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.