RÈNYI ENTROPY AND CALIBRATION OF DISTRIBUTION TAILS

The popular principle of Shannon entropy maximization subject to moment constraints yields distributions with light tails. Therefore, it is not suitable for estimation of probability distributions with heavy tails, which arise in various applications, including financial engineering, reliability theory, and climatology. However, with Renyi entropy in place of Shannon entropy, the principle yields distributions with heavy tails. This work obtains a general form of such distributions and proposes a novel method for estimating parameters of generalized Pareto distribution (GPD), which follows from Renyi entropy maximization subject to a constraint on expected value (this particular case was solved by Bercher (2008)). It also shows how a conditional tail GPD can be estimated based on quantile and CVaR (expected shortfall) regressions and, as an illustration, estimates a conditional tail GPD for Fidelity Magellan Fund return as a functions of stock indices. Quantile and CVaR regressions arc implemented with the Portfolio Safeguard (PSG) optimization package, which has precoded errors for quantile and CVaR regressions.