Suppose that {Xn} is a strongly mixing process with unknow marginal density f(x) and that we estimate f(x) by a kernel estimator [fcirc]n(x|hn)and want to achive the MISE no larger than some preassigned postive number w. However,the appropriate sample size n*depends on a functional of the unknow density function. Therefore some sequential procedure is required and we adopt a fully sequential procedure. In this paper we investigate the asymptotic properties of the procedure and show that the producure is asymptotically efficient in a certain sense as w→0. The results are almost the same in the i.i.d. setting. our result extend a class of models to which the methodology can be applied. For example economic variable,experiments on a single subject in which obervation are not indepent, and so on.