ON THE PROBLEM OF NECESSARY CONDITIONS ENSURING UNIFORM CONVERGENCE OF KERNEL DENSITY ESTIMATES

Let X_1,…,X,be a sequence of p-dimensional iid.random vectors with a commondistribution F(x).Denote the kernel estimate of the probability density of F(if it exists)by_n(x)=n~(-1)h~_n(-p)K((x-X_i)/h_n)Suppose that there exists a measurable function g(x)and h_n>0,h_n→0 such thatlim sup丨f_n(x)-g(x)丨=0 a.s.Does F(x)have a uniformly continuous density function f(x)and f(x)=g(x)?This paperdeals with the problem and gives a sufficient and necessary condition for generalp-dimensional case.

ON THE PROBLEM OF NECESSARY CONDITIONS ENSURING UNIFORM CONVERGENCE OF KERNEL DENSITY ESTIMATES

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.