Confidence intervals for a location parameter of a shifted exponential

Let $X_1,X_2,\dots,X_n$ be independent and identically distributed random variables with the probability density function

$$f(x\mid\theta)=\begin{cases}e^{-(x-\theta)},&\text{if }x\ge\theta,\\0&\text{otherwise,}\end{cases}$$

where $\theta\in\mathbb R$ is the unknown parameter. Further, let $Y=\min(X_1,X_2,\dots,X_n)$. Which of the following are confidence intervals for $\theta$ with the confidence coefficient $(1-\alpha)$, where $\alpha\in(0,1)$?