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I was wondering if any of the members of this community would like to share his/her intuition about completeness in statistics. For the sake of "completeness", here's the definition, taken from Wikipedia:

Consider a random variable $X$ whose probability distribution belongs to a parametric family of probability distributions $P_\theta$ parametrized by $\theta$.

The statistic $T$ is said to be complete for the distribution of $X$ if for every measurable function $g$ (which must be independent of $\theta$) the following implication holds: $$E(g(T(X))) = 0 \mbox{ for all }\theta \mbox{ implies that }P_\theta(g(T(X)) = 0) = 1\mbox{ for all }\theta.$$

Now, to give you an idea of what kind of answer I am looking for, here's how I think about (minimal) sufficiency: I think of a statistic as a partition of the sample space. In that context, a statistic is sufficient for $\theta$ if this partition does not result in a loss of "information" about $\theta$; it is minimal sufficient if it is the coarsest partition which does not result in a loss of information (superlatives carry a uniqueness connotation, which I am ignoring here).

Note: I am aware that a very similar question has been asked before, but I'm looking for a different answer.

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  • Also asked on MSE: http://math.stackexchange.com/questions/681577/intuition-behind-statistical-completeness – M Turgeon Apr 20 '14 at 19:23
  • @kjetil the question you link to is a duplicate of the question I mentioned in my original post. Which I said was **not** the same question as mine. – M Turgeon Feb 21 '16 at 13:18
  • But I think the answers to that question answers your question here. Anyhow, it is unclear what sort of answer you are after. Can you please explain what is unsatisfactory with the already given answers? – kjetil b halvorsen Feb 21 '16 at 13:20
  • @kjetil As mentioned in the question above (albeit somewhat implicitely), I would like an answer in terms of partition of the sample space – M Turgeon Feb 21 '16 at 13:22
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    Well, but i'm quite sure such an explanation cannot be given. From Bahadurs's theorem, a sufficient complete statistic is neccesarily minimal sufficient, **but the reverse imlication is untrue**. So you cannot expect to capture completeness only through properties of partitions of the sample space. – kjetil b halvorsen Feb 21 '16 at 13:25

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