I was aolving the following problem:
Find the statistics of the following density that is complete: $$\text{Let } y_k \text{ be i.i.d random variables with the density: }\\f(y_k;x)=(x-1)y_k^{-x}\mathbb{1}_{\{y_k \geq 1\}} ; k=1,\dots ,N$$.
Here is how I attempted this question:
From Neyman-Fisher Vactorization we know that a function $T(y)$ is sufficient if we can write the given density in the form: $f(y;x) = g(T(y),x).h(y)$ where $g$ depends on $y$ only through $T(y)$.
I can write the density of the function y as:
$f(y,x) = \prod_{k=1}^N f(y_k,x) = (x-1)^N \big\{ \prod y_k\big\}^{-x} \mathbb1_{\{\text{min}y_k \geq 1\}}$.
Comparing it with the fisher form I can write $T(y) = \prod_{k=1}^N y_k, h(y) = \mathbb1_{\{\text{min}y_k \geq 1\}} $ and $g(T(y),x) = (x-1)^N \big\{ T(y)\big\}^{-x}$.
Further to prove the completeness of the statistics I have to prove that there is only one function that is unbiased. How to find that function and prove its uniqueness?
Any hint in this regard will be helpful. Thank You!