 Solution

The likelihood function for the shifted exponential distribution isIt increases strictly up to the smallest observation and is zero beyond it. Thus shifted exponential maximum likelihood givesFor $t\ge 0$, independence and the survival function of the exponential distribution give the exact order statistic lawIn particular, ${\hat{\theta }}_n\ge \theta$ almost surely and ${\mathbb{P}}_{\theta }(|{\hat{\theta }}_n-\theta |>\epsilon )=e^{-n\epsilon }\to 0$. This proves statistical consistency. The exact endpoint limit for a shifted exponential distribution is stronger than an asymptotic approximation:The limit is one-sided and nonnormal; the moving support endpoint explains why a regular square-root-$n$ asymptotic normality of a maximum likelihood estimator theorem does not apply here.