= Shifted exponential maximum likelihood
{title2=$\hat\mu=X_{(1)},\ \hat\lambda=n/\sum_i(X_i-X_{(1)})$}
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.
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