A NOTE ON THE ASYMPTOTIC NORMALITY OF SEQUENTIAL DENSITY ESTIMATORS

Let ＄f_{n}(x)＄ be the recursive kernel estimators of an unknown density function ＄f(x)＄ at a given point ＄x＄ . Also, let ＄N(t)(t>0)＄ be a family of positive integer-valued random variables. We consider the sequential estimators ＄f_{N(t)}(x)＄ . In this paper, under certain regularity conditions on ＄N(t)＄ we shall show that ＄(N(t)h_{N(t)}^{p})^{1/2}(f_{N(i)}(x)-f(x))＄ is asymptotically normally distributed as ＄t＄ tends to infinity. Our conditions on ＄N(t)＄ generalize those given by Carroll [2], Stute [9] and Isogai [6]