  The modeling and identification of time series data with a long memory are
important in various fields. The streamflow discharge is one such example that
can be reasonably described as an aggregated stochastic process of randomized
affine processes where the probability measure, we call it reversion measure,
for the randomization is not directly observable. Accurate identification of
the reversion measure is critical because of its omnipresence in the aggregated
stochastic process. However, the modeling accuracy is commonly limited by the
available real-world data. One approach to this issue is to evaluate the upper
and lower bounds of a statistic of interest subject to ambiguity of the
reversion measure. Here, we use the Tsallis Value-at-Risk (TsVaR) as a convex
risk measure to generalize the widely used entropic Value-at-Risk (EVaR) as a
sharp statistical indicator. We demonstrate that the EVaR cannot be used for
evaluating key statistics, such as mean and variance, of the streamflow
discharge due to the blowup of some exponential integrand. In contrast, the
TsVaR avoids this issue because it requires only the existence of some
polynomial, not exponential moment. As a demonstration, we apply the
semi-implicit gradient descent method to calculate the TsVaR and corresponding
Radon-Nikodym derivative for time series data of actual streamflow discharges
in mountainous river environments.
