Empirical cost distribution: A scenario approach to the construction of probability boxes with application to channel equalization

Based on recent results on randomized min-max convex problems, we present a technique for building probability boxes that envelop the probability distribution of the costs incurred by a sample-based solution. The proposed technique is data-based in that the probability boxes are built based on the same data that are used to obtain the solution. The construction is distribution-free, and no specific assumption is made about the probability distribution of the data. For concreteness, the proposed technique is presented on a channel equalization problem where a finite-impulse response equalizer is designed to minimize distortion. The presence of uncertainty affecting the channel is described by a sample of uncertainty instances, and the equalizer is chosen according to a min-max logic over this sample. A probability box, built according to the proposed technique, characterizes the performance of this equalizer when it is applied to other uncertain channel instances than those seen in the sample.