Show that if a function of a sufficient statistic is ancillary, then the sufficient statistic is not complete.

termegolz6

termegolz6

Answered question

2022-07-16

Show that if a function of a sufficient statistic is ancillary, then the sufficient statistic is not complete.

Answer & Explanation

grocbyntza

grocbyntza

Beginner2022-07-17Added 25 answers

Suppose that a function of a sufficient statistic f ( T ) is ancillary. Then its pdf doesn't depend on θ, call it g T . T is complete if for all E θ ( f ( T ) ) = c implies that f ( T ) = c. We have that a b f ( T ) g T d θ = c. This is f ( T ) g T ( b a ) = c, so f ( T ) = c g T ( b a ) which is not c. Therefore there exists a θ such that E ( f ( T ) ) = c but f ( T ) = c g T ( b a ) c. Hence T is not complete. Thus, if f ( T ) is ancillary then T is not complete.

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