Method challenge

The true distribution of $j$-th explanatory variable is unknown and we have 2 alternatives:

1. To assume that $g^j(z)$ is a uniform distribution over the range of variable $X^j$ for $k$ selected values (all unique or equidistant grid) of the $j$-th explanatory variable.

$$\widehat{vip}_{CP}^{j,uni}(\underline{x}_*) = \frac 1k \sum_{l=1}^k |h^{j}_{x_*}(z_l) - f(\underline{x}_*)|$$

2. To use all observations in the dataset $n$ to estimate the empirical distribution of $X^{j}$, despite it might require more computation time.

$$\widehat{vip}_{CP}^{j,emp}(\underline{x}_*) = \frac 1n \sum_{i=1}^n |h^{j}_{\underline{x}_*}(x^{j}_i) - f(\underline{x}_*)|$$