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

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

References

[1]	J.-R. Barra, "Mathematical bases of statistics" , Acad. Press (1981) (Translated from French)
[2]	L. Schmetterer, "Introduction to mathematical statistics" , Springer (1974) (Translated from German)

References

[a1]	E.L. Lehmann, "Theory of point estimation" , Wiley (1983)
[a2]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)