1. ## Complete Statistic

Let $\displaystyle X_1,\cdots,X_n$ be a random sample from $\displaystyle N(\mu,\mu^2)$. Show that $\displaystyle T=(\sum_{i=1}^n X_i,\sum_{i=1}^n {X_i}^2) \mbox{is minimum sufficient but not complete}$.

so, I did the first part to show that T is a minimum sufficient statistic. But I am confused on how to show that it is not complete. How can I find a non zero function using t that does not depend on mu? I am confused on the expected value part. Could anyone help?
Thanks.

2. Originally Posted by chutiya
Let $\displaystyle X_1,\cdots,X_n$ be a random sample from $\displaystyle N(\mu,\mu^2)$. Show that $\displaystyle T=(\sum_{i=1}^n X_i,\sum_{i=1}^n {X_i}^2) \mbox{is minimum sufficient but not complete}$.

so, I did the first part to show that T is a minimum sufficient statistic. But I am confused on how to show that it is not complete. How can I find a non zero function using t that does not depend on mu? I am confused on the expected value part. Could anyone help?
Thanks.
Hint: It looks like $\displaystyle \sum X_i$ and $\displaystyle \sum X_i ^ 2$ can be used to estimate the same thing. Use this to get an unbiased estimate of 0.

3. Ok. so $\displaystyle \sum_{i=1}^n X_i \sim N(n\mu, n\mu^2)$ and $\displaystyle E[(\sum_{i=1}^n X_i)^2]=n\mu^2+n^2\mu^2$

then find $\displaystyle E_{\mu}[\sum_{i=1}^n {X_i}^2-(\sum_{i=1}^n {X_i})^2] = 0$

is this right or am I lost?

4. Originally Posted by chutiya
Ok. so $\displaystyle \sum_{i=1}^n X_i \sim N(n\mu, n\mu^2)$ and $\displaystyle E[(\sum_{i=1}^n X_i)^2]=n\mu^2+n^2\mu^2$

then find $\displaystyle E_{\mu}[\sum_{i=1}^n {X_i}^2-(\sum_{i=1}^n {X_i})^2] = 0$

is this right or am I lost?
You have the right idea, but you are going to need to scale one of those statistics to make the expectation 0 (they don't have the same expectation right off the bat, but the expectations are proportional).

5. so if i do this:

$\displaystyle E_{\mu}[(n+1)\sum_{i=1}^n {X_i}^2-(\sum_{i=1}^n {X_i})^2] = [(n+1)\times n\mu^2] - [n\mu^2+n^2\mu^2]=0$

does this work?

6. $\displaystyle E_\mu \sum X_i ^ 2 = \sum E_\mu X_i ^ 2 = \sum (\mu^2 + \mu^2)$

$\displaystyle E_\mu \left(\sum X_i \right)^2 = n\mu ^ 2 + n^2 \mu^2$