Exact Nuisance-Free Testing at the Ewens Boundary of the Pitman–Yor Partition, with a Matching Efficiency Bound

The Ewens sampling formula sits at the boundary α=0 of the two-parameter Pitman–Yor partition, and deciding whether data lie on it is an inferential question in which the strength θ is unknown. We give tests of α=0 that need no knowledge of θ. Conditioning on the block count, which is sufficient for the strength, gives a randomized score test with exact level at every sample size, and its score agrees, to the order of the local experiment, with the efficient one. The local experiment is anisotropic: the discount and the strength are learned at the rates (logn)−3/2 and (logn)−1/2, and after each coordinate is rescaled by its own rate, the experiment is a Gaussian shift on a half-plane cone for α≥0 and on the whole plane once the Dirichlet–multinomial branch α<0 is admitted. Profiling out the strength divides the discount information by four. The asymptotic score test attains the one-sided and two-sided local power bounds; whether the exact test attains them is open. At each fixed alternative with 0<α<1 and θ>0, the type II errors of both tests decay like n−(θ+α), which is the fastest rate available to any test whose level is controlled over all strengths. Simulations and an analysis of Williams’ Rothamsted moth abundances illustrate the tests.