Modified least squares processes (MLSP’s) and self-randomized MLSP’s are introduced in D[0, 1] for the slope in linear structural and functional error-in-variables models (EIVM’s). Sup-norm approximations in probability
and, as a consequence, functional central limit theorems (CLT’s) are established for the data-based self-normalized versions
of these MLSP’s and self-randomized MLSP’s. The MLSP’s are believed to be new types of objects of study, and the invariance
principles for them constitute new asymptotics, in EIVM’s. Moreover, the obtained data-based functional CLT’s for the MLSP’s
open up new possibilities for constructing various asymptotic confidence intervals (CI’s) for the slope that are named functional
asymptotic CI’s here. Three special examples of such CI’s are given.