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6. Sufficiency, Completeness, and Ancillarity 316

- (F - 32) between the two units, Fahrenheit and Celsius.
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At this point, one may ask the following question. What is the relevance of
such special families of distributions in the context of ancillarity? It may help
if one goes back to the Examples 6.5.1-6.5.5 and thinks through the process
of how we had formed some of the ancillary statistics. Suppose that X , ...,
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X are iid random variables having the common pdf f(x), indexed by some
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appropriate parameter(s). Then, we can conclude the following.

One should not, however, get the impression that (6.5.8)-(6.5.10) list the
unique or in some sense the “best” ancillary statistics. These summary state-
ments and ancillary statistics should be viewed as building blocks to arrive at
many forms of ancillary statistics.

6.5.2 Its Role in the Recovery of Information

In Examples 6.5.6-6.5.7, we had seen how ancillary statistics could play
significant roles in conjunction with non-sufficient statistics. Suppose that
T is a non-sufficient statistic for θ and T is ancillary for θ. In other
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words, in terms of the information content, < I (θ) where X is the
X
whole data and = 0 for all θ ∈ Θ. Can we recover all the information
contained in X by reporting T while conditioning on the observed value of
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T ? The answer is: we can do so and it is a fairly simple process. Such a
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process of conditioning has far reaching implications as emphasized by