In the matched-pairs experiment for testing the effect of a treatment, suppose that only the differences Zi = Yi − Xi are observable. The Z’s are assumed to be a sample from an unknown continuous distribution, which under the hypothesis of no treatment effect is symmetric with respect to the origin. Under the alternatives it is symmetric with respect to a point ζ > 0. Determine the test which among all unbiased tests maximizes the power against the alternatives that the Z’s are a sample from N(ζ,σ2) with ζ > 0.

[Under the hypothesis, the set of statistics (n i=1 Z2 i ,...,n i=1 Z2n i ) is sufficient; that it is complete is shown as the corresponding result in Theorem 5.8.1.

The remainder of the argument follows the lines of Section 11.]