In order to estimate θ, we can generally use any sufficient statistic, but it is more practical to use the sufficient statistic with the lowest dimension. Since all statistics are functions of the samples, a minimal sufficient statistic is defined as follows.

Definition: Minimal Sufficient Statistic Sufficient statistic T for parameter θ is minimal if it can be expressed as a function of every other sufficient statistic for θ.

Since a function of the N samples cannot increase the dimension (a function by definition has a unique value for each value of its argument), it is clear that a minimal sufficient statistic must have the lowest dimension possible of all sufficient statistics. If the minimal sufficient statistic is related to another sufficient statistic via a one-to-one function, then both are minimal: they are equivalent. It is important to note that the definition above has two requirements: (i) T must be sufficient and (ii) T is a function of all other sufficient statistics for θ.

Example 9.9. For the uniform pdf , the following single statistic is not minimal sufficient:

(9.27) Numbered Display Equation

Even though T is a function of the sufficient statistics {X

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