Sufficiency and Completeness¶

Overview¶

For the Normal model, we identify sufficient and complete statistics.

Sufficient Statistics for Normal¶

By the factorization theorem, \((\bar{X}, S^2)\) is jointly sufficient for \((\mu, \sigma^2)\) in the Normal model.

Completeness¶

A sufficient statistic \(T\) is complete if \(E[g(T)] = 0\) for all \(\theta\) implies \(g(T) = 0\) a.s. Completeness ensures the UMVUE is unique.

Lehmann–Scheffé Theorem¶

If \(T\) is complete and sufficient, then any unbiased function of \(T\) is the unique UMVUE. For the Normal model:

\(\bar{X}\) is the UMVUE of \(\mu\)
\(S^2\) is the UMVUE of \(\sigma^2\)