CUMULANT GENERATING FUNCTION AND TAIL PROBABILITY APPROXIMATIONS FOR KENDALLS SCORE WITH TIED RANKINGS

Robillard's approach to obtaining an expression for the cumulant gener ating function of the null distribution of Kendall's S-statistic, when one ranking is tied, is extended to the general case where both ranki ngs are tied. An expression is obtained for the cumulant generating fu nction and it is used to provide a direct proof of the asymptotic norm ality of the standardized score, S/root Var(S), when both rankings are tied. The third cumulant of S is derived and an expression for exact evaluation of the fourth cumulant is given. Significance testing in th e general case of tied rankings via a Pearson type I curve and an Edge worth approximation to the null distribution of S is investigated and compared with results obtained under the standard normal approximation as well as the exact distribution obtained by enumeration.