Minimal sufficient statistic

A statistic $ X $ which is a sufficient statistic for a family of distributions $ {\mathcal P} = \{ { {\mathsf P} _ \theta } : {\theta \in \Theta } \} $ and is such that for any other sufficient statistic $ Y $, $ X = g ( Y ) $, where $ g $ is some measurable function. A sufficient statistic is minimal if and only if the sufficient $ \sigma $- algebra it generates is minimal, that is, is contained in any other sufficient $ \sigma $- algebra.

The notion of a $ {\mathcal P} $- minimal sufficient statistic (or $ \sigma $- algebra) is also used. A sufficient $ \sigma $- algebra $ {\mathcal B} _ {0} $( and the corresponding statistic) is called $ {\mathcal P} $- minimal if $ {\mathcal B} _ {0} $ is contained in the completion $ \overline{ {\mathcal B} }\; $, relative to the family of distributions $ {\mathcal P} $, of any sufficient $ \sigma $- algebra $ {\mathcal B} $. If the family $ {\mathcal P} $ is dominated by a $ \sigma $- finite measure $ \mu $, then the $ \sigma $- algebra $ {\mathcal B} _ {0} $ generated by the family of densities

$$ \left \{ { p _ \theta ( \omega ) = \frac{d p }{d \mu } ( \omega ) } : {\theta \in \Theta } \right \} $$

is sufficient and $ {\mathcal P} $- minimal.

A general example of a minimal sufficient statistic is given by the canonical statistic $ T = ( T _ {1} \dots T _ {n} ) $ of an exponential family

$$ p _ \theta ( \omega ) = \ C ( \theta ) \mathop{\rm exp} \ \sum _ { j } Q _ {j} ( \theta ) T _ {j} ( \omega ). $$

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