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

• 成果类型: Article
• 署名作者: VALZ, PD; MCLEOD, AI; THOMPSON, ME
• 署名单位: Western University (University of Western Ontario); University of Waterloo
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176324460
• 发表日期: 1995
• 页码: 144-160
• 关键词: correlation-coefficient

摘要

Robillard's approach to obtaining an expression for the cumulant generating function of the null distribution of Kendall's S-statistic, when one ranking is tied, is extended to the general case where both rankings are tied. An expression is obtained for the cumulant generating function and it is used to provide a direct proof of the asymptotic normality 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 the general case of tied rankings via a Pearson type I curve and an Edgeworth 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.