Data-based Quantitative Analysis under Nonlinear Expectations

Abstract: 

Traditionally, a real-life random sample is often treated as measurements resulting from an i.i.d. sequence of random variables or, more generally, as an outcome of either linear or nonlinear regression models driven by an i.i.d. sequence. In many situations, however, this standard modeling approach fails to address the complexity of real-life random data. We argue that it is necessary to take into account the uncertainty hidden inside random sequences that are observed in practice.

To deal with this issue, we introduce a robust nonlinear expectation to quantitatively measure and calculate this type of uncertainty. The corresponding fundamental concept of a `nonlinear i.i.d. sequence’ is used to model a large variety of real-world random phenomena. We give a robust and simple algorithm, called `phi-max-mean,’ which can be used to measure such type of uncertainties, and we show that it provides an asymptotically optimal unbiased estimator to the corresponding nonlinear distribution.