Maximum Likelihood Estimation of a Translation Parameter of a Truncated Distribution

Abstract

Let $f_\theta(x) = f(x - \theta), \theta, x\in R$, where $f(x) = 0$ for $x \leqq 0$ and let $\hat{\theta}_n$ be the maximum likelihood estimate (MLE) of $\theta$ based on a sample of size $n$. If $\alpha = \lim f'(x)$ exists as $x \rightarrow 0$, and $0 < \alpha < \infty$, then under some regularity conditions, it is shown that $\alpha_n(\hat{\theta}_n - \theta)$ has an asymptotic standard normal distribution where $2\alpha_n^2 = \alpha n \log n$ and that if $\theta$ is regarded as a random variable with a prior density, then the posterior distribution of $\alpha_n(\theta - \hat{\theta}_n)$ converges to normality in probability.