## The Annals of Statistics

### Estimating a Distribution Function with Truncated Data

Michael Woodroofe

#### Abstract

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.

#### Article information

Source
Ann. Statist. Volume 13, Number 1 (1985), 163-177.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346584

Digital Object Identifier
doi:10.1214/aos/1176346584

Mathematical Reviews number (MathSciNet)
MR773160

Zentralblatt MATH identifier
0574.62040

JSTOR