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PQ.png

I read some examples about how to show statistic

complete, but I still have no idea about the two dimension case, If you could help me, I will be very appreciate.

PQ.png

A Gamma random variable has pdf
1
fy (y; a, B) =
[(a)Bay" e-y/B, Oky <0o
0,
otherwise
Suppose that we have a random sample of size n from a Gamma distribution. The two
dimension sufficient statistic for o and B is ([Yi, EYi).
(a) Is the sufficient statistic also complete? (i.e. is this a nice exponential family of
distributions?)
(b) Find the method of moments estimators for a and B.
(c) Suppose that a = 2 is known.
(i) Find the maximum likelihood estimator of B.
(ii) Can you argue that BMLE is the UMVUE of 3 based on the Lehmann-Scheffe
Theorem?

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