Adaptive stable distribution and Hurst exponent by method of moments moving estimator for nonstationary time series
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| Format: | Preprint |
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2025
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| author | Duda, Jarek |
| author_facet | Duda, Jarek |
| contents | 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 $F_t=\sum_{τ<t} (1-η)^{t-τ} \ln(ρ_θ(x_τ))$ 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 $m_p=E[|x-μ|^p]$ absolute central moments, updated by $m_{p,t+1} = m_{p,t} + η(|x_t-μ_t|^p-m_{p,t})$. 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 $ρ(x) \sim 1/|x|^{α+1}$ behavior, controlling probability of potentially dangerous extreme events. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_05354 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Adaptive stable distribution and Hurst exponent by method of moments moving estimator for nonstationary time series Duda, Jarek Methodology Machine Learning Econometrics 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 $F_t=\sum_{τ<t} (1-η)^{t-τ} \ln(ρ_θ(x_τ))$ 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 $m_p=E[|x-μ|^p]$ absolute central moments, updated by $m_{p,t+1} = m_{p,t} + η(|x_t-μ_t|^p-m_{p,t})$. 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 $ρ(x) \sim 1/|x|^{α+1}$ behavior, controlling probability of potentially dangerous extreme events. |
| title | Adaptive stable distribution and Hurst exponent by method of moments moving estimator for nonstationary time series |
| topic | Methodology Machine Learning Econometrics |
| url | https://arxiv.org/abs/2506.05354 |