In this study, a variety of statistical models of regression type is considered. In some models observations are to be added by the respective design points if the latter are observable. Our models contain discontinuities of the response functions. In the case of a one-dimensional factor - an input variable - these discontinuities are presented by some change points. In the multivariate factor models, the discontinuities become functions. In either case, the principal objective of the statistical estimation is this discontinuity, a point or a function. Another feature of our models is that they may include a nuisance parameter. Typically, the dimension of the nuisance parameter grows proportionally to the number of observations. The nuisance parameter can be deterministic or stochastic. In the latter case, it is governed by a stochastic process. The statistical analysis in our study is asymptotical as the sample size increases to infinity.