The Annals of Statistics

Nonparametric Estimation in the Presence of Length Bias

Y. Vardi

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Abstract

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.

Article information

Source
Ann. Statist., Volume 10, Number 2 (1982), 616-620.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176345802

Digital Object Identifier
doi:10.1214/aos/1176345802

Mathematical Reviews number (MathSciNet)
MR653536

Zentralblatt MATH identifier
0491.62034

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 62D05: Sampling theory, sample surveys 60F05: Central limit and other weak theorems

Keywords
Empirical distribution function biased sampling maximum likelihood weighted distribution

Citation

Vardi, Y. Nonparametric Estimation in the Presence of Length Bias. Ann. Statist. 10 (1982), no. 2, 616--620. doi:10.1214/aos/1176345802. https://projecteuclid.org/euclid.aos/1176345802


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