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Let $M={\min }_i{x}_i$ and $S={\sum }_i(x_i-M)$. On the likelihood support $0<\mu \le M$, the log likelihood is $n\log \lambda -\lambda {\sum }_i(x_i-\mu )$. For fixed $\lambda$ it increases with $\mu$, so $\hat{\mu }=M$. Maximizing in $\lambda$ then gives the shifted exponential maximum likelihood estimatesThese finite estimates exist almost surely for $n\ge 2$. For one observation, or an all-equal sample, the likelihood is unbounded as $\lambda \to \infty$.

The minimum has $M-\mu \sim \mathrm{Exp}(n\lambda )$, so $E\hat{\mu }=\mu +1/(n\lambda )$: its bias is positive. By exponential memorylessness, $S\sim \mathrm{Gamma}(n-1,\text{rate }\lambda )$. For $n>2$, integration of the gamma density gives $E(1/S)=\lambda /(n-2)$, and henceAt $n=2$ this inverse moment is infinite, so a finite bias does not exist. Finally $E\overline{X}=\mu +1/\lambda =\beta$, so $\hat{\beta }$ is unbiased. The positivity assertions are therefore subject to the sample-size qualifications above.