提出动态估计非平稳时间序列的稳定分布参数与赫斯特指数的新方法。
Adaptive stable distribution and Hurst exponent by method of moments moving estimator for nonstationary time series
- 基于指数加权移动平均，对每个时刻独立估计参数，避免传统模型偏差。
- 在道琼斯指数数据上成功追踪α参数变化，揭示市场极端风险演化趋势。
- 适合金融风控、高频交易等需实时评估市场稳定性场景。
真实时间序列的非平稳性要求模型自适应调整。传统方法如ARMA-ARCH假设某种固定依赖结构，易引入偏差。本文采用新颖的移动估计方法：对每个时刻t，通过加权局部似然函数 $F_t = \sum_{τ<t} (1-η)^{t-τ} \ln(ρ_θ(x_τ))$ 估计参数，旧数据权重呈指数衰减。实践中可通过指数移动平均（EMA）实现，例如绝对中心矩 $m_p = E[|x-μ|^p]$ 更新公式为 $m_{p,t+1} = m_{p,t} + η(|x_t-μ_t|^p - m_{p,t})$。本文聚焦于α稳定分布的应用，其参数α影响赫斯特指数，可用于自适应估计。在道琼斯工业指数（DJIA）数据上，不仅追踪了均值μ和尺度参数σ的变化，还连续估计了α参数，反映尾部行为 $ρ(x) \sim 1/|x|^{α+1}$，从而评估极端事件发生的概率，为市场稳定性提供动态监测能力。
原文摘要 · Abstract (English)
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.
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