A General Correlation Coefficient for Directional Data and Related Regression Problems
提出一个适用于双变量角度分布和一般流形上双变量分布的相关系数ρ2，研究了其性质、与指数分布族的联系，以及在独立性假设下样本系数的渐近分布不依赖于边际分布，并考察了由完全依赖导出的回归模型。
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.