Questioning normality: A study of wavelet leaders distribution

The motivation of this article is to estimate multifractality classification and model selection parameters: the first-order scaling exponent c1 and the second-order scaling exponent (or intermittency coefficient) c2. These exponents are derived from wavelet leaders, which are fundamental tools in applied multifractal analysis. While most estimation methods, particularly Bayesian approaches, assume log-normality, we challenge this hypothesis by statistically testing the normality of log-leaders. Upon rejecting this assumption, we propose a novel and more flexible model based on log-concave distributions. We validate this model on well-known stochastic processes, including fractional Brownian motion, the multifractal random walk, and the canonical Mandelbrot cascade, as well as on real-world marathon runner data. Under the log-normality hypothesis, we revisit the estimation procedure for c1, providing confidence intervals, and for c2, applying it to fractional Brownian motions with various Hurst indices and to the multifractal random walk. Finally, we establish several theoretical results on the distribution of log-leaders in random wavelet series, which align with our numerical findings.