Minimal sufficient statistic

A statistic which is a sufficient statistic for a family of distributions and is such that for any other sufficient statistic , , where is some measurable function. A sufficient statistic is minimal if and only if the sufficient -algebra it generates is minimal, that is, is contained in any other sufficient -algebra.

The notion of a -minimal sufficient statistic (or -algebra) is also used. A sufficient -algebra (and the corresponding statistic) is called -minimal if is contained in the completion , relative to the family of distributions , of any sufficient -algebra . If the family is dominated by a -finite measure , then the -algebra generated by the family of densities

is sufficient and -minimal.

A general example of a minimal sufficient statistic is given by the canonical statistic of an exponential family

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)