1. ## statistics problem

Let X1,X2,....Xn be a random sample from
f(x; u , v) = {v/(2*Pi*x^3)}^(1/2) *Exp[-v(x-u)^2/(2*u^2*x)] where u and v are parameters and x>0

Find a complete sufficient statistic for (u,v) and use it to find the Minimum Variance Unbiased Estimator of u.

and I'll look at it

3. Let $X1,X2,....Xn$ be a random sample from $f(x; u,v)=(v/2\pi x^3)^{1/2}e^{-v(x-u)^2/2u^2x},$where $x>0$.
Find a complete sufficient statistic for (u,v) and use it to find the Minimum Variance Unbiased Estimator of $u$.

please take a look at it. thank you

4. If I read this correctly, the exponent needed ()...

$L(x)=\bigg({v\over 2\pi}\biggr)^{n/2} \Pi_{i=1}^n x_i^{-3/2}exp\biggl({-v\over 2u^2}\sum_{i=1}^n(x_i-u)^2/x_i\biggr)$

$=\bigg({v\over 2\pi}\biggr)^{n/2} \Pi_{i=1}^n x_i^{-3/2}exp\biggl({-v\over 2u^2}\sum_{i=1}^n(x_i-2u+u^2x_i^{-1})\biggr)$

$=\bigg({v\over 2\pi}\biggr)^{n/2} \Pi_{i=1}^n x_i^{-3/2}exp\biggl({-v\sum_{i=1}^nx_i\over 2u^2}+{nv\over u}-{v\sum_{i=1}^nx_i^{-1}\over 2}\biggr)$

So our joint suff stats are $\sum_{i=1}^nx_i$ and $\sum_{i=1}^nx_i^{-1}$