Inadmissible likelihood ratio test. In many applications in which a UMP invariant test exists, it coincides with the likelihood ratio test. That this is, however, not always the case is seen from the following example. Let P1,..., Pn be n equidistant points on the circle x 2 + y2 = 4, and Q1,..., Qn on the circle x 2 + y2 = 1. Denote the origin in the (x, y) plane by O, let 0 < α ≤ 1 2 be fixed, and let (X, Y ) be distributed over the 2n + 1 points P1,..., Pn, Q1,..., Qn, O with probabilities given by the following table:
Pi Qi O H α/n (1 − 2α)/n α
K pi /n 0 (n − 1)/n, where pi = 1. The problem remains invariant under rotations of the plane by the angles 2kπ/n (k = 0, 1,..., n − 1). The rejection region of the likelihood ratio test consists of the points P1,..., Pn, and its power is 1/n. On the other hand, the UMP invariant test rejects when X = Y = 0 and has power (n − 1)/n.