Recovery of conditional distribution, quantile, and regression functions from moments

In the framework of probabilistic moment problem, the moment-recovered approximations of conditional distributions, quantiles and regression functions are derived. Some of their asymptotic properties are investigated, including evaluations of uniform and L1-errors of corresponding approximations. In addition, the properties of smoothed versions of approximations based on Bernstein polynomials are presented. The asymptotic behavior of empirical counterparts of constructions applied to several indirect models, like right-censored and mixture models are discussed as well.