2. Let $X_1, \dots, X_n$ be a random sample from the distribution with the p.d.f.:
$f(x \mid \theta) = c^{-(x-\theta)/3} I(x > \theta)/3, \quad -\infty < \theta < \infty$ (20)
a) Show that $X_{(1)} = \min(X_i)$ is a CSS(Complete Sufficient Statistic) for $\theta$.(10)
(hint : $Y = X_{(1)} \sim p(Y < y \mid \theta) = G(y \mid \theta) = 1 - c^{-n(y-\theta)/3}, \quad y > 0$)
b) Find the MVUE of $\theta$.(10)
(hint: $X_i = U_i + \theta$, $U_i \sim f(u) = (1/3)c^{-u/3}I_{(0,\infty)}(u)$, $U_{(1)} \sim G(u) = 1 - c^{-nu/3}, 0 < u < \infty$)
