ON THE PROBLEM OF NECESSARY CONDITIONS ENSURING UNIFORM CONVERGENCE OF KERNEL DENSITY ESTIMATES
Citation：
Cheng ping.ON THE PROBLEM OF NECESSARY CONDITIONS ENSURING UNIFORM CONVERGENCE OF KERNEL DENSITY ESTIMATES[J].Chinese Annals of Mathematics B,1984,5(3):357~362
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Authors：
Cheng ping;
Abstract：
Let $\[{X_1}, \cdots ,{X_n}\]$ be a sequentse of p-dimensional iid. random vectors with a common distribution F(x). Denote the kernel estimate of the probability density of F(if it exists) by
Suppose that there exists a measurable function g(x) and $\[{h_n} > 0,{h_n} \to 0\]$ such that
$$\[\mathop {\lim \sup }\limits_{n \to \infty } \left| {{f_n}(x) - f(x)} \right| = 0\begin{array}{*{20}{c}}
{a.s}&{}
\end{array}\]$$
Does F(x) have a uniformly continuous density fuuction f(x) and f(x)=g(x)? This paper deals with the problem and gives a sufficient and necessary condition for general p-dimensional case.
