Qualitative stability of stochastic programs with applications in asymptotic statistics

Abstract

The paper aims at drawing attention to the potential that qualitative stability theory of stochastic programming bears for the study of asymptotic properties of statistical estimators. Stability theory of stochastic programming yields a unifying approach to convergence (almost surely, in probability, and in distribution) of solutions to random optimization problems and can hence be applied to many estimation problems. Non-unique solutions of the underlying optimization problems, constraints for the solutions, and discontinuous objective functions can be dealt with. Making use of stability results, it is often possible to extend existing consistency statements and assertions on the asymptotic distribution, especially to non-standard cases. In this paper we will exemplify how stability results can be employed. For this aim existing results are supplemented by assertions which convert the stabilit results in a form which is more convenient for asymptotic statistics. Furthermore, it is shown, how εn-optimal solutions can be dealt with. Emphasis is on strong and weak consistency and asymptotic distribution in the case of non-unique solutions. M-estimation, constrained M-estimation for location and scatter, quantile estimation and the behavior of the argmin functional for càdlàg processes are considered.