On the Maximum Deviation of the Sample Density

Abstract

Let $X_1, X_2, \cdots$ denote independent random variables having a common density $f$. The present paper considers estimates of $f$ of the following form ([8] and [10]): \begin{align}\tag{1.1}f_n(x) = \tau \int G(x, \tau(x - y)) dF_n(y) \\ = (\tau/n) \sum^n_{i = 1} G(x, \tau(x - X_i))\end{align} where $F_n$ denotes the sample distribution function of $X_1, \cdots, X_n, \tau = \tau(n) \rightarrow \infty$ and $\tau = 0(n)$ as $n \rightarrow \infty$, and $G$ is a non-negative function defined on $R^2$ satisfying regularity conditions to be listed in Section 2. More precisely, it considers the asymptotic behavior as $n \rightarrow \infty$ of the maximum deviation of $f_n(x)$ from $f(x)$ where $x$ varies in a compact interval which without loss of generality we take to be $\lbrack -1, 1\rbrack$. The main result, Theorem 3.1, states that under regularity conditions \begin{equation*}\tag{1.2}p\lim_{n \rightarrow \infty} \max_{| x| \leqq 1} (n/2\tau \log \tau)^{\frac{1}{2}} |(f_n(x) - f(x))/ |G_x |_2(f(x))^{\frac{1}{2}}| = 1\end{equation*} where $G_x$ denotes an $x$-section of $G$--i.e. $G_x(y) = G(x, y), y \epsilon R^1$--and $|\cdot |_p$ denotes the norm in $L_p = L_p(R^1$, Lebesgue measure). Also, a limiting distribution related to (1.2) is computed, and some sufficient conditions are given for the almost sure convergence of $\max_{|x| \leqq 1}| f_n(x) - f(x)|$ to zero as $n \rightarrow \infty$. Since under mild regularity conditions $f_n(x)$ and $f_n(y), x \neq y$, are asymptotically, independently, normally distributed when suitably normalized ([5]), one might expect (1.2) to follow from its analogue for normal random variables ([2]) by an argument involving the weak convergence of stochastic processes. However, having been unable to verify the necessary compactness conditions ([9]) with respect to any topology which makes the maximum functional in (1.2) continuous, we have adopted a more elementary approach. This approach uses a theorem on the large deviations of sums of independent, identically distributed random vectors to estimate the relevant probabilities directly. In order to develop the primary topic of the paper as quickly as possible, we have postponed the proof of the theorem on large deviations until Section 5, while using it in Sections 3 and 4 to prove our main theorems. Section 2 presents some preliminary material. We are aware of only two other papers which have considered the uniform convergence of $f_n$ to $f$, namely [8] and [7].