Significance points for some tests of uniformity on the sphere^†

^†This work was supported in part by the United States Air Force, Office of Scientific Research under grant No. AFOSR–79–0043.

Abstract

Beran (1968) and Giné (1975) 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 method (Keilsori and Nunn (1979), Keilson, Nunn and Sumita (1981), and Sumita (1981)).

Keywords:

• Small samples
• biné's statistics
• Weighted sums of independent Chi—square variates
• convolution
• the Laguerre transform method

^†This work was supported in part by the United States Air Force, Office of Scientific Research under grant No. AFOSR–79–0043.

^†This work was supported in part by the United States Air Force, Office of Scientific Research under grant No. AFOSR–79–0043.

Notes

^†This work was supported in part by the United States Air Force, Office of Scientific Research under grant No. AFOSR–79–0043.