Say we have a shifted exponential distribution with common density $$f(x|\theta)=\left\{\begin{matrix} e^{-(x-\theta)} & x\geq \theta\\ 0 & x<\theta \end{matrix}\right.$$
We have $\theta$ a real number as the unknown shift parameter and $\textbf{X}=(X_1,...,X_n)$ a random sample. If $X_{(1)}=min \left \{ X_1,...,X_n \right \}$, then the density $f_{(1)}(x)=ne^{-n(x-\theta)}$, where $x\geq \theta$.
I have tried to determine if this estimator $\hat{\theta}=X_{(1)}$ is unbiased or not. I got that $E[\hat{\theta}]=e^{-n\theta}(\theta+\frac{1}{n})$ which is not equal to $\theta$ so there is bias, but I am not sure if I have done this correctly.
If this estimator isn't biased, how does one then determine an unbiased estimator by making an adjustment to the estimator $\hat{\theta}=X_{(1)}$.