On the Estimation of the Probability Density, I

Abstract

Estimators of the form $\hat f_n(x) = (1/n) \sum^n_{i=1} \delta_n(x - x_i)$ of a probability density f(x) are considered, where x1 ⋯ xn is a sample of n observations from f(x). In Part I, the properties of such estimators are discussed on the basis of their mean integrated square errors E[∫(fn(x) - f(x))2dx] (M.I.S.E.). The corresponding development for discrete distributions is sketched and examples are given in both continuous and discrete cases. In Part II the properties of the estimator $\hat f_n(x)$ will be discussed with reference to various pointwise consistency criteria. Many of the definitions and results in both Parts I and II are analogous to those of Parzen [1] for the spectral density. Part II will appear elsewhere.