Differential entropy estimation with a Paretian kernel: Tail heaviness and smoothing

Differential entropy extends the concept of entropy to continuous probability distributions, measuring the uncertainty associated with a continuous random variable. In financial data analysis, accurately estimating differential entropy is pivotal for understanding market dynamics and assessing risk. Traditional methods often fall short when dealing with the heavy-tailed distributions characteristic of financial returns. This paper introduces a novel approach to differential entropy estimation employing a Paretian kernel function adept at handling tail heaviness’s intricacies. By incorporating an additional smoothing parameter, the Pareto exponent, our method offers flexibility in adjusting to light and heavy-tailed distributions. We compare our approach against established estimators through a comprehensive Monte Carlo simulation, demonstrating its superior performance in various scenarios. Applying our method to foreign exchange market data further illustrates its practical utility in identifying stochastic regimes and enhancing financial analysis. Our findings advocate for integrating the Paretian kernel estimator into the toolkit of financial analysts and researchers for a more nuanced understanding of market behavior.