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.
Publié le : 1995-02-14
Classification:
60-04,
Cumulant generating function of Kendall's score,
hypergeometric distribution,
Kendall's rank correlation with ties in both rankings,
asymptotic normality,
normal,
Edgeworth and Pearson curve approximations,
62G10,
60C05,
60E10,
60E20,
62G20
@article{1176324460,
author = {Valz, Paul D. and McLeod, A. Ian and Thompson, Mary E.},
title = {Cumulant Generating Function and Tail Probability Approximations for Kendall's Score with Tied Rankings},
journal = {Ann. Statist.},
volume = {23},
number = {6},
year = {1995},
pages = { 144-160},
language = {en},
url = {http://dml.mathdoc.fr/item/1176324460}
}
Valz, Paul D.; McLeod, A. Ian; Thompson, Mary E. Cumulant Generating Function and Tail Probability Approximations for Kendall's Score with Tied Rankings. Ann. Statist., Tome 23 (1995) no. 6, pp. 144-160. http://gdmltest.u-ga.fr/item/1176324460/