# Thread: statistics problem

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.

please help me on this one

2. please type in tex
and I'll look at it

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

please take a look at it. thank you

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

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

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

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

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