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Significance points for some tests of uniformity on the sphere. (English) Zbl 0533.62046

Summary: R. J. Beran [J. Appl. Probab. 5, 177-195 (1968; Zbl 0174.502)] and E. Giné [Ann. Stat. 3, 1243-1266 (1975; Zbl 0322.62058)] have proposed several omnibus tests for uniformity on the unit sphere in three dimensional Euclidean space. While several authors have contributed to providing approximate percentage points for the limiting distributions, no tables of the limiting distributions, percentage points thereof, or finite sample distributions or percentage points have been available. In this paper we fill this gap by:
1) finding the exact distributions of the statistics of Beran and Giné for \(n=2\); 2) presenting some percentage points for selected small and moderate sample sizes obtained by Monte-Carlo methods; 3) evaluating numerically the cumulative distribution functions and significance points of the limiting distributions via the Laguerre transform methods.

MSC:

62H10 Multivariate distribution of statistics
62Q05 Statistical tables
65C05 Monte Carlo methods
62E20 Asymptotic distribution theory in statistics
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