Algorithm 2:

Level α joint confidence bands of ${\widehat{\beta }}_r(s)$

Data: βr,Var(βr), βr(1),…, βr(B), Ns.

Result: Joint confidence bands of $\left\{{\widehat{\beta }}_r(s),s\in \mathcal{S}\right\}$.

1. Perform Functional Principal Component Analysis (FPCA) on [βr(1),…,βr(B)]T. Derive the mean function μ = [μ(s1),…,μ(sL)]T, eigenvalues λ1,…,λL and eigenfunctions ψ1,…,ψL, where ψk = [ψk(s1),…,ψk(sL)]T, k = 1,…,L;

for n = 1,…, Ns do

2. Simulate ${\xi }_{nk}~\mathcal{N}\left(0,{\lambda }_k\right)$ for k = 1,…,KT. Calculate ${\beta }_{r,n}=\mu +\sum _{k=1}^{K_T}{\xi }_{nk}{\psi }_k$;

3. Calculate $u_n={\text{max}}_{s_l\in \mathcal{S}}\{|{\beta }_{r,n}-{\beta }_r|/\sqrt{\text{diag}\left(\text{Var}\left({\beta }_r\right)\right)}\}$;

end

4. Obtain q1−α, the (1−α) empirical quantile of $\left\{u_1,\dots ,u_{N_s}\right\}$;

5. The joint confidence interval at $s_l\in \mathcal{S}$ is calculated as ${\widehat{\beta }}_r\left(s_l\right)\pm q_{1-\alpha }\sqrt{\text{Var}{\left({\beta }_r\right)}_{(l,l)}}$. The upper and lower bounds of the joint confidence bands can be smoothed.