"The estimation of monotone support boundaries is relatively unexplored and still in full development. This research project examines two partial frontier models which provide a valid alternative for purely stochastic ones, that are known to be sensitive to model misspecification, and for completely envelopment models, which often suffer from lack of robustness and precision. The development of mathematical properties under these two recent models is, however, often a lot harder than under the other ones. This project tries to solve this difficulty by attacking many unsolved issues.The added value of this study is three-fold. First, we contribute to the recent literature on robust frontier modeling by tackling the vexing defect of non-monotonicity of large empirical partial frontiers. Moreover, we will regularize both partial frontier estimators by relying on a conditional Generalized Pareto model. Other improvements include the extension to fractional expected-maximum frontiers and to non-positive data.Second, we contribute to the expanding literature on isotonic estimation of a multivariate monotone function by analyzing projection-type versions of its unconstrained estimator. In most studies employing this technique, the difficult question of developing the asymptotic distributional behavior remains unsolved. We will show here that the projected isotonic estimators are free of charge.Finally, we will introduce a new class of specific probability-weighted moments, ‘xpectiles’ called, which parallels the class of quantiles. It will be motivated via several angles, and is expected to afford an appropriate theory that better displays the interesting features of the population distribution.Each of these three research ideas, coming from different areas in statistics is an independent and challenging research project on its own. The methodologies developed and the results found will be important both for a mathematical statistics and an econometrics audience."