9.7 COMPLETE STATISTIC
Although T = [X(1), X(N)]T is minimal sufficient for θ of the uniform distribution in Example 9.12, and together they contain as much information about θ as the original samples, it turns out they also contain information that is not relevant for estimating θ. This irrelevant information is called ancillary, and any component of a minimal sufficient statistic that is ancillary is called an ancillary statistic. A minimal sufficient statistic that has no ancillary component is called a complete minimal sufficient statistic. We provide two definitions of an ancillary statistic.
Definition: Ancillary Statistic Statistic V is ancillary if its distribution is independent of θ.
An ancillary statistic does not contain any information about parameter θ. This is essentially the opposite property of a sufficient statistic which contains the same information about θ as the original samples.
Definition: First-Order Ancillary Statistic Statistic V is first-order ancillary if is independent of θ.
An ancillary statistic is also first-order ancillary: if its distribution is independent of θ, then its expectation does not depend on θ.
Example 9.16. For the uniform pdf , observe that the statistic V = X(N)−X(1) is first-order ancillary because its mean does not depend on θ: