= Unbiased endpoint estimator for a shifted exponential sample
{title2=$\widehat\theta=X_{(1)}-(n\lambda)^{-1}$}
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 $\theta\ge0$, the estimator can be negative; truncation at zero introduces bias.
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