## The Annals of Statistics

- Ann. Statist.
- Volume 23, Number 1 (1995), 144-160.

### Cumulant Generating Function and Tail Probability Approximations for Kendall's Score with Tied Rankings

Paul D. Valz, A. Ian McLeod, and Mary E. Thompson

#### Abstract

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.

#### Article information

**Source**

Ann. Statist., Volume 23, Number 1 (1995), 144-160.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176324460

**Digital Object Identifier**

doi:10.1214/aos/1176324460

**Mathematical Reviews number (MathSciNet)**

MR1331661

**Zentralblatt MATH identifier**

0822.62037

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G10: Hypothesis testing

Secondary: 60C05: Combinatorial probability 60E10: Characteristic functions; other transforms 60E20 62G20: Asymptotic properties

**Keywords**

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

#### Citation

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. 23 (1995), no. 1, 144--160. doi:10.1214/aos/1176324460. https://projecteuclid.org/euclid.aos/1176324460