  The real life time series are usually nonstationary, bringing a difficult
question of model adaptation. Classical approaches like ARMA-ARCH assume
arbitrary type of dependence. To avoid 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 for one or multiple powers
$p\in\mathbb{R}^+$ using $m_{p,t+1} = m_{p,t} + \eta (|x_t-\mu_t|^p-m_{p,t})$.
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. While standard ARMA-ARCH approaches
provide evolution of $\mu$ and $\sigma$, here we also get evolution of $\nu$
describing $\rho(x)\sim |x|^{-\nu-1}$ tail shape, probability of extreme events
- which might turn out catastrophic, destabilizing the market.
