Tests of Independence

Testing for Interdependence

Testing independence between two of more components of a random vector is an important problem in statistics. For sake of simplicity, suppose that the law of each component is continuous. In the bivariate case, for testing independence between random variables X ₁ and X ₂, most of the tests proposed initially were based on some dependence measure ρ, taking usually value 0 under the null hypothesis of independence. Once a random sample (X ₁₁, X ₁₂), …, (X _n1, X _n2) is collected, that is, the pairs (X _i1, X _i2), i = 1, …, n, are independent observations of (X ₁, X ₂), an estimator $\hat{{\rho }}_{n}$ of ρ is obtained and it is compared with the value of ρ under the null hypothesis. In general, $\hat{{\rho }}_{n}$ must be a “good” estimator of ρ in the sense that as n → ∞, ${n}^{1/2}\left (\hat{{\rho }}_{n} - \rho \right )$ ↝$N\left (0,{\sigma }_{0}^{2}\right )$, where “↝Under the null hypothesis ” denotes convergence in law, and σ ₀is the limiting...