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
- 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 fθ exists, S(X) is minimal sufficient if and only if
- is independent of θ : S(x) = S(y)
This follows as a direct consequence from Fisher's factorization theorem stated above.
A case in which there is no minimal sufficient statistic was shown by Bahadur, 1954. 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 ) are all discrete or are all continuous.
If there exists a minimal sufficient statistic, and this is usually the case, then every complete sufficient statistic is necessarily minimal sufficient(note that this statement does not exclude the option of 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 is a minimal sufficient statistic if is discrete or has a density function.
Read more about this topic: Sufficient Statistic
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