complete sufficient = minimal sufficient?

[leading]

1. As any statistics textbook points out, a complete sufficient
statistic is necessarily minimal sufficient.
Conversely is minimal sufficient statistic also complete sufficient?
2. If G is a complete sufficient statistic, and f is a function such
that f(G) is a sufficient statistic, is f(G) also complete?
Thanks

[Herman Rubin]

In article <062ffa47-fc03-40e4...@c30g2000hsa.googlegroups.com>,

leading <litin...@gmail.com> wrote:
>1. As any statistics textbook points out, a complete sufficient
>statistic is necessarily minimal sufficient.
>Conversely is minimal sufficient statistic also complete sufficient?

No. The minimal sufficient statistic for the Cauchy
distribution under translation is the ordered sample, and
if the sample size is sufficiently large, the expected
value of the X_[j]+X_[n+1-j] for any j>=3 is twice the
center. If we have a choice of j's we can get two unbiased
estimates of the same parameter, so the expected value of
the difference is 0.

>2. If G is a complete sufficient statistic, and f is a function such
>that f(G) is a sufficient statistic, is f(G) also complete?
>Thanks

Trivially. You have less choice of functions to have
expected value 0.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558