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.

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