Adaptive exponential power distribution with moving estimator for nonstationary time series

While standard estimation assumes that all datapoints are from probability distribution of the same fixed parameters $θ$, we will focus on maximum likelihood (ML) adaptive estimation for nonstationary time series: separately estimating parameters $θ_T$ for each time $T$ based on the earlier values $(x_t)_{t<T}$ using (exponential) moving ML estimator $θ_T=\arg\max_θl_T$ for $l_T=\sum_{t<T} η^{T-t} \ln(ρ_θ(x_t))$ and some $η\in(0,1]$. Computational cost of such moving estimator is generally much higher as we need to optimize log-likelihood multiple times, however, in many cases it can be made inexpensive thanks to dependencies. We focus on such example: $ρ(x)\propto \exp(-|(x-μ)/σ|^κ/κ)$ exponential power distribution (EPD) family, which covers wide range of tail behavior like Gaussian ($κ=2$) or Laplace ($κ=1$) distribution. It is also convenient for such adaptive estimation of scale parameter $σ$ as its standard ML estimation is $σ^κ$ being average $\|x-μ\|^κ$. By just replacing average with exponential moving average: $(σ_{T+1})^κ=η(σ_T)^κ+(1-η)|x_T-μ|^κ$ we can inexpensively make it adaptive. It is tested on daily log-return series for DJIA companies, leading to essentially better log-likelihoods than standard (static) estimation, with optimal $κ$ tails types varying between companies. Presented general alternative estimation philosophy provides tools which might be useful for building better models for analysis of nonstationary time-series.