Algorithm 2.

Cross-fitted inference on VIM value ψ_0,s (valid in non-null settings)

1:	generate $B_n\in {\{1,\dots ,K\}}^n$ by sampling uniformly from {1, …, K} with replacement, and for j = 1, …, K, denote by Dj the subset of observations with index in $S_j≔\{i:B_{n,i}=j\}$;
2:	for k = 1, …, K do
3:	using only data in ∪j≠kDj, construct estimators fk,n of f0 and fk,n,s of f0,s;
4:	using only data in Dk construct empirical distribution estimator Pk,n of P0;
5:	with $n_k≔{\sum }_{i=1}^{n}I\{i\in S_k\}$, compute ${\psi }_{k,n,s}:=V(f_{k,n},P_{k,n})-V(f_{k,n,s},P_{k,n})$ and
	${\tau }_{k,n,s}^2≔\frac{1}{n_k}\sum _{i\in S_k}{\{\dot{V}(f_{k,n},P_{k,n};{\delta }_{Z_i}-P_{k,n})-\dot{V}(f_{k,n,s},P_{k,n};{\delta }_{Z_i}-P_{k,n})\}}^2;$
6:	end for
7:	compute estimator ${\psi }_{n,s}^{\ast }≔\frac{1}{K}{\sum }_{k=1}^K{\psi }_{k,n,s}$ of ψ0,s;
8:	compute estimator ${\tau }_{n,s,\ast }^2:=\frac{1}{K}{\sum }_{k=1}^K{\tau }_{k,n,s}^2$ of the asymptotic variance ${\tau }_{0,s}^2$ of $n^{1/2}({\psi }_{n,s}^{*}-{\psi }_{0,s})$.