A note on estimates in stochastic programming

Abstract

In this paper we shall deal with statistical estimates in stochastic programming problems. The estimate problem is a very important one. Namely, from a mathematical point of view many practical optimization problems with a random element lead to deterministic optimization problems depending on the random element through probability measure only. Moreover, this probability measure is hardly ever known in real-life situations. We consider the case when the theoretical distribution function is completely unknown. Then the empirical distribution function usually replaces the theoretical one in order to obtain some estimates of the optimal value and the optimal solution, at least. These estimates will just be the subject of our investigation. In detail, we shall mostly deal with the case of dependent samples. Evidently, the dependent samples appear more often in real-life situations than the independent ones. First, we shall briefly mention the consistence results. Further, we shall study the convergence rate with respect to various types of random samples dependence. We compare these results with those achieved earlier, including some results on asymptotic distribution. We shall recognize that the convergence rate is “the best one” for the independent case, of course. However, we shall introduce the dependent types with the same convergence rate.