endfor

The first method, explained in Section A1.4.1, consists of directly calibrating a sequence of nested two-sided intervals, as outlined in Section 3.3. The second method, explained in Section A1.4.2, consists of separately calibrating two sequences of lower and upper one-sided confidence intervals, each adopting the significance level α/2 instead of α. Pu j=l ˆϕj(x)amongthefeasible ones with minimal |u l|, whenever the optimization problem does not have a unique solution. Therefore, we can assume without loss of generality that (1) has a unique solution; if that is not the case, we can break the ties at random by adding a little noise to ˆϕ. For any integer T 1, consider an increasing sequence tτ [0,1], for τ {0,...,T}. A nested sequenceofT intervalsindexedbyτ {0,...,T},whichmaybewrittenintheformof St = ˆLm,α(Xm+1;tτ), ˆUm,α(Xm+1;tτ), for appropriate lower and upper endpoints ˆLm,α(Xm+1;tτ) and ˆUm,α(Xm+1;tτ), respectively, is then constructed from (1) as follows.