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Qualitative stability of stochastic programs with applications in asymptotic statistics. (English) Zbl 1093.62032
Summary: 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.
We will exemplify how stability results can be employed. For this aim existing results are supplemented by assertions which convert the stability results in a form which is more convenient for asymptotic statistics. Furthermore, it is shown, how \(\varepsilon_n\)-optimal solutions can be dealt with. Emphasis is on strong and weak consistency and asymptotic distributions 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.

MSC:
62F12 Asymptotic properties of parametric estimators
90C15 Stochastic programming
62E20 Asymptotic distribution theory in statistics
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