  Every observation may follow a distribution that is randomly selected in a
class of distributions. It is called the distribution uncertainty. This is a
fact acknowledged in some research fields such as financial risk measure. Thus,
the classical expectation is not identifiable in general.In this paper, a
distribution uncertainty is defined, and then an upper expectation regression
is proposed, which can describe the relationship between extreme events and
relevant covariates under the framework of distribution uncertainty. As there
are no classical methods available to estimate the parameters in the upper
expectation regression, a two-step penalized maximum least squares procedure is
proposed to estimate the mean function and the upper expectation of the error.
The resulting estimators are consistent and asymptotically normal in a certain
sense.Simulation studies and a real data example are conducted to show that the
classical least squares estimation does not work and the penalized maximum
least squares performs well.
