*Bounty: 50*

*Bounty: 50*

While in standard (“static”) e.g. ML estimation we assume that all values are from a distribution of the same parameters, in practice we often have nonstationary time series: in which these parameters can evolve in time.

It is usually considered by using sophisticated models like GARCH conditioning sigma with recent errors and sigmas, or Kalman filters – they assume some arbitrary hidden mechanism.

I have recently worked on a **simpler and more agnostic way: use moving estimator**, like loglikelihood with exponentially weakening weights of recent values:

$$theta_T=argmax_theta l_Tqquad textrm{for}qquad l_T= sum_{t<T}eta^{T-t} ln(rho_theta(x_t)) $$

intended to estimate local parameters, separately on each position. We **don’t assume any hidden mechanism, only shift the estimator.**

For example it turns out that EPD (exponential power distribution) family $rho(x) propto exp(-|x|^kappa)$, which covers Gaussian ($kappa=2$) and Laplace ($kappa=1$) distributions, can have cheaply made such moving estimator (plots below), getting much better loglikelihood for daily log-returns of Dow Jones companies (100 years DJIA, 10 years individual), even exceeding GARCH: https://arxiv.org/pdf/2003.02149 – just using the $(sigma_{T+1})^kappa=eta (sigma_{T})^kappa+(1-eta)|x-mu|^kappa$ formula: **replacing estimator as average with moving estimator as exponential moving average**:

I have also MSE moving estimator for adaptive least squares linear regression: page 4 of https://arxiv.org/pdf/1906.03238 – can be used to get adaptive AR without Kalman filter, also analogous approach for adaptive estimation of joint distribution with polynomials: https://arxiv.org/pdf/1807.04119

**Are such moving estimators considered in literature?**

**What applications they might be useful for?**