The real life time series are usually nonstationary, bringing a difficult question of model adaptation. Classical approaches like GARCH assume arbitrary type of dependence. To prevent such bias, we will focus on recently proposed agnostic philosophy of moving estimator: in time $t$ finding parameters optimizing e.g. $F_t=\sum_{\tau<t} (1-\eta)^{t-\tau} \ln(\rho_\theta (x_\tau))$ moving log-likelihood, evolving in time. It allows for example to estimate parameters using inexpensive exponential moving averages (EMA), like absolute central moments $E[|x-\mu|^p]$ evolving with $m_{p,t+1} = m_{p,t} + \eta (|x_t-\mu_t|^p-m_{p,t})$ for one or multiple powers $p\in\mathbb{R}^+$. Application of such general adaptive methods of moments will be presented on Student's t-distribution, popular especially in economical applications, here applied to log-returns of DJIA companies.

翻译：现实中的时间序列通常具有非平稳性，这带来了模型自适应的棘手问题。经典方法如GARCH假设依赖关系具有任意类型。为避免此类偏差，我们聚焦于近期提出的不可知论移动估计方法：在时间$t$寻找参数以优化例如$F_t=\sum_{\tau<t} (1-\eta)^{t-\tau} \ln(\rho_\theta (x_\tau))$的移动对数似然函数，该函数随时间演化。例如，该方法允许使用廉价的指数移动平均（EMA）来估计参数，如绝对中心矩$E[|x-\mu|^p]$通过$m_{p,t+1} = m_{p,t} + \eta (|x_t-\mu_t|^p-m_{p,t})$演化，其中幂指数$p\in\mathbb{R}^+$可为一个或多个。我们将以学生t分布为例展示此类通用自适应矩方法的应用，该分布在经济学应用中尤为常见，本文将其应用于道琼斯工业平均指数成分股的对数收益率。