Open Access
June, 1965 Estimation of Probability Density
V. K. Murthy
Ann. Math. Statist. 36(3): 1027-1031 (June, 1965). DOI: 10.1214/aoms/1177700074


Assuming that the distribution being sampled is absolutely continous, Parzen [3] has established the consistency and asymptotic normality of a class of estimators $\{f_n(x)\}$ based on a random sample of size $n$, for estimating the probability density. In this paper, we relax the assumption of absolute continuity of the distribution $F(x)$ and show that the class of estimators $\{f_n(x)\}$ still consistently estimate the density at all points of continuity of the distribution $F(x)$ where the density $f(x)$ is also continuous. It is further shown that the sequence of estimators $\{f_n(x)\}$ are asymptotically normally distributed. The extension of these results to the bi-variate and essentially the multi-variate case with applications and a discussion on the construction of higher dimensional windows will be presented at the International Symposium in Multivariate Analysis to be held in Dayton, Ohio during June 1965.


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V. K. Murthy. "Estimation of Probability Density." Ann. Math. Statist. 36 (3) 1027 - 1031, June, 1965.


Published: June, 1965
First available in Project Euclid: 27 April 2007

zbMATH: 0142.15805
MathSciNet: MR177473
Digital Object Identifier: 10.1214/aoms/1177700074

Rights: Copyright © 1965 Institute of Mathematical Statistics

Vol.36 • No. 3 • June, 1965
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