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 $\theta$.