Algorithm 1.

Selection of the tuning parameter 7 using cross-validation

Step 1 Pre-determine a grid of points for $\gamma$ in [0,1], denoted as ${\gamma }^{\left(g\right)},g=1,\cdots ,G$, and set each ${cv}_g=0$.

Step 2 Randomly assign the $K$ strata into $M$ folds, leaving one fold for testing and the others for training. Set q = 1.

Step 2.1 While $q\le M$, use the qth training set to compute the de-biased lasso estimator with ${\gamma }^{\left(g\right)},g=1,\cdots ,G$, denoted as ${\widehat{b}}^{\left(gq\right)}$, and define the active set ${\widehat{A}}^{\left(gq\right)}$.

Step 2.2 Define the thresholded de-biased lasso estimator ${\widehat{b}}_{thres}^{\left(gq\right)}={\widehat{b}}^{\left(gq\right)}\cdot 1(j\in {\widehat{A}}^{\left(gq\right)})$, i.e. setting components of $\widehat{b}(gq)$ outside the active set $\widehat{A}(gq)$ to 0.

Step 2.3 Compute the negative log partial likelihood on the qth testing set ${\ell }^{\left(q\right)}({\widehat{b}}_{thres}^{\left(gq\right)})$.

Step 2.4 Set ${cv}_g\leftarrow {cv}_g+N^{\left(q\right)}{\ell }^{\left(q\right)}({\hat{b}}_{thres}^{\left(gq\right)})$, for $g=1,\cdots ,G$, where $N^{\left(q\right)}$ is the total number of observations in the qth testing set.

Step 2.5 Set $q\leftarrow q+1$ and go to Step 2.1.

Step 3 Let $\widehat{g}=\mathrm{a}\mathrm{r}\mathrm{g}{\mathrm{m}\mathrm{i}\mathrm{n}}_g c{v}_g$. The final output tuning parameter value is ${\gamma }^{\left(\widehat{g}\right)}$.