniform (in $x$) strong consistency of the estimates (1) is valid. In this respect, our results are very similar in the case $m = 1$ to those of Nadaraya [4], although the method of proof and conditions imposed are different. Theorem 3 concerns the estimation of the unique mode of the density $f(x)$ when it exists.