SMOOTH TESTS OF GOODNESS-OF-FIT FOR DIRECTIONAL AND AXIAL DATA

In this paper we develop, for directional and axial data, smooth tests of goodness-of-fit for relationally symmetric distributions against g eneral families of embedding alternatives constructed from complete or thonormal bases of functions. These families generalize a proposal of Beran (1979) based on spherical harmonics. Combined with Rao's score l est, our alternatives yield simple test strategies. We present a metho d for constructing an orthonormal basis adapted to the case where the alternatives are first assumed to be rotationally symmetric and then f or more general situations. As an example of the versatility of our me thod, the results are applied to the problem of testing goodness-of-fi t for the uniform, the von Mises-Fisher-Langevin, and the Scheiddegger -Dimroth-Watson distributions. It is shown that the proposed test stra tegy encompasses and generalizes many of the approaches that have so f ar been proposed For these distributions. Moreover, our method allows for easy adaptation to more complex alternatives than those previously available. Ln addition, the test statistic can be broken into parts t hat may be used to detect specific departures from the null hypothesis . (C) 1997 Academic Press.