5. Completeness

Question 1

Completeness

Answer 1

A stat. model is said to be complete if for E[g(x)]=0 then Pr{g(x)=0}=1, a statistic is complete if it induces a complete model

Question 2

Transformation of COMP stat

Answer 2

if T is complete then T*=g(T) is complete

Question 3

Basu’s th.

Answer 3

If T is COMP SUFF for the statistical model, and V is ancillary, then T and V are stat. independent

Question 4

Proof of Basu’s th.

Answer 4

Remembering that Pr{X in B} = E[1_B(X)] then we can write Pr{V in B} =cB and Pr{V in B|T=t} =hB, both depending on B but not on @.
Then E[hB] = E[E[1_B(V)|T]] = E[1_B(V)] = cB,
then e{hB - cB] = 0, but since T is complete this implies that hB = cB, proving independence