Sankhya: The Indian Journal of Statistics

2003, Volume 65, Pt. 2, 333--355

Inference For Wrapped Symmetric $\Alpha$-Stable Circular Models

By

RICCARDO GATTO, University of Bern, Bern, Switzerland and S. RAO JAMMALAMADAKA, University of California, Santa Barbara, USA

SUMMARY: This article provides accurate approximations for the distribution of the length of the resultant as well as for the conditional distribution for the circular mean given the resultant length, when the data come from a wrapped symmetric $\alpha$-stable model. Since the latter distribution is asymptotically independent of the concentration parameter for a given value of the resultant length, it can be used for inference on the mean direction when the concentration parameter is unknown. The value of the saddlepoint methods lies in making such asymptotics available for very small sample sizes. Besides  possessing important theoretical properties, this class of circular models is very rich and includes the wrapped normal and the wrapped Cauchy distributions as special cases. These distributional results allow one to employ this broader class of parametric distributions instead of the von Mises distribution, as is typically done with circular data.

AMS (1991) subject classification. 62H11, 62E17, 60E10.

Key words and phrases. Angular distribution, concentration parameter, conditional inference, Fourier coefficients, heavy-tailed distribution, influence function, mean direction, trigonometric method of moments estimator, resultant length, saddlepoint approximation.

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