The Distribution of Length and Components of the Sum of $n$ Random Unit Vectors

Abstract

Statistical problems involving angular observations may arise in diverse scientific fields, either from direct measurement of angles--say the direction of winds or of glacial pebbles or of fracture planes--or they may arise from the measurement of times reduced modulo some period and converted into angles--say time of day when train wrecks occur. Specifically, we consider a set of $n$ points $\xi_\nu$ situated on a unit circle and assumed to constitute a sample from a distribution having the p.d.f. $g(\xi)$, where $0 \leqq \xi < 2\pi$. Let the $n$ random unit vectors thus defined have the components $\sin \xi_\nu$ and $\cos \xi_\nu$, and set \begin{equation*}\tag{0.1} V = \sum \cos \xi_\nu,\quad W = \sum \sin \xi_\nu,\quad R = \sqrt{V^2 + W^2}.\end{equation*} Let $P(r, n)$ be the probability that $R \leqq r$, and let $Q = 1 - P(r, n)$. This paper shows how the statistics $V$ and $R$ provide tests for the uniform distribution $g(\xi) = 1/2\pi$. The distribution of $R$ on the hypothesis of uniformity was derived by Kluyver as a solution to Pearson's random walk problem, and is tabulated here for use in significance tests. The distribution of $V$ is derived here, but has not been calculated. To illustrate the type of tests that might employ the statistics $R$ and $V$, consider a carnival wheel--first from the standpoint of the punter who suspects bias, second from the standpoint of the mechanic who has attempted to introduce bias. The punter, by studying the performance of the wheel, might wish to answer two questions: first, does the wheel differ credibly from an unbiased wheel? second, what is the direction and extent of the bias, if any? An answer to the first question is obtainable from the distribution of $R$, and an answer to the second has been provided by Mises. The mechanic, on the other hand--because he knows the direction of the bias, if there is a bias--might better use the statistic $V$ as a test of his success, and he might appropriately modify the Mises approach in estimating the extent of the bias.