Sufficient statistics with nuisance parameters.

(i) A statistic T is said to be partially sufficient for θ in the presence of a nuisance parameter η if the parameter space is the direct product of the set of possible

θ- and η-values, and if the following two conditions hold:

(a) the conditional distribution given T = t depends only on η;

(b) the marginal distribution of T depends only on θ. If these conditions are satisfied, there exists a UMP test for testing the composite hypothesis H : θ = θ0 against the composite class of alternatives θ = θ1, which depends only on T .

(ii) Part (i) provides an alternative proof that the test of Example 3.8.1 is UMP.
[Let ψ0(t) be the most powerful level α test for testing θ0 against θ1 that depends only on t, let φ(x) be any level-α test, and letψ(t) = Eη1 [φ(X) | t]. Since Eθiψ(T ) = Eθi,η1φ(X), it follows that ψ is a level-α test of H and its power, and therefore the power of φ, does not exceed the power of ψ0.]
Note. For further discussion of this and related concepts of partial sufficiency see Fraser (1956), Dawid (1975), Sprott (1975), Basu (1978), and Barndorff-Nielsen (1978).