Figure 1:

HAL in the univariate setting. Benkeser and van der Laan [2016] conducted a simulation, drawing 500 independent copies of $X$ from a $\mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(-4,4)$ distribution and 500 copies of $ϵ$ from a $\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}\left(\mathrm{0,1}\right)$ distribution where $Y=2sin(\pi /2|X\left|\right)+ϵ$ so that ${‖{\psi }_0‖}_v$. The HAL basis functions included $n$ indicators at observed data points ${\theta }_j\left(x\right)=I\left(x\ge {\tilde{x}}_j\right)$ for $j=1,\dots ,n$ and selected the bound on the variation norm via ten-fold cross-validation from 100 possible bounds from 0 to 350.