Estimated stochastic programs with chance constraints

We will consider a non-parametric estimation procedure for chance-constrained stochastic programs where the random parameters appear on the right-hand side of linear constraints for the decision variable. The assumed independence of the components of the random right-hand side data results in stochastic programs with a separability structure in the constraints. We estimate the unknown probability distribution of the random right-hand side data via isotonic regression estimates of increasing hazard rates. Our choice of the estimates was motivated by the relationship between logarithmic concave measures and increasing hazard rate distributions. We establish large deviation results for optimal values and optimal solutions sets of the estimated programs. Finally, we discuss the numerical treatment of the estimated chance-constrained programs and report on a test run.