Correcting the trigonometric moments and resultant lengths for grouped circular data

Abstract We revisit the problem of correcting the trigonometric moments and the resultant lengths that are obtained from grouped circular data. In circular measurements as elsewhere, it is common in many practical applications to aggregate the data into intervals either for convenience, or because of lack of accuracy in measuring, and then assuming that all the observations in a given interval lie at the midpoint of that interval, for calculating needed statistics. It then becomes important to ask for formulae necessary to adjust for the error introduced by such approximations, and this problem has been considered earlier in Greenwood (1959) and Mardia (1972) focusing on the first two trigonometric moments, because they are the ones used mostly in practice. The proofs provided there also rely on other outside results. We take it considerably further with our principal contributions being twofold: (i) unlike earlier derivations, we provide very simple and straightforward derivations which are complete and hold for trigonometric moments of any order and (ii) make recommendations, based on extensive simulations, as to when and where such corrections make a significant difference. To further support our mathematical derivations, we provide extensive simulations using data that may come from 3 potentially common models, namely, the von Mises, the Generalized von Mises, and the Sine-skewed von Mises, and assess the performance of the correction formulae in these contexts. Our results demonstrate that the corrected trigonometric moments yield more accurate estimates, especially when the underlying distribution that generated the data deviate considerably from uniformity. The methods are illustrated using cross-bedding azimuth data from the Kamthi Formation, showing that grouping resolution can materially affect inference on directional concentration.