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June, 1963 On the Estimation of the Probability Density, I
G. S. Watson, M. R. Leadbetter
Ann. Math. Statist. 34(2): 480-491 (June, 1963). DOI: 10.1214/aoms/1177704159


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 $x_1 \cdots x_n$ 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 $E\lbrack\int(f_n(x) - f(x))^2dx\rbrack$ (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.


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G. S. Watson. M. R. Leadbetter. "On the Estimation of the Probability Density, I." Ann. Math. Statist. 34 (2) 480 - 491, June, 1963.


Published: June, 1963
First available in Project Euclid: 27 April 2007

zbMATH: 0113.34504
MathSciNet: MR148149
Digital Object Identifier: 10.1214/aoms/1177704159

Rights: Copyright © 1963 Institute of Mathematical Statistics

Vol.34 • No. 2 • June, 1963
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