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Title: Parameter estimation for a Cauchy family of distributions on the sphere Authors:  Shogo Kato - Institute of Statistical Mathematics (Japan) [presenting]
Peter McCullagh - University of Chicago (United States)
Abstract: A Cauchy family of distributions on the sphere is proposed as a spherical extension of the wrapped Cauchy family on the circle. Some properties of the proposed family, especially those related to parameter estimation, are discussed. Three estimators for the spherical Cauchy family are presented, namely, a method of moments estimator, the maximum likelihood estimator, and an asymptotically efficient estimator. The method of moments estimator and the asymptotically efficient estimator are expressed in closed form. A simple algorithm is presented to estimate the maximum likelihood estimate numerically. The EM algorithm is also available for maximum likelihood estimation by transforming the spherical Cauchy family into a $t$-family on the Euclidean space via the stereographic projection. Asymptotic properties of the proposed estimators are considered. A simulation study is carried out to compare the estimators in terms of their performance for finite sample sizes.