Partial martingale difference correlation

We introduce the partial martingale difference correlation, a scalar-valued measure of conditional mean dependence of $Y$ given $X$, adjusting for the nonlinear dependence on $Z$, where $X$, $Y$ and $Z$ are random vectors of arbitrary dimensions. At the population level, partial martingale difference correlation is a natural extension of partial distance correlation developed recently by Székely and Rizzo [14], which characterizes the dependence of $Y$ and $X$, after controlling for the nonlinear effect of $Z$. It extends the martingale difference correlation first introduced in Shao and Zhang [10] just as partial distance correlation extends the distance correlation in Székely, Rizzo and Bakirov [13]. Sample partial martingale difference correlation is also defined building on some new results on equivalent expressions of sample martingale difference correlation. Numerical results demonstrate the effectiveness of these new dependence measures in the context of variable selection and dependence testing.