Testing Conditional Independence using Conditional Martingale Transforms

This paper investigates testing conditional independence between Y and Z given λθ(X) for some θ ∈ Θ ⊂ R, for a function λθ(·) known upto a parameter θ ∈ Θ. First, the paper proposes a new method of conditional martingale transforms under which tests are asymptotically pivotal and asymptotically unbiased against √ n-converging Pitman local alternatives. Second, the paper performs an analysis of asymptotic power comparison using the limiting Pitman efficiency. From this analysis, it is found that there is improvement in power when we use nonparametric estimators of conditional distribution functions in place of true ones. When both Y and Z are continuous, the improvement in power due to the nonparametric estimation disappears after the conditional martingale transform, but when either Y or Z is binary, this phenomenon of power improvement reappears even after the conditional martingale transform. We perform Monte Carlo simulation studies to compare the martingale transform approach and the bootstrap approach.