Complete Statistic

**chutiya** · February 13th 2011, 02:25 PM

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

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.

**theodds** · February 13th 2011, 02:36 PM

Originally Posted by chutiya

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

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 $\sum X_i$ and $\sum X_i ^ 2$ can be used to estimate the same thing. Use this to get an unbiased estimate of 0.

**chutiya** · February 13th 2011, 02:52 PM

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

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

is this right or am I lost?

**theodds** · February 13th 2011, 03:00 PM

Originally Posted by chutiya

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

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

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).

**chutiya** · February 13th 2011, 03:11 PM

so if i do this:

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

does this work?

**theodds** · February 13th 2011, 03:20 PM

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

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

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