Page 335 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 312

Example 6.5.6 (Example 6.5.1 Continued) Suppose that X , X are iid N(θ,
2
1
1) where θ is the unknown parameter, –∞ < θ < ∞. The statistic T = X – X 2
1
1
is ancillary for θ. Consider another statistic T = X . One would recall from
1
2
Example 6.4.4 that (θ) = 1 whereas I (θ) = 2, and so the statistic T can
X
2
not be sufficient for θ. Here, while T is not sufficient for θ, it has some
2
information about θ, but T itself has no information about θ. Now, if we are
1
told the observed value of the statistic T = (T , T ), then we can reconstruct
2
1
the original data X = (X , X ) uniquely. That is, the statistic T = (T , T ) turns
2
1
2
1
out to be jointly sufficient for the unknown parameter θ. !
Example 6.5.7 This example is due to D. Basu. Suppose that (X, Y) is a
bivariate normal variable distributed as N (0, 0, 1, 1, ρ), introduced in Section
2
3.6, where the unknown parameter is the correlation coefficient ρ ∈ (–1, 1).
Now consider the two statistics T = X and T = Y. Since T and T have
2
2
1
1
individually both standard normal distributions, it follows that T is ancillary
1
for ρ and so is T . But, note that the statistic T = (T , T ) is minimal sufficient
2
2
1
for the unknown parameter ρ. What is remarkable is that the statistic T has
1
no information about ρ and the statistic T has no information about ρ, but the
2
statistic (T , T ) jointly has all the information about ρ. !
1 2
The situation described in (6.5.2) has been highlighted in the Example
6.5.6 where we note that (T , T ) is sufficient for θ, but (T , T ) is not minimal
1
2
1
2
sufficient for θ. Instead, 2T - T is minimal sufficient in the Example 6.5.6.
1
2
The situation described in (6.5.3) has been highlighted in the Example 6.5.7
where we note especially that (T , T ) is minimal sufficient for θ. In other
2
1
words, there are remarkable differences between the situations described by
these two Examples.
Let us now calculate the information where ρ is the
correlation coefficient in a bivariate normal population.