Kendall's Tau

Definition

Let $(X_1,X_1),(X_2,X_2),\dots ,(X_n,X_n)$ be a Random Sample from a bivariate distribution with cdf $F(x,y)$ We want to test $H_0:X,Y$ are independent

Let $(X_1,X_1),(X_2,X_2)$ be independent pairs with the same bivariate distribution. If $\mathrm{sgn}\{(X_1-X_2)(Y_1-Y_2)\}=1$, then we say these pairs are concordant If $\mathrm{sgn}\{(X_1-X_2)(Y_1-Y_2)\}=-1$, then pairs are discordant

Kendall’s $\tau$ is defined as $\tau =P[\mathrm{sgn}\{(X_1-X_2)(Y_1-Y_2)\}=1]-P[\mathrm{sgn}\{(X_1-X_2)(Y_1-Y_2)\}=-1]$

$\tau$ is bounded by $\pm 1$. If $X,Y$ are independent, then $\tau =0$

Consider another statistic $K={\left(\genfrac{}{}{0ex}{}{n}{2}\right)}^{-1}\underset{i<j}{\sum \sum }\mathrm{sgn}\{(X_1-X_2)(Y_1-Y_2)\}$

$E[K]=\tau$, so it is an unbiased estimator of $\tau$

Facts

Let $(X_1,X_1),(X_2,X_2),\dots ,(X_n,X_n)$ be a Random Sample from a bivariate distribution with cdf $F(x,y)$, and $K={\left(\genfrac{}{}{0ex}{}{n}{2}\right)}^{-1}\underset{i<j}{\sum \sum }\mathrm{sgn}\{(X_1-X_2)(Y_1-Y_2)\}$, then under $H_0:X,Y$ are independent

• $K$ is distribution free with a symmetric pdf
• $E_{H_0}(K)=0$
• ${\mathrm{Var}}_{H_0}(K)=\frac{2(2n+1)}{9n(n-1)}$
• $\frac{K}{\sqrt{{\mathrm{Var}}_{H_0}(K)}}\stackrel{D}{\to }N(0,1)$