How to show T(x) isn't a sufficient statistic?

metodystap9

metodystap9

Answered question

2022-08-09

How to show T ( x ) isn't a sufficient statistic?

Answer & Explanation

Jewel Brooks

Jewel Brooks

Beginner2022-08-10Added 15 answers

We know that ( X ( 1 ) , X ( n ) ) is minimal sufficient for θ. Recall that T ( X ) is minimal sufficient iff, for any sufficient S, then T = ψ ( S ) a.s. in the whole family U ( θ , 2 θ ) for a Borel ψ. Now suppose that ( X ¯ , S 2 ) is sufficient; then there should be a ψ s.t. ( X ( 1 ) , X ( n ) ) = ψ ( X ¯ , S 2 ); however there is no such ψ, which would imply that ( X ( 1 ) , X ( n ) ) is not minimal sufficient (but it is). We conclude that ( X ¯ , S 2 ) is not sufficient.

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