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I have some trouble understanding complete sufficient statistics?

Let $T=\Sigma x_i$ be a sufficient statistic.

If $E[g(T)]=0$ with probability 1, for some function $g$, then it is a complete sufficient statistic.

But what does this mean? I've seen examples of uniform and Bernoulli (page 6 http://amath.colorado.edu/courses/4520/2011fall/HandOuts/umvue.pdf), but it's not intuitive, I got more confused seeing the integration.

Could someone explain in a simple and intuitive way?

kjetil b halvorsen
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user13985
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1 Answers1

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Essentially, it means that no non-trivial function of the statistic has constant mean value.

This may not be very enlighthening in itself. Perhaps one way of looking at the utility of such notion is in connection with the theorem of Lehmann-Scheffé (Cox-Hinkley, Theoretical Statistics, p. 31): "In general, if a sufficient statistic is boundedly complete it is minimal sufficient. The converse is false."

Intuitively, if a function of $T$ has mean value not dependent on $\theta$, that mean value is not informative about $\theta$ and we could get rid of it to obtain a sufficient statistic "simpler". If it is boundedly complete ans sufficient, no such "simplification" is possible.

F. Tusell
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  • thanks. How I see it is: you find the expectation of your unbiased estimator, say $\delta$. Set the expectation of $\delta$ equal to zero. And the only way to get that is let $\delta=0$. And that's $\delta$ would be complete sufficient. – user13985 Nov 05 '12 at 19:30
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    Thanks for the answer! (1) "if a function of T has mean value not dependent on θ, that mean value is not informative about θ", how could we "get rid of it to obtain a sufficient statistic simpler"? (2) Why does completeness ["ensures that the parameters of the probability distribution representing the model can all be estimated on the basis of the statistic: it ensures that the distributions corresponding to different values of the parameters are distinct"](http://en.wikipedia.org/wiki/Completeness_(statistics))? please also see my question here http://stats.stackexchange.com/q/53107/1005. – Tim Sep 04 '14 at 13:37