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March, 1985 Estimating a Distribution Function with Truncated Data
Michael Woodroofe
Ann. Statist. 13(1): 163-177 (March, 1985). DOI: 10.1214/aos/1176346584


Let $\mathscr{P}$ be a finite population with $N \geq 1$ elements; for each $e \in \mathscr{P}$, let $X_e$ and $Y_e$ be independent, positive random variables with unknown distribution functions $F$ and $G$; and suppose that the pairs $(X_e, Y_e)$ are i.i.d. We consider the problem of estimating $F, G$, and $N$ when the data consist of those pairs $(X_e, Y_e)$ for which $e \in \mathscr{P}$ and $Y_e \leq X_e$. The nonparametric maximum likelihood estimators (MLEs) of $F$ and $G$ are described; and their asymptotic properties as $N \rightarrow \infty$ are derived. It is shown that the MLEs are consistent against pairs $(F, G)$ for which $F$ and $G$ are continuous, $G^{-1}(0) \leq F^{-1}(0)$, and $G^{-1}(1) \leq F^{-1}(1). \sqrt N \times$ estimation error for $F$ converges in distribution to a Gaussian process if $\int^\infty_0 (1/G) dF < \infty$, but may fail to converge if this integral is infinite.


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Michael Woodroofe. "Estimating a Distribution Function with Truncated Data." Ann. Statist. 13 (1) 163 - 177, March, 1985.


Published: March, 1985
First available in Project Euclid: 12 April 2007

zbMATH: 0574.62040
MathSciNet: MR773160
Digital Object Identifier: 10.1214/aos/1176346584

Primary: 62F20
Secondary: 62G05

Keywords: Asymptotic distributions , consistency , maximum likelihood estimation , nonparametric

Rights: Copyright © 1985 Institute of Mathematical Statistics


Vol.13 • No. 1 • March, 1985
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