Editing Sufficient statistic (section)

==Minimal sufficiency==
A sufficient statistic is '''minimal sufficient''' if it can be represented as a function of any other sufficient statistic. In other words, ''S''(''X'') is '''minimal sufficient''' if and only if<ref>Dodge (2003) — entry for minimal sufficient statistics</ref>
#''S''(''X'') is sufficient, and
#if ''T''(''X'') is sufficient, then there exists a function ''f'' such that ''S''(''X'') = ''f''(''T''(''X'')).

Intuitively, a minimal sufficient statistic ''most efficiently'' captures all possible information about the parameter ''θ''.

A useful characterization of minimal sufficiency is that when the density ''f''<sub>''θ''</sub> exists, ''S''(''X'') is '''minimal sufficient''' if and only if{{Citation needed|reason=The classical result (e.g., Theorem 6.2.13 in Casella and Berger) only gives a sufficient condition, not a characterization|date=September 2023}}
:<math>\frac{f_\theta(x)}{f_\theta(y)}</math> is independent of ''θ'' :<math>\Longleftrightarrow</math> ''S''(''x'') = ''S''(''y'')

This follows as a consequence from [[#Fisher–Neyman factorization theorem|Fisher's factorization theorem]] stated above.

A case in which there is no minimal sufficient statistic was shown by Bahadur, 1954.<ref>Lehmann and Casella (1998), ''Theory of Point Estimation'', 2nd Edition, Springer, p 37</ref> However, under mild conditions, a minimal sufficient statistic does always exist. In particular, in Euclidean space, these conditions always hold if the random variables (associated with <math>P_\theta</math> ) are all discrete or are all continuous.

If there exists a minimal sufficient statistic, and this is usually the case, then every [[Completeness (statistics)|complete]] sufficient statistic is necessarily minimal sufficient<ref>Lehmann and Casella (1998), ''Theory of Point Estimation'', 2nd Edition, Springer, page 42</ref> (note that this statement does not exclude a pathological case in which a complete sufficient exists while there is no minimal sufficient statistic). While it is hard to find cases in which a minimal sufficient statistic does not exist, it is not so hard to find cases in which there is no complete statistic.

The collection of likelihood ratios <math>\left\{\frac{L(X \mid \theta_i)}{L(X \mid \theta_0)}\right\}</math> for <math>i = 1, ..., k</math>, is a minimal sufficient statistic if the parameter space is discrete <math>\left\{\theta_0, ..., \theta_k\right\}</math>.