Abstract
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=argmax_θ l_T for l_T=Σ_t<T η^T-t ln(ρ_θ (x_t)) and some η∈(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)∝ 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.