A general correlation coefficient for directional data and related regression problems

SUMMARY A correlation coefficient p2 iS proposed for bivariate angular distributions and for bivariate distributions on general manifolds. In the cylindrical case p2 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 p2 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 p2 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 p2 are examined. A numerical example is given.

[1]  Richard A. Johnson,et al.  Some Angular-Linear Distributions and Related Regression Models , 1978 .

[2]  Kanti V. Mardia,et al.  A spherical correlation coefficient robust against scale , 1978 .

[3]  Kanti V. Mardia,et al.  A Model for Cylindrical Variables with Applications , 1978 .

[4]  K. Mardia,et al.  The von Mises–Fisher Matrix Distribution in Orientation Statistics , 1977 .

[5]  M. Puri,et al.  Problems of Association for Bivariate Circular Data and a New Test of Independence. , 1977 .

[6]  K. V. Mardia,et al.  Linear-Circular Correlation Coefficients and Rhythmometry , 1976 .

[7]  R. Johnson,et al.  Measures and models for angular correlation and angular-linear correlation. [correlation of random variables] , 1976 .

[8]  M. Giné,et al.  Invariant Tests for Uniformity on Compact Riemannian Manifolds Based on Sobolev Norms , 1975 .

[9]  Kanti V. Mardia,et al.  Multisample Tests for Multimodal and Axial Circular Populations , 1973 .

[10]  K. Mardia Statistics of Directional Data , 1972 .

[11]  Edward D. Rothman,et al.  Tests of Coordinate Independence for a Bivariate Sample on a Torus , 1971 .

[12]  K. Mardia Measures of multivariate skewness and kurtosis with applications , 1970 .

[13]  Gould Al A Regression Technique for Angular Variates , 1969 .

[14]  A. James Distributions of Matrix Variates and Latent Roots Derived from Normal Samples , 1964 .

[15]  J. Hooper,et al.  Simultaneous Equations and Canonical Correlation Theory , 1959 .

[16]  J. Mackenzie,et al.  The estimation of an orientation relationship , 1957 .

[17]  K. Pillai Some New Test Criteria in Multivariate Analysis , 1955 .