This is the main function that calculates the p-value associated with a
nonparametric kernel test of association between the kernel and continuous
outcome variable. A null model (where the kernel is not associated with the
outcome) is initially fit. Then, the variance of
\(Y_{i}|X_{i}\) is used as the basis for the
score test,
$$S\left(\rho\right) = \frac{Q_{\tau}\left(\hat{\beta_0},\rho\right)-\mu_Q}{\sigma_Q}.$$
However,
because \(\rho\) disappears under the null hypothesis, we run a grid search over a range of values of \(\rho\) (the bounds
of which were derived by Liu et al. in 2008). This grid search gets the upper bound for the score test's p-value.
This function is tailored for the underlying model
$$y_{i} = h\left(z_{i}\right) + e_{i},$$
where
\(h\left(\cdot\right)\) is
the kernel function, \(z_{i}\) is a multidimensional array of variables, and \(y_{i}\) is a continuous outcome taking values in
in the real numbers.
The function returns an numeric p-value for the kernel score test of association.