= Exact endpoint limit for a shifted exponential distribution
{title2=$n\lambda(Y_{(1)}-\theta)\sim\operatorname{Exp}(1)$}
For independent observations $Y_i=\theta+E_i$ with $E_i$ having <exponential distribution> of known rate $\lambda>0$, the <likelihood> increases in $\theta$ until the smallest observation, so the <maximum-likelihood estimator> is $Y_{(1)}$. Its survival function is $\mathbb P(Y_{(1)}-\theta>t)=e^{-n\lambda t}$ for $t\ge0$. Therefore $n\lambda(Y_{(1)}-\theta)$ has exactly the unit-rate <exponential distribution> for every sample size. This gives <statistical consistency> and a one-sided nonnormal limit with rate $n$, illustrating how a support endpoint changes usual regular likelihood asymptotics.
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