Algorithm 2. Asymptotic confidence interval

$$\begin{array}{cc}\hfill & \mathrm{Input}:\text{matrices:}X,Z\text{and}Q,\text{estimate:}\widehat{\mathcal{B}}=\left[\begin{array}{c}\hfill {\widehat{\beta }}^{\text{gP}}\hfill \\ \hfill {\widehat{b}}^{\text{gP}}\hfill \end{array}\right],\text{optimal parameters:}{\lambda }_Q\text{and}{\lambda }_R,\text{function:}\psi \hfill \\ \hfill & 1.\text{Define:}\mathcal{X}≔[X,Z],\theta ≔\mathcal{X}\widehat{\mathcal{B}}\text{and}\Psi ≔\text{diag}\left\{{\psi }^{″}({\theta }_1),\dots ,{\psi }^{″}({\theta }_n)\right\}\hfill \\ \hfill & 2.\text{Construct}m+p\text{by}m+p\text{matrix:}\mathcal{Q}≔\phantom{[}[\frac{0_{m\times m}}{0_{p\times m}}\mid \frac{0_{m\times p}}{{\lambda }_{Q}Q+{\lambda }_R{\mathrm{I}}_p}]\hfill \\ \hfill & 3.\text{Calculate the variance of}\widehat{\mathcal{B}}:{\text{var}}_a\widehat{\mathcal{B}}≔{\left({\mathcal{X}}^{\mathsf{T}}\Psi \mathcal{X}+\mathcal{Q}\right)}^{-1}{\mathcal{X}}^{\mathsf{T}}\Psi \mathcal{X}{\left({\mathcal{X}}^{\mathsf{T}}\Psi \mathcal{X}+\mathcal{Q}\right)}^{-1}\hfill \\ \hfill & 4.\text{Define the asymptotic confidence interval (CI) for}{\mathcal{B}}_i\text{as}\hfill \\ \hfill & C{I}_{asmp}({\mathcal{B}}_i)≔\left({\widehat{\mathcal{B}}}_i-1.96\cdot \sqrt{({\text{var}}_a\widehat{\mathcal{B}}{)}_{ii}},{\widehat{\mathcal{B}}}_i+1.96\cdot \sqrt{({\text{var}}_a\widehat{\mathcal{B}}{)}_{ii}}\right)\hfill \end{array}$$