Adaptive exponential power distribution with moving estimator for nonstationary time series
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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=max_θ l_T for l_T=∑_t<Tη^T-tln(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: exponential power distribution (EPD) ρ(x)∝(-|(x-μ)/σ|^κ/κ) 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, surprisingly 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.
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