Combining Estimates of Tectonic Plate Rotations:: An Extension of Welch's Method to Spherical Regression

Abstract

The relative motion between two diverging tectonic plates is a rotation of the sphere. Given measurements of points on the boundaries of the plates, the rotation can be estimated by minimizing a function which is asymptotically (as the concentration parameter of the data distribution goes to infinity) the sum of squared residuals of a linear regression. The linear approximation permits construction of an asymptotic confidence region for the rotation. To estimate the relative motion between plates that converge, it is necessary to combine two or more rotations of diverging plates, and previous methods required the assumption that separate data sets for all of the rotation estimates have the same concentration parameter. This assumption is frequently contradicted by data, indicating heteroscedasticity in the linear regression model. One successful approach to the problem in the linear model due to Welch, involves sample size asymptotics. A similar solution in the non- linear model thus depends on two kinds of asymptotics. We examine two types of general spherical regression models where the parameter estimate is obtained by maximizing or minimizing a particular function. We establish conditions under which the function converges to a residual sum of squares of linear regression as both concentration parameter and sample size go to infinity. Applying the double-convergence asymptotics to tectonic plate data yields asymptotic confidence regions for combined rotation estimates. The confidence region constructions we propose are appropriate for small to moderate sample sizes. Two kinds of confidence regions are constructed, one of which uses all the data for the separate rotation estimates; the other is a conservative approximation of the first, using only summary statistics from the separate estimates. Simulation runs indicate that both of the new constructions produce confidence regions much more consistent with nominal size, particularly when sample sizes are very different and concentration parameters of the data sets are very unequal.