## The Annals of Statistics

### Maximum Likelihood Estimation of a Translation Parameter of a Truncated Distribution

#### Abstract

$f(x)$ is a uniformly continuous density which equals zero for negative values of $x$, has a right-hand derivative equal to $\alpha$ at $x = 0$, where $0 < \alpha < \infty$, and satisfies certain regularity conditions. $X_1,\cdots, X_n$ are independent random variables with the common density $f(x - \theta), \theta$ an unknown parameter. Let $\hat{\theta}_n$ denote the maximum likelihood estimator of $\theta$, and define $\alpha_n$ by the equation $2\alpha_n^2 = \alpha n \log n$. It was shown by Woodroofe that the asymptotic distribution of $\alpha_n(\hat{\theta}_n - \theta)$ is standard normal. It is shown in the present paper that $\hat{\theta}_n$ is an asymptotically efficient estimator of $\theta$.

#### Article information

Source
Ann. Statist., Volume 1, Number 5 (1973), 944-947.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342515

Digital Object Identifier
doi:10.1214/aos/1176342515

Mathematical Reviews number (MathSciNet)
MR341727

Zentralblatt MATH identifier
0271.62043

JSTOR