On the Smoothing of Probability Density Functions

Summary We consider the estimation of a probability density function by linear smoothing of the observed density. A basis for estimation is obtained by assuming that the ordinates of the true density function have a prior distribution such that adjacent ordinates are highly correlated. An equation determining the optimum weighting function under these circumstances is derived, and solved in special cases. The properties of the estimate are discussed and the asymptotic behaviour of its mean squared deviation D 2 given. It is shown that as the sample size N increases, D 2 cannot decrease faster than N –1.