# Talk:Completeness (statistics)

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## Rename

In line with the recently renamed sufficient statistic, how about we rename this complete statistic? I think it is more descriptive.—3mta3 (talk) 19:40, 25 November 2009 (UTC)

But, if you look at the subsection "Completeness of the family" you will see that it is the family of distributions that is said to be complete, not a "statistic". Thus your suggested title would be more restrictive than necessary. As to "more descriptive", both are equally descriptive, but of different things. Melcombe (talk) 10:01, 26 November 2009 (UTC)
The rename is good and this above argument is not right! The definition of a complete statistic should be state it is complete for a specific family. rather than just for all theta. I mean it should be for all theta in Theta. The big Theta is the specified family. Jackzhp (talk) 14:50, 25 March 2011 (UTC)

## completeness implies sufficiency?

We have one example showing that a statistic is sufficient and complete, and two examples showing that a statistic is sufficient but not complete. If the statement that "completeness does not entail sufficiency" is true, then we still need one more example to show a statistic is complete but not sufficient. I feel that the statement is not true, but I am not able to prove it. Can someone please prove it is either way (true or not true). Jackzhp (talk) 15:03, 25 March 2011 (UTC)

Consider any family P and the statistic T(X) = 5 (i.e. your statistic always estimates 5, no matter what data is given). Then E(g(T)) = 0 for all $\theta$ implies that p(g = 0) = 1 for all $\theta$. So T is complete, but it isn't sufficient. Phillipk999 (talk) 07:01, 30 October 2011 (UTC)

## Distraction about sufficiency: Needs work before being reinserted

If a complete sufficient statistic exists, then it need not be minimal sufficient.[citation needed] However, we have the following result: suppose T is a minimal sufficient statistic (and under the mild conditions mentioned above, such T can be found), and suppose U is a complete sufficient statistic. Then U is a minimal sufficient statistic (note: completeness does not necessarily imply sufficiency[citation needed] and sufficiency does not necessarily imply completeness). Taking this fact into account, the family Pθ of distributions is called complete if and only if its minimal sufficient statistic is complete.

### Heuristic approach

A sufficient statistic retains at least enough information from the data to estimate θ. A complete statistic retains no irrelevant information in estimating θ (it is possible a complete statistic may retain no information). If the intersection of these two groups exists, it will contain complete sufficient statistics. In other words, it contains efficient statistics that retain as much information as possible from the data and will retain no irrelevant information.

### Counterexample 1

This example will show that the statistic (X1,X2) is sufficient but not complete for the model. Again suppose (X1, X2) are independent, identically distributed random variables, normally distributed with expectation θ and variance 1.

Then

$g((X_1,\ X_2)) = X_1 - X_2\,\!$

is an unbiased estimator of zero (Observe that it's distribution does not depend on the parameter. This means that it is an ancillary statistic). Therefore the pair (X1, X2) itself is not a complete statistic. (Even though the pair (X1, X2) is a sufficient statistic when the sample size is 2).

### Counterexample 2

This example shows that the statistic X is sufficient but not complete for the model. Let U follow Uniform[−½, ½]. Let X = U + θ, so that the distribution of X is parametrized by the mean θ = E(X).

Then if g(x) = sin(2πx), then E(g(X)) = 0 irrespective of θ. Therefore X itself is not a complete statistic for θ.

### Example 3

This example is to show a statistic is complete but not sufficient. ....

### Comments on the above section(s)

If X1 and X2 are two independent random variables with Bernoulli(theta) distribution. Isn't X1 complete and not sufficient? A similar example is to take a sample from Normal(theta,1) and choose as your statistic the mean of a sub-sample. —Preceding unsigned comment added by 128.147.0.41 (talk) 21:51, 28 March 2011 (UTC)

It should suffice to present examples from standard books, e.g. Bickel Doksum or Lehmann, without inventing new examples. Using standard examples will save everybody time.
Also, it seems to me that the article suffered from an over-emphasis on sufficiency.
It seems that this material would be better suited to an article comparing and contrasting the two concepts.  Kiefer.Wolfowitz  (Discussion) 22:00, 28 March 2011 (UTC)
There seems to be confusion. In standard books "completeness" is defined for families of distributions. The concept of "complete sufficient statistic" is only ever considered in the context where the statistic is already known or assumed to be sufficient. E.g. texts say "If a sufficent statistic (satisfies certain conditions) then it is called complete" e.g Cox & Hinkley Theortetical Statistics page 30. I have found no reference for "complete statistic" (none in article either) although one might suppose that this is a statistic whose family of distributions is complete. Melcombe (talk) 16:36, 18 May 2011 (UTC)
And if the discussion is to include things like ancillary statistics, it would be good to have definitions of what sufficiency and completeness means where the nuisance parameters are shown explicitly. Melcombe (talk) 16:44, 18 May 2011 (UTC)
What does Lehmann say, Melcombe? Of course, Lehmann's TSP is the canonical reference. Your quote displays already some inadequacies of Cox and Hinkley. 20:41, 18 May 2011 (UTC)

## The 1st sentence of the example 1

what is the example for? I put it there, why remove it?? Jackzhp (talk) 22:34, 8 April 2011 (UTC)

## Complete class theorem

There is a redirect page that points to this page (I cannot figure out how to produce the redirect page directly, so click on the "redirected from" line at the top of the page).

But this "Completeness (statistics)" page is completely unrelated to the complete class theorem in decision theory. The page on admissible decision rule references this page, as does the decision theory page, but the references and discussion on this page give no help.

This redirect page should probably be removed. Better, someone should convert it into an actual article on the complete class theorem. At the very least, the redirect page should point to a "to be done" article on the complete class theorem, not to the "Completeness (statistics)" page. Bill Jefferys (talk) 00:39, 26 October 2011 (UTC)