Let the joint distribution of (𝑿, 𝒀) be bivariate normal with mean vector $\begin{pmatrix} 0 \\ 0\end{pmatrix}$ and variance-covariance matrix $\begin{pmatrix} 1 & 𝝆 \\ 𝝆 & 1 \end{pmatrix}$, where $βˆ’πŸ < 𝝆 < 𝟏$. Let $𝚽_{𝝆}(𝟎, 𝟎) = 𝑷(𝑿 ≀ 𝟎, 𝒀 ≀ 𝟎)$. Then what will be Kendall’s 𝝉 coefficient between 𝑿 and 𝒀 equal to?

Since I am new to statistics I have no idea where to start?


1 Answer 1


Originally, Kendall's tau, also called rank correlation, is a statistical measure that can be applied to a discrete set of observed data.

In the more recent literature about dependency modelling with Copulas which became popular in mathematical finance the following definition of Kendall's tau is given.

Let $\Phi_\rho(x,y),\Phi(x),\Phi(y)$ be the bivariate and the univariate CDFs of the standard normal distribution. Then the Gaussian Copula is defined as $$ C_\rho(x,y)=\Phi_\rho(\Phi^{-1}(x),\Phi^{-1}(y)) $$ Kendall's tau is then defined as \begin{align} \rho_\tau&=\mathbb E\Big[{\rm sign}[(X-\tilde{X})(Y-\tilde{Y})]\Big]\\ &=\mathbb P\Big[(X-\tilde{X})(Y-\tilde{Y})>0\Big]-P\Big[(X-\tilde{X})(Y-\tilde{Y})<0\Big]\,. \end{align} where $(X,Y)$ is bivariate standard normal, and $(\tilde{X},\tilde{Y})$ has the same distribution but is independent of $(X,Y).$ It can be shown (see [1] and duplicate) that $$ \rho_\tau=4\int_0^1\int_0^1C_\rho(x,y)\,dC_\rho(x,y)-1=\frac{2}{\pi}\arcsin\rho\,. $$

[1] M. Haugh, An Introduction to Copulas. IEOR E4602: Quantitative Risk Management Spring 2016. http://www.columbia.edu/


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