# Adaptive stable distribution and Hurst exponent by method of moments moving estimator for nonstationary time series

arXiv: https://arxiv.org/abs/2506.05354 · HTML: https://mlsanity.com/paper/2506.05354/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 10 references that are in the map.
- First posted on arXiv 20 May 2025.

## Builds on (most cited first, 4 of 4)

- [Exploiting statistical dependencies of time series with hierarchical correlation reconstruction](https://mlsanity.com/paper/1807.04119/) (2018)
- [Adaptive exponential power distribution with moving estimator for nonstationary time series](https://mlsanity.com/paper/2003.02149/) (2020)
- [Parametric context adaptive Laplace distribution for multimedia compression](https://mlsanity.com/paper/1906.03238/) (2019)
- [Adaptive Student's t-distribution with method of moments moving estimator for nonstationary time series](https://mlsanity.com/paper/2304.03069/) (2023)

## Abstract (arXiv)

Nonstationarity of real-life time series requires model adaptation. In classical approaches like ARMA-ARCH there is assumed some arbitrarily chosen dependence type. To avoid their bias, we will focus on novel more agnostic approach: moving estimator, which estimates parameters separately for every time t: optimizing Ft=sumτ<t (1-η)t-τ ln(ρθ (xτ)) local log-likelihood with exponentially weakening weights of the old values. In practice such moving estimates can be found by EMA (exponential moving average) of some parameters, like mp=E[|x-μ|p] absolute central moments, updated by mp,t+1 = mp,t + η (|xt-μt|p-mp,t). We will focus here on its applications for alpha-Stable distribution, which also influences Hurst exponent, hence can be used for its adaptive estimation. Its application will be shown on financial data as DJIA time series - beside standard estimation of evolution of center μ and scale parameter σ, there is also estimated evolution of α parameter allowing to continuously evaluate market stability - tails having ρ(x) ~ 1/|x|α+1 behavior, controlling probability of potentially dangerous extreme events.