The concept of a minimal sufficient statistic, which captures nothing more than the essential. Suppose X has a sufficient statistic T. Must it also have a minimal sufficient statistic?

Zaiden Soto

Zaiden Soto

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2022-08-20

The concept of a minimal sufficient statistic, which captures nothing more than the essential.
Suppose X has a sufficient statistic T. Must it also have a minimal sufficient statistic?

Answer & Explanation

Wilson Riggs

Wilson Riggs

Beginner2022-08-21Added 8 answers

No. You have a minimial sufficient statistic iff your parametric distribution f ( X ; θ ) can be factorized into h ( X ) g ( θ ; T ( X ) ), then the T ( X ) is the minimal sufficient statistic or can be reduced to one. Moreover, it is pretty common to fail to satisfy this condition, e.g., Weibull distribution which density function is given by
f ( x ; θ ) = θ x θ 1 λ θ exp { x θ / λ θ } , x 0 ,
where θ > 0 , θ 1 is the unknown shape parameter, cannot be factorized w.r.t θ.

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