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I am trying to show that a normal distribution with parameters $\mu = 0$ and variance $\theta$ is not complete. I am looking for a function $u(x)$ that is not equal to 0 such that $\mathbb E(u(x)) = 0$.

I have done some research on this problem and I have found that $\bar{X}$ and $S$ (sample standard deviation) are independent and can help yield me a function that will give me $\mathbb E(u(x)) = 0$. I know that $\bar{X}$ is normally distributed with mean 0 and variance $\theta/n$. I know that $S^2$ has a chi-squared distribution. I was trying to take an expectation $\mathbb E(\bar{X} S^2)$ to yield a value of 0, but I am not sure how exactly to do that.

Also, would this approach be correct in showing that it is not complete for $\theta$?

EDIT: An iid sample is taken and theta > 0.

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    I have tried to rewrite your post to use math notation. It would be useful for you to learn enough $\LaTeX$ to do it yourself. Looking at examples here should be enough.2012-01-26
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    Note that there is *still more work to be done*. In particular you use both $\theta$ and $\sigma^2$ to refer to the variance. So, you should settle on a single notation. Also, *completeness* is a property of a *family* of distributions, not a single one. Other notes: (1) $S^2$ is *not* distributed chi-squared, but a scaled version of it is, (2) a "function" like $\bar{X} \sigma^2$ will not help you since it is not a *statistic*. **Please** make edits to your question accordingly. Cheers. :)2012-01-26
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    @icobes: It might be best not to materially change the question (even when the edit is small) unless you have confirmation that is what the OP intended. :)2012-01-26
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    My apologies! Will refrain from doing it in the future :)2012-01-26
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    @icobes: Just meant as a word of caution. You have probably correctly interpreted the OP's intent. Still, my opinion (which others may not agree with) is that it is better to allow the OP to edit such things where appropriate.2012-01-26
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    Thank you all for making my post more readable. Any help on how to proceed with this expectation would be great. Also, would X-bar * S^2 now be a correct statistic to take an expectation of?2012-01-26
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    Speaking of $\bar{X}$ and $S^2$ makes it look as if you're talking about an i.i.d. sample. Is that what you have in mind? I assumed at first you were talking about a single obervation. If you've got $X_1,\ldots,X_n\sim\operatorname{i.i.d.}$ with any distribution at all, the statistic $X_1-X_2$ is already an unbiased estimator of $0$, so you've got lack of completeness.2012-01-26

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