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Remarks on Some Nonparametric Estimates of a Density Function by Murray Rosenblatt; Richard A Davis; Keh-Shin Lii; Dimitris N Politis is a book available to read on EtoBox.

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This note discusses some aspects of the estimation of the density function of a univariate probability distribution. All estimates of the density function satisfying relatively mild conditions are shown to be biased. The asymp totic mean square error of a particular class of estimates is evaluated. Estimates of the density function. Let Xi, • • • , X n be independent and identically distributed random variables with continuous density function f(y).It can easily be shown thatis not an unbiased estimate of/(y). Suppose to the contrary that ( )for all continuous / and all y. Condition (1) implies that for each y 9a sufficient statistic for the problem. But then (2) fsiviXw-tXJdy is a symmetric estimate of F(b) -F(a) -[ f(y) dy. Moreover, (2) is an unbiased estimate of F(b) -F(a), since E f SiviXw-tXJdy\*\* f ES(y;X 1} • • -7 X n ) dy = f f(y) dy = F(b) -F(a)

Author

Murray Rosenblatt; Richard A Davis; Keh-Shin Lii; Dimitris N Politis

Publisher

Springer New York

Published

2011

Language

EN

ISBN

9781441983398

Subjects

Mathematics, Stem