The minimum of n exponentials has rate n

Let $X_1,X_2,\dots ,X_n$ be a random sample from an exponential distribution with probability density function f(x|$\theta )=e^{(}-(x-\theta ))$ if $x\ge \theta$, and 0 otherwise, where $\theta \in \mathbb{R}$ is an unknown parameter. If $X_{(1)}=\min${$X_1,X_2,\dots ,X_n$}, then which of the following statements are true?