On the Estimation of the Probability Density, I

Estimators of the form fₙ(x) = (1/n) ∑ⁿᵢ₌₁ δₙ(x - xᵢ) 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 fₙ(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.