Sequential Estimation of a Continuous Probability Density Function and Mode

The purpose of this paper is to discuss statistical properties of an estimate of a probability density function based on the first n observations under the assumption of continuity or uniform continuity of the probability density function in case where we observe a sequence of random vectors which come from a population with the probability density function. Let X1, X2,X3,••• be a sequence of independent identically distributed m-dimensional random vectors having a probability density function f(x). Suppose the first n observations be denoted by X1, X2, ••• Xn. Then a natural estimate of the probability density function f(x) may be denoted as follows for a suitable positive constant h