Joint Colloquium: Shige Peng -- Data-based Quantitative Analysis under Nonlinear Expectations

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.