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February, 1995 Cumulant Generating Function and Tail Probability Approximations for Kendall's Score with Tied Rankings
Paul D. Valz, A. Ian McLeod, Mary E. Thompson
Ann. Statist. 23(1): 144-160 (February, 1995). DOI: 10.1214/aos/1176324460


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, $S/ \sqrt{\operatorname{Var}(S)}$, 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.


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Paul D. Valz. A. Ian McLeod. Mary E. Thompson. "Cumulant Generating Function and Tail Probability Approximations for Kendall's Score with Tied Rankings." Ann. Statist. 23 (1) 144 - 160, February, 1995.


Published: February, 1995
First available in Project Euclid: 11 April 2007

zbMATH: 0822.62037
MathSciNet: MR1331661
Digital Object Identifier: 10.1214/aos/1176324460

Primary: 62G10
Secondary: 60C05 , 60E10 , 60E20 , 62G20

Keywords: 60-04 , asymptotic normality , Cumulant generating function of Kendall's score , Edgeworth and Pearson curve approximations , hypergeometric distribution , Kendall's rank correlation with ties in both rankings , normal

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 1 • February, 1995
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