Unbiased endpoint estimator for a shifted exponential sample ($\hat{\theta }=X_{(1)}-(n\lambda {)}^{-1}$)

For independent $X_i=\theta +Y_i$ with $Y_i$ exponentially distributed at known rate $\lambda$, the joint likelihood factors through the sample minimum $X_{(1)}$. Its excess over the endpoint is exponential with rate $n\lambda$, because $\mathbb{P}(\min Y_i>t)=e^{-n\lambda t}$. Thus the displayed estimator is an unbiased estimator, with variance $(n\lambda {)}^{-2}$. Even if $\theta \ge 0$, the estimator can be negative; truncation at zero introduces bias.