So say I have $x_1, x_2... x_n$ random sample with common pdf having a single unknown parameter $\theta$. The question is worded as follows: Find UMVUE for P(x>a). Is this problem just a matter of finding UMVUE for $\theta$ then substituting to the pdf?
Edit:
The common pdf is $f(x)=\theta x^{-2}$ when $x>\theta$ and 0 otherwise.
I tried getting both the MLE and the Method of Moments estimator for $\theta$ but could not. I was able to get $P(x>a)$ as follows $\int_a^{\infty}\theta x^{-2}dx=\theta/a$ when $a>\theta$ and 1 otherwise. So, I need to find the UMVUE for $\theta/a$ and that is where I am stuck.
Edit:
I have made further progress. Realizing that the minimum of the sample gives the only useful information about $\theta$. I obtained the distribution of the minimum $m$ as $f(m)=\frac{n\theta^n}{m^{n+1}}$. $E(m)=\frac{n}{n-1}\theta$ which means that $\frac{n-1}{n}m$ is unbiased for $\theta$.
My question is, can I say that the minimum is sufficient without using the factorization theorem and just arguing that given a sample, the most that can be drawn from it in relation to the parameter as defined is it's minimum? Also, is the minimum complete due to Pareto being exponential family?