Title

Length Biased Density Estimation Of Fibres

Keywords

kernel estimator; Length biased density; optimal asymptotic mean square error; order of bias and variance

Abstract

Cox (1969) discussed several procedures used in sampling of textile fibres. One such procedure is called " length biased " or weighted sampling and occurs when the chance of selection is proportional to fibre length. Cox considered the problem of estimating the unweighted distribution function Fat a fixed x > 0 and compared the asymptotic variance of estimators based on length biased samples with those based on unweighted samples. Consideration here is devoted to estimating the probability density function f at a fixed x > 0 based on length biased samples.It is shown, under suitable regularity conditions, that thesquare of the bias of the weighted estimator is less (greater) than the square of the bias of the Parzen (1962)-Rosenblatt (1956) kernel estimator of f(x) based on unweighted observations when (f'(x)/x)(f(2)(x)+f'(x))< 0(>0) and n is sufficiently large. Moreover, the variance of the length biased estimator isless (greater) than that of the unweighted estimator when x > μ(x < μ) for allnsufficiently large, where μ denotesthe mean with respect to f. An optimal window width hn((x) is given which makes the asymptotic meansquare error of the length biased estimator a minimum. Under regularity assumptions, it is shown thatthe optimal asymptotic mean square error of the lengthbiased estimator at x is less than that for the unweighted estimator exactly when (μ/x)3|g(2)(x)/f(2)(x)| 1. Moreover, simulations are undertaken to compare the two estimators for several sample sizes. © 1991, Taylor & Francis Group, LLC. All rights reserved.

Publication Date

1-1-1991

Publication Title

Journal of Nonparametric Statistics

Volume

1

Issue

1-2

Number of Pages

127-141

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1080/10485259108832515

Socpus ID

0040579913 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/0040579913

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