Improved Pivotal Methods for Constructing Confidence Regions With Directional Data

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Abstract
The importance of pivoting is well established in the context of nonparametric confidence regions. It ensures enhanced coverage accuracy. However, pivoting for directional data cannot be achieved simply by rescaling. A somewhat cumbersome pivotal method, which involves passing first into a space of higher dimension, has been developed by Fisher and Hall for samples of unit vectors. Although that method has some advantages over nonpivotal techniques, it does suffer from certain drawbacks—in particular, the operation of passing to a higher dimension. Here we suggest alternative pivotal approaches, the implementation of which does not require us to increase the intrinsic dimension of the data and which in practice seem to achieve greater coverage accuracy. These methods are of two types: new pivotal bootstrap techniques and techniques that exploit the “implicit pivotalness” of the empirical likelihood algorithm. Unlike the method proposed by Fisher and Hall, these methods are also applicable to axial data and lead to the first pivotal, small-sample nonparametric confidence methods for mean principal or polar axes.

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