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Guest
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Hi,
I am reading an article where the pdf of three param family
(A,B,sigma) for the random sample X = [x1,...,xn] is written as the
normal pdf :
(1) f(X|A,B,sigma) = {1/[2*pi*sigma^2]^n} *
exp{ -[ X*X + g(A,B) + T1(X,A,B) +
T2(X,A,B)] / (2*sigma^2)}
In the paper, the authors state that T1 and T2 are (jointly)
sufficient statistics for the parameter B, and invoke the fact that
the pdf being from the exponential family of pdfs to allow them to
"read off" the sufficient statistics.
I have a problem with this claim. The only factorization thm of this
type applying to exponential pdfs having a collection of distribution
params Q, I know says that if the pdf has the form:
(2) f(X|Q) = H(X) * G(Q) * exp( Sum i = 1 to M of Wi(Q)*Ti(X) )
Then the T1(X),...TM(X) are jointly sufficient stats for Q.
(from Casella and Berger, Stat Inference thm 6.2.10)
Since it is a factorization thm, I would have to believe that that eqn
(2) would be a sufficient and neceesary condtn for a sufficient
statistic
Given the above statement, I don't see how the pdf in (1) can lead to
the conclusion that T1 and T2 are sufficient stats for B since upon
factorization of the T1 and T2 portions of the pdf, I am left with two
factors, a factor {1/[2*pi*sigma^2]^n} , which is like G(Q) in eqn
(2), but then the factor exp{ -[ X*X/(2*sigma^2)]} , which is *not*
like the H(X) factor in (2).
Is this conclusion wrong or is the author using some type of relaxed
condition that since B is not in the exp{ -[ X*X/(2*sigma^2)]}
factor, T1 and T2, while not sufficient stats for {A,B and sigma}, are
sufficient stats for A and B.
TIA,
Matt Brenneman |
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