Question

Let the joint distribution of \( (X,Y) \) be bivariate normal with mean vector 

and variance-covariance matrix 

, where \( -1<\rho<1 \). Let \( \Phi_\rho(0,0) = P(X \leq 0, Y \leq 0) \). Then the Kendall’s \( \tau \) coefficient between \( X \) and \( Y \) equals

• A. \( 4\Phi_\rho(0,0) - 1 \)
• B. \( 4\Phi_\rho(0,0) \)
• C. \( 4\Phi_\rho(0,0) + 1 \)
• D. \( \Phi_\rho(0,0) \)

Hint:

In the bivariate normal case, the Kendall’s \( \tau \) coefficient can be computed using the joint distribution function \( \Phi_\rho(0,0) \), which captures the probability of both variables being less than or equal to 0.

Answer 1

The Correct Option is A

The Kendall’s \( \tau \) coefficient is a measure of the ordinal association between two variables, and it can be related to the joint distribution of \( X \) and \( Y \) in the bivariate normal case. For bivariate normal variables with correlation \( \rho \), the Kendall’s \( \tau \) coefficient is given by the formula: \[ \tau = 4\Phi_\rho(0,0) - 1, \] where \( \Phi_\rho(0,0) \) is the bivariate normal cumulative distribution function evaluated at \( (0,0) \). Since the joint distribution is bivariate normal, the formula above applies directly, and hence the Kendall’s \( \tau \) coefficient is \( 4\Phi_\rho(0,0) - 1 \). Final Answer: \[ \boxed{\text{(A) } 4\Phi_\rho(0,0) - 1}. \]