Sufficient, minimal and correct estimator

Question

Let $X$ a random variable with density $$f(x)=\lambda\theta^\lambda x^{-(\lambda+1)},$$ with $x\geq \theta$, $\lambda>0$ and $\theta>0$.

Let $S=\min\{X_1,\cdots, X_n\}$.

I have the following two questions:

1. Is $S$ a sufficient and minimal estimator of the parameter $\theta$?
2. Is $S$ a unbiased estimator?

I can prove that $S$ is sufficient and that $S$ is the estimator that comes from the maximum likelihood method. I don't know how to prove if $S$ is sufficient or correct.

I also found the distribution of $S$ that, if I am correct, is $P(S\leq x)=\Big(1-\frac{\theta^\lambda}{x^\lambda}\Big)^n$.

Answer 1

If $X \ge \theta$, then $\Pr[X < \theta] = 0$, and similarly, $S = \min\{X_1, \ldots, X_n\}$ implies $S \ge \theta$ and $\Pr[S < \theta] = 0$. Moreover, $\Pr[S > \theta] > 0$: why? What does this tell you about $\operatorname{E}[S]$?

As for minimal sufficiency, use the property that $S$ is minimal sufficient if $$\frac{f_\theta(\boldsymbol x)}{f_\theta(\boldsymbol y)}$$ is independent of $\theta$ if and only if $S(\boldsymbol x) = S(\boldsymbol y)$ for samples $\boldsymbol x$ and $\boldsymbol y$.