Deriving Kendall's Tau from The Population Perspective (Copulas)

Mar 22, 2025·

Jiyuan (Jay) Liu

· 5 min read

Introduction

Kendall’s tau is one of the most important measures of rank correlation, providing a robust way to quantify the strength of monotonic relationships between variables. While many practitioners are familiar with its computational definition involving concordant and discordant pairs, fewer know about its elegant representation through copulas:

$$\tau = 4 \int_0^1 \int_0^1 C(u,v) \, dC(u,v) - 1$$

This formula reveals deep connections between concordance, probability theory, and the geometry of dependence structures. Let’s derive this beautiful result step by step.

Starting Point: The Copula Formula

We want to understand why Kendall’s tau can be expressed as:

$$\tau = 4 \int_0^1 \int_0^1 C(u,v) \, dC(u,v) - 1$$

where $C(u,v)$ is the copula function that captures the dependence structure between two random variables.

Step 1: Definition of Kendall’s Tau

Kendall’s tau measures the probability of concordance minus the probability of discordance for two random pairs:

$$\tau = P[\text{concordance}] - P[\text{discordance}]$$

For two independent pairs $(X_1, Y_1)$ and $(X_2, Y_2)$ from the same bivariate distribution, we have:

Concordance: $(X_1 - X_2)(Y_1 - Y_2) > 0$
Discordance: $(X_1 - X_2)(Y_1 - Y_2) < 0$

Therefore:

$$\tau = P[(X_1 - X_2)(Y_1 - Y_2) > 0] - P[(X_1 - X_2)(Y_1 - Y_2) < 0]$$

Step 2: Expressing in Terms of Joint Distribution

Since concordance and discordance are complementary events (ignoring ties of probability 0):

$$\tau = 2P[(X_1 - X_2)(Y_1 - Y_2) > 0] - 1$$

The concordance probability can be written as:

$$P[(X_1 - X_2)(Y_1 - Y_2) > 0] = P[X_1 < X_2, Y_1 < Y_2] + P[X_1 > X_2, Y_1 > Y_2]$$

By symmetry of the independent pairs:

$$P[X_1 < X_2, Y_1 < Y_2] = P[X_1 > X_2, Y_1 > Y_2]$$

So:

$$P[(X_1 - X_2)(Y_1 - Y_2) > 0] = 2P[X_1 < X_2, Y_1 < Y_2]$$

Step 3: The Key Integral Representation

Let $F$ and $G$ be the marginal CDFs of $X$ and $Y$ respectively, and let $H(x,y)$ be their joint CDF. By Sklar’s theorem:

$$H(x,y) = C(F(x), G(y))$$

The probability $P[X_1 < X_2, Y_1 < Y_2]$ can be expressed as:

$$P[X_1 < X_2, Y_1 < Y_2] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} H(x,y) \, dH(x,y)$$

Understanding the Integral: The Expected Value of the CDF

Key insight: This integral is computing the expected value of the CDF:

$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} H(x,y) \, dH(x,y) = E[H(X,Y)]$$

Here’s the intuition:

Take a random point $(X_2, Y_2)$ from your distribution
Ask: “What’s the probability that another independent point $(X_1, Y_1)$ falls in the ‘southwest’ quadrant relative to $(X_2, Y_2)$?”
That probability is exactly $H(X_2, Y_2)$
The integral computes the expected value of this probability over all possible positions of the second point

Rigorous Derivation

Law of total expectation:
$$P[X_1 < X_2, Y_1 < Y_2] = E[P[X_1 < X_2, Y_1 < Y_2 \mid X_2, Y_2]]$$
Conditional probability: Given $(X_2, Y_2) = (s,t)$, we need:
$$P[X_1 < s, Y_1 < t \mid X_2 = s, Y_2 = t] = P[X_1 < s, Y_1 < t] = H(s^-, t^-)$$
where $H(s^-, t^-)$ denotes the left-continuous version of $H$ at $(s,t)$. For continuous distributions, this is just $H(s,t)$.
Taking expectation:
$$P[X_1 < X_2, Y_1 < Y_2] = E[H(X_2, Y_2)] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} H(x,y) \, dH(x,y)$$

Step 4: Change of Variables to Unit Square

Transform to uniform margins using $U = F(X)$ and $V = G(Y)$. Then:

$H(x,y) = C(F(x), G(y)) = C(u,v)$ where $u = F(x)$, $v = G(y)$
$dH(x,y) = dC(u,v)$

The integral becomes:

$$P[X_1 < X_2, Y_1 < Y_2] = \int_0^1 \int_0^1 C(u,v) \, dC(u,v)$$

Step 5: Final Assembly

Substituting back into our expression for $\tau$:

$$\tau = 2P[(X_1 - X_2)(Y_1 - Y_2) > 0] - 1$$$$= 2 \cdot 2P[X_1 < X_2, Y_1 < Y_2] - 1$$$$= 4P[X_1 < X_2, Y_1 < Y_2] - 1$$$$= 4 \int_0^1 \int_0^1 C(u,v) \, dC(u,v) - 1$$

Verification with Examples

Independence Case

For independent variables, $C(u,v) = uv$:

$$\tau = 4 \int_0^1 \int_0^1 uv \, d(uv) - 1 = 4 \int_0^1 \int_0^1 uv \, du \, dv - 1$$$$= 4 \int_0^1 u \left[\frac{v^2}{2}\right]_0^1 du - 1 = 4 \int_0^1 \frac{u}{2} du - 1 = 4 \cdot \frac{1}{4} - 1 = 0$$

Perfect Positive Dependence

For the comonotonicity copula $C(u,v) = \min(u,v)$:

$$\int_0^1 \int_0^1 \min(u,v) \, d\min(u,v) = \int_0^1 \int_0^u u \, dv \, du + \int_0^1 \int_u^1 v \, du \, dv$$$$= \int_0^1 u^2 \, du + \int_0^1 v(1-v) \, dv = \frac{1}{3} + \frac{1}{6} = \frac{1}{2}$$

Therefore: $\tau = 4 \cdot \frac{1}{2} - 1 = 1$ ✓

Key Insights and Implications

1. Copula Independence

The formula shows that $\tau$ depends only on the copula, not the marginal distributions. This is why Kendall’s tau is invariant under strictly increasing transformations of the variables.

2. Geometric Interpretation

The integral $\int_0^1 \int_0^1 C(u,v) \, dC(u,v)$ measures the “volume” under the copula surface weighted by the copula itself. This geometric perspective provides intuition about how different dependence structures affect the correlation measure.

3. Range Properties

Since $0 \leq C(u,v) \leq 1$ and $\int_0^1 \int_0^1 dC(u,v) = 1$, we have:

$$0 \leq \int_0^1 \int_0^1 C(u,v) \, dC(u,v) \leq 1$$

This gives us the familiar range $-1 \leq \tau \leq 1$.

4. Connection to Other Measures

This derivation technique can be extended to other dependence measures, revealing deep connections in the theory of copulas and rank correlation.

Conclusion

This derivation elegantly connects the concordance-based definition of Kendall’s tau with the analytical framework of copulas. The key insight that the integral represents the expected value of the CDF provides both computational power and theoretical understanding.

The formula $\tau = 4 \int_0^1 \int_0^1 C(u,v) \, dC(u,v) - 1$ is not just a computational tool—it’s a window into the fundamental nature of dependence, showing how rank correlation emerges naturally from the geometry of probability distributions.

Understanding this connection opens doors to advanced topics in dependence modeling, extreme value theory, and multivariate statistics, making it an essential result for anyone working with copulas and dependence structures.