a doubt on (complete) sufficient statistics

[peter]

Let S(rho) be an nxn matrix with diagonal entries equal to 1 and off-diagonal entries equal to rho.
Consider the following regression model with "intraclass" correlation matrix:

y|X ~ N(X*beta,sigma^2*S(rho)),

where y is an nx1 vector containing the observations on the "dependent" variable, X is an nxk full rank matrix, and rho is such that S(rho) is positive definite.

Derive the minimal sufficient statistic for this model. Is it complete?

I have some doubts on this, so thanks in advance for your help.

[Ken Pledger]

In article
<1616979150.57336.12584...@gallium.mathforum.org>,
peter <ps...@hotmail.com> wrote:

> ....
> Consider the following regression model ....

Your problem might perhaps get more expert attention in the
<sci.stat.math> news group.

Ken Pledger.

[Peter Steele]

I think it may be useful if I clarify my doubts.

My main doubt is that it seems to me that the for the above model, the
dimension of the minimal sufficient statistic is smaller than the
dimension of the parameter space.

Let us consider the following simplified version of the above model

y|X ~ N(mu*e, sigma^2*S(rho)),

where e is a vector of all ones, and mu, sigma^2, and rho are scalar
unknown parameter.

Letting I denote the nxn identity matrix and ' denote transposition,
the inverse of S(rho) can be written as

S^(-1)=a(rho)*I+b(rho)*ee',

for some scalar functions a(rho) and b(rho).

Thus, by expanding the term

(y-mu*e)' S^(-1) (y-mu*e)

appearing in the density of y|X, and using the the factorization
theorem, we get that the three statistics y'y, y'ee'y, and y'e are
sufficient. But y'ee'y is the square of y'e, so the minimal sufficient
statistic has dimension 2.

What are the implication of the fact that the dimension of minimal
sufficient statistic (i.e., 2) is lower than the dimension of the
parameter space (i.e., 3). It seems to me that one consequence is that
the model does not contain enough information about the parameters mu,
sigma^2 and rho, although I don't understand how to make this
statement precise (because it seems to me that mu, sigma^2 and rho are
identified)

Any thoughts on this would be very useful.

[Ray Koopman]

I made essentially the same proposal a couple years ago:
http://groups.google.ca/group/sci.stat.math/msg/2e45a841f0e31499
I don't think all three parameters can be estimated.

[Peter Steele]

I too think that we will not be able to estimate mu, sigma^2 and rho
in a sensible way, although the three parameters are identified. The
only way of producing sensible estimates is probably to assume that at
least one of the three parameters is known.

Are there general known properties of models having a minimal
sufficient statistic of dimension smaller than the dimension of the
parameter space? I have never come across models of this type before
so I'm very curious.