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6. Sufficiency, Completeness, and Ancillarity 318

so that we have

In other words, the information about ρ contained in the conditional distribu-
tion of T | T = v, v ∈ ℜ, is given by
1 2

which depends on the value v unlike what we had in the Example 6.5.10.
Then, the information contained in (X, Y) will be given by

In other words, even though the statistic X tells us nothing about ?, by aver-
aging the conditional (on the statistic Y) information in X, we have recovered
the full information about ρ contained in the whole data (X, Y). Refer to the
Example 6.5.8. !

6.6 Completeness
Consider a real valued random variable X whose pmf or pdf is given by f(x; θ)
for x ∈ χ and θ ∈ Θ. Let T = T(X) be a statistic and suppose that its pmf or pdf
is denoted by g(t; θ) for t ∈ T and θ ∈ Θ.
Definition 6.6.1 The collection of pmf’s or pdf’s denoted by {g(t; θ): θ ∈
Θ} is called the family of distributions induced by the statistic T.
Definition 6.6.2 The family of distributions {g(t; θ): θ ∈ Θ}, induced
by a statistic T, is called complete if and only if the following condition