Generalized divergences for statistical evaluation of uncertainty in long-memory processes

Environmental variables such as streamflow discharge and water quality indices vary stochastically over time and often exhibit long (subexponential) memory. Their dynamics are described by stochastic differential equations that are superposed with respect to a probability measure of the reversion speeds, called the reversion measure. The reversion measure governs their statistics; however, as its accurate estimation is not always successful owing to data limitations, possible misspecifications should be assumed. In particular, bounding their statistics from below and above should be performed robustly within a unified mathematical framework, which still remains challenging. We propose bounding the statistics of long-memory processes by a pair of risk measures based on the Kaniadakis or α divergence, which are generalizations of the Kullback–Leibler divergence. We obtain a series of mathematical results for the upper- and lower-bounding divergence risk measures of the reversion measure based on these divergences in proper Orlicz spaces, along with their continuous embedding relationships. These results clarify the advantages and disadvantages of the two divergences in applications. A multifractional limit of the long-memory process under misspecification is also considered to investigate behavior of the scaling function and its correspondence with an anomalous diffusion. We also show that the recently proposed generalized Lorenz curve allows for a deeper comprehension of the bounds. Finally, we apply our mathematical framework to hourly streamflow discharge data with a record length of about 20 years.