$f(x, \theta)= \frac{\theta}{x^2}$ with $x\geq\theta$ and $\theta>0$, find the MLE Let $X$ be a random variable with density $$f(x, \theta)= \frac{\theta}{x^2}$$ with $x\geq\theta$ and $\theta>0$.
a) Show if $S=\min\{x_1,\cdots, x_n\}$ is a sufficient statistics and if it is minimal.
b) Find the Maximum Likelihood Estimator of $\theta$ and tell if it is unbiased.
c) Find the distribution of $S$ and tell if there is an unbiased estimator of the form $cS$ for some $c$.
attempt: There are several problems. $S$ doens't look like a sufficient statistics cause $L(\theta|x_1, \cdots, x_n)= \frac{\theta^n}{(x_1\cdots x_n)^2}$ doesn't seem to be known if we know the minimum. Morover there is no maximum for that function so I can't find MLE.
Thanks!

Comment. This is for intuition only. It seems your conversation with @V.Vancak (+1) has taken care of (a). 
Hints for the rest: Using the 'quantile' (inverse CDF) method, an observation $X$ from your Pareto distribution can be simulated as 
$X = \theta/U,$ where $U$ is standard uniform.
In the simulation let $\theta = 2$ and $n = 10.$ Sample a million sample minimums $S.$ Then the average of the $S$'s approximates $E(S).$ Clearly the minimum is
a biased estimator of $\theta.$ In this example $E(S) \approx 2.222 \pm 0.004.$ This makes intuitive sense because the minimum
must always be at least a little larger than $\theta.$
m = 10^5; th = 2;  n = 10
s = replicate(m, min(th/runif(n)))
mean(s) 
## 2.221677      # aprx E(S)


I will leave the mathematical derivation of $E(S)$ and looking for an 'unbiasing' constant
(that may depend on $n$) to you.
Addendum: (One more hint per Comments.) According to Wikipedia the CDF of $X$ is  $F_X(x) = 1 - \theta/x,$ for $x > \theta.$ [This is the CDF I inverted in order to simulate, noting that $U = 1 - U^\prime$ is standard uniform if $U^\prime$ is.] 
Thus for $n \ge 2,$
$$1 - F_S(s) = P(X > s) = P(X_1 > s, \dots, X_n > x) 
= P(X_1 > s) \cdots P(X_n > s) = (\theta/s)^n.$$
So $F_S(s) = P(S \le s) = 1 - (\theta/s)^n,$ for $s > \theta.$
From there you should be able able to find $f_S(s),\,$ $E(S),$
and the unbiasing constant.
