The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
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On the Application of Multivariate Kendall's Tau and Its Interpretation

다차원 캔달의 타우의 통계학적 응용과 그의 해석

  • Lee, Woojoo (Department of Statistics, Inha University) ;
  • Ahn, Jae Youn (Department of Statistics, Ewha Womans University)
  • 이우주 (인하대학교 통계학과) ;
  • 안재윤 (이화여자대학교 통계학과)
  • Received : 2013.04.02
  • Accepted : 2013.05.15
  • Published : 2013.06.30
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Abstract

We study multivariate extension of Kendall's tau and its statistical interpretation. There exist various versions of multivariate Kendall's tau, for example Scarsini (1984), Joe (1990) and Genest et al. (2011); however, few of them mention its lower bounds. For the bivariate case, the Fr$\acute{e}$chet-Hoeffding lower bound can achieve the lower bound of Kendall's tau. However in the multivariate case, the Fr$\acute{e}$chet-Hoeffding lower bound itself does not exist as a distribution, which makes the interpretation of Kendall's tau unclear when it has negative value. In this paper, we explain sufficient conditions to achieve the lower bound of Kendall's tau and provide real data examples that provide further insights into the interpretation for the lower bounds of Kendall's tau.

본 논문에서는 캔달의 타우(Kendall's tau)의 다차원으로의 확장과 그의 통계적 추론 및 해석에 대해 알아본다. 특히 다차원 캔달의 타우가 음의 값을 가질 때 의미를 해석하기 위해, 그것의 하한이 얻어지는 경우를 직관적으로 이해할 수 있도록 변수들간의 관계의 관점에서 설명하여본다. 또한 다차원 캔달의 타우를 실제 사례에 적용해 본후, 최근 Lee와 Ahn에서 연구된 d-countermonotonicity와 partially m-countermonotonic와 같은 새로운 개념을 통하여 캔달의 타우가 음의 값이 가질 때의 의미에 대해서 논의한다.

Keywords

Acknowledgement

Supported by : 인하대학교

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