Independence Test for Bivariate Copula Data

This function returns the p-value of a bivariate asymptotic independence test based on Kendall's $\tau$.

BiCopIndTest(u1, u2)

Arguments

u1, u2: Data vectors of equal length with values in $[0,1]$.

Value

statistic: Test statistic of the independence test.
p.value: P-value of the independence test.

Details

The test exploits the asymptotic normality of the test statistic $$\texttt{statistic} := T = \sqrt{\frac{9N(N - 1)}{2(2N + 5)}} \times |\hat{\tau}|, $$ where $N$ is the number of observations (length of u1) and $\hat{\tau}$ the empirical Kendall's tau of the data vectors u1 and u2. The p-value of the null hypothesis of bivariate independence hence is asymptotically $$\texttt{p.value} = 2 \times \left(1 - \Phi\left(T\right)\right), $$ where $\Phi$ is the standard normal distribution function.

References

Genest, C. and A. C. Favre (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrologic Engineering, 12 (4), 347-368.

Author

Jeffrey Dissmann

Examples

set.seed(123)
## Example 1: Gaussian copula with large dependence parameter
cop <- BiCop(1, 0.7)
dat <- BiCopSim(500, cop)

# perform the asymptotic independence test
BiCopIndTest(dat[, 1], dat[, 2])
#> $statistic
#> [1] 14.92841
#> 
#> $p.value
#> [1] 0
#> 

## Example 2: Gaussian copula with small dependence parameter
cop <- BiCop(1, 0.01)
dat <- BiCopSim(500, cop)

# perform the asymptotic independence test
BiCopIndTest(dat[, 1], dat[, 2])
#> $statistic
#> [1] 1.611854
#> 
#> $p.value
#> [1] 0.1069937
#>