Statistical evaluation of a long-memory process using the generalized entropic Value-at-Risk

TL;DR

This paper introduces a novel statistical approach using Tsallis Value-at-Risk to evaluate long-memory processes like streamflow discharge, overcoming limitations of traditional entropic measures and providing robust bounds for key statistics.

Contribution

It generalizes the entropic Value-at-Risk to Tsallis VaR, enabling effective evaluation of long-memory time series with limited data and polynomial moments.

Findings

TsVaR avoids exponential integrand blowup issues

Application of semi-implicit gradient descent for TsVaR calculation

Successful evaluation of streamflow data in mountainous rivers

Abstract

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.