Smooth test for uniformity on the surface of a unit sphere

In this paper, we suggest the smooth test procedures for testing uniformity on the surface of a unit sphere against the embedding alternatives based on spherical harmonics. The limiting distribution of the test statistic under null hypothesis is obtained, the order of the alternatives is suggested. The analysis of the usual classes of distribution shows that the embedding alternatives represents a sufficiently broad class of distribution on the surface of a unit sphere. Furthermore the smooth test is suitable for detecting arbitrary alternatives being approximated by the embedding alternatives.