A general correlation coefficient for directional data and related regression problems

Abstract

A correlation coefficient ρ2 is proposed for bivariate angular distributions and for bivariate distributions on general manifolds. In the cylindrical case ρ2 is the coefficient of Mardia (1976), and for the bivariate angular case it is a modified version of the correlation coefficient of Mardia & Puri (1978). Some properties of ρ2 are examined and compared with those of other bidirectional correlation coefficients. In particular, this coefficient is found to be closely connected with important exponential families of distributions. Further, the asymptotic distribution of the sample version of ρ2 under the hypothesis of independence does not depend on the marginal distributions. Thus it is asymptotically robust against concentration in the bivariate angular case. The regression models arising from complete dependence as measured by ρ2 are examined. A numerical example is given.