Let (x₁, x₂, ⋯, xₙ) be independent realizations of a random variable taking values on a circle C of unit circumference, and let Tₙ = n⁻¹ ∫¹₀ ∑ⁿⱼ₌₁ f(x + xⱼ) - n ² dx, where $f(x)$ is a probability density on C, f ε L₂ 0, 1, and the addition x + xⱼ is performed modulo 1. Tₙ is used to test whether the observations are uniformly distributed on C. It includes as special cases several other statistics previously proposed for this purpose by Ajne, Rayleigh and Watson. The main results of the paper are the asymptotic distributions of Tₙ under fixed alternatives to uniformity and under sequences of local alternatives to uniformity. A characterization is found for those alternatives against which Tₙ, with specified $f(x)$, gives a consistent test. The approximate Bahadur slope of Tₙ is calculated from the asymptotic null distribution; however, an example indicates that this slope may not always reflect the power of Tₙ reliably. A Monte Carlo simulation for a special case of Tₙ suggests that a fair approximation to the power of Tₙ may be obtained from its mean and variance under the alternative.
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Rudolf Beran (1969) studied this question.