Statistical Inference — confidence coefficient for a shifted exponential threshold

Let $x_1,x_2,…,x_n$ be a random sample from $e^{-(x-\theta)}$, $x\ge\theta$, $\theta>0$. What is the confidence coefficient of the interval $\left(x_{(1)}-\frac{\ln4}{n},\,x_{(1)}+\frac{\ln2}{n}\right)$ for $\theta$, where $x_{(1)}=\min(x_1,x_2,…,x_n)$?