We propose new methods to construct ambiguity sets consisting of probability density functions bounded by upper and lower bands. Our approaches belong to nonparametric ones. We will post our work soon.
We consider a stochastic program where the distribution is not completely known and only a set of historical data and some information of the shape, e.g., the modes, of the distribution are available. We apply a data-driven approach to construct an ambiguity set consisting of all the probability densitiy functions that could have generated the available data and possess the same shape characteristics. Our decision-making problem then hedges against the worst-case scenario distribution over the ambiguity set. The resulting distributionally robust optimization is computationally intractable due to the fact that it has both infinitely many constraints and infinitely many decision variables. To this end, we propose an effective approximiation method and accordingly provide a theoretical approximation bound which depends on the parameter of the problem. We compare our approach with an existing approach based on a hypoethesis-test ambiguity set over empricial experiments. The numerical results illustrate the strength of our approach over data sets of small sample sizes.