Let $x_{1}, \ldots, x_{n}$ be independent and identically distributed observations from a distribution with probability density function

$$f(x)= \begin{cases}\lambda e^{-\lambda(x-\mu)}, & x \geqslant \mu \\ 0, & x<\mu\end{cases}$$

where $\lambda$ and $\mu$ are unknown positive parameters. Let $\beta=\mu+1 / \lambda$. Find the maximum likelihood estimators $\hat{\lambda}, \hat{\mu}$ and $\hat{\beta}$.

Determine for each of $\hat{\lambda}, \hat{\mu}$ and $\hat{\beta}$ whether or not it has a positive bias.