Shifted exponential maximum likelihood ($\hat{\mu }=X_{(1)},\hat{\lambda }=n/{\sum }_i(X_i-X_{(1)})$)

A shifted exponential distribution has location estimate equal to the sample minimum and rate estimate given above, for a nondegenerate sample of at least two observations. The minimum exceeds the true location on average by $1/(n\lambda )$. The residual sum above the minimum is gamma with shape $n-1$, so the rate estimate has expectation $n\lambda /(n-2)$ for $n>2$ and infinite expectation at $n=2$. The estimated location plus reciprocal rate is the unbiased sample mean.

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 $t\ge 0$. 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.