Open Access
June, 1982 Nonparametric Estimation in the Presence of Length Bias
Y. Vardi
Ann. Statist. 10(2): 616-620 (June, 1982). DOI: 10.1214/aos/1176345802


We derive the nonparametric maximum likelihood estimate, $\hat{F}$ say, of a lifetime distribution $F$ on the basis of two independent samples, one a sample of size $m$ from $F$ and the other a sample of size $n$ from the length-biased distribution of $F$, i.e. from $G_F(x) = \int^x_0 u dF(u)/\mu, \mu = \int^\infty_0 x dF(x)$. We further show that $(m + n)^{1/2}(\hat{F} - F)$ converges weakly to a pinned Gaussian process with a simple covariance function, when $m + n \rightarrow \infty$ and $m/n \rightarrow$ constant. Potential applications are described.


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Y. Vardi. "Nonparametric Estimation in the Presence of Length Bias." Ann. Statist. 10 (2) 616 - 620, June, 1982.


Published: June, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0491.62034
MathSciNet: MR653536
Digital Object Identifier: 10.1214/aos/1176345802

Primary: 62G05
Secondary: 60F05 , 62D05

Keywords: Biased sampling , Empirical distribution function , maximum likelihood , weighted distribution

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 2 • June, 1982
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