Statistica Sinica

Grace S. Shieh, Richard A. Johnson and Edward W. Frees

Abstract:A class of new statistics for testing independence of bivariate circular data is obtained by averaging a ``weighted Kendall's tau'' over its marginals. The averaging is done by first fixing the two origins, calculating a weighted Kendall's tau rank statistic and then averaging over cyclic permutations of the two sets of ranks. These statistics are based on ranks, are distribution-free, and are invariant under different choices of origin and rotation. They could, for instance, be applied to testing independence of bird flight and prevailing wind direction. We obtain the asymptotic distribution of our rank statistic as a special case of a general class of statistics, called weighted degenerate U-statistics. LetZ_{1,...,}Z_{n}be i.i.d. r.v.'s withE(Z_{1})=0 and Var(Z_{1})=1, andE(Z_{1}^{4}<∞). We define a weighted degenerate U-statistic asWU_{n}=Σ_{i≠j}d_{ijn}h(Z_{i},Z_{j}) . Here, {d_{ijn}} are non-stochastic weights andhis degenerate in the sense that Var[E(h(Z_{1},Z_{2}|Z< sub>2)]=0 . Under regularity conditions, the limit distribution ofWU_{n}is shown to be a linear combination of independent chi square random variables. It is interesting that a special case (using equal weights) of our general procedures, a Circular Kendall's tau, turns out to be equivalent to a statistic proposed by Fisher and Lee (1982). A power study and an application are presented.

Key words and phrases:Directional data, Kendall's tau, limit distribution, rank correlation.