Algorithm 3.

Sample-split, cross-fitted inference on VIM value ψ_0,s

1:	generate $B_n\in {\{1,\dots ,2K\}}^n$ by sampling uniformly from {1, …, 2K} with replacement, and for j = 1, …, 2K, denote by Dj the set of observations with index in $S_j≔\{i:B_{n,i}=j\}$ and $n_j≔|D_j|$;
2:	for k = 1, …, 2K do
3:	using only data in ${\cup }_{j\ne k}{D}_j$, construct estimators fk,n of f0 and fk,n,s of f0,s;
4:	using only data in Dk, construct estimator Pn,k of P0;
5:	if k is odd, compute ${\eta }_{k,n}^2≔\frac{1}{n_k}{\sum }_{i\in S_k}\dot{V}{(f_{k,n},P_{k,n};{\delta }_{Z_i}-P_{k,n})}^2$ and $v_{k,n}≔V(f_{k,n},P_{k,n})$;
6:	if k is even, compute ${\eta }_{k,n,s}^2≔\frac{1}{n_k}{\sum }_{i\in S_k}\dot{V}{(f_{k,n,s},P_{k,n};{\delta }_{Z_i}-P_{k,n})}^2$ and $v_{k,n,s}≔V(f_{k,n,s},P_{k,n})$;
7:	end for
8:	compute $v_n^{*}≔\frac{1}{K}{\sum }_{k=1}^K{v}_{2k-1,n}$, $v_{n,s}^{*}≔\frac{1}{K}{\sum }_{k=1}^K{v}_{2k,n,s}$ and estimator ${\psi }_{n,s}^{*}≔v_n^{*}-v_{n,s}^{*}$ of ψ0,s;
9:	compute ${\eta }_n^2≔\frac{1}{K}{\sum }_{k=1}^K{\eta }_{2k-1,n}^2$, ${\eta }_{n,s}^2≔\frac{1}{K}{\sum }_{k=1}^K{\eta }_{2k,n,s}^2$ and estimator ${\omega }_{n,s}≔{\eta }_n^2/(n-n_s)+{\eta }_{n,s}^2/n_s$ of the variance of ${\psi }_{n,s}^{*}$;
10:	to test $H_0:{\psi }_{0,s}\in [0,\beta ]$ vs $H_1:{\psi }_{0,s}>\beta$ at level 1−α, reject H0 in favor of H1 iff $p_n≔1-\Phi (t_n)<\alpha$ with $t_n≔{\omega }_{n,s}^{-1/2}({\psi }_{n,s}^{*}-\beta )$ and Φ the standard normal distribution function.