Adaptive exponential power distribution with moving estimator for nonstationary time series

• URL: http://arxiv.org/abs/2003.02149v2
• Date: Mon, 23 Mar 2020 16:26:16 GMT
• Title: Adaptive exponential power distribution with moving estimator for nonstationary time series
• Authors: Jarek Duda
• Abstract summary: We will focus on maximum likelihood (ML) adaptive estimation for nonstationary time series. We focus on such example: $rho(x)propto exp(-|(x-mu)/sigma|kappa/kappa)$ exponential power distribution (EPD) family. It is tested on daily log-return series for DJIA companies, leading to essentially better log-likelihoods than standard (static) estimation.
• Score: 0.8702432681310399
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: While standard estimation assumes that all datapoints are from probability distribution of the same fixed parameters $\theta$, we will focus on maximum likelihood (ML) adaptive estimation for nonstationary time series: separately estimating parameters $\theta_T$ for each time $T$ based on the earlier values $(x_t)_{t<T}$ using (exponential) moving ML estimator $\theta_T=\arg\max_\theta l_T$ for $l_T=\sum_{t<T} \eta^{T-t} \ln(\rho_\theta (x_t))$ and some $\eta\in(0,1]$. 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: $\rho(x)\propto \exp(-|(x-\mu)/\sigma|^\kappa/\kappa)$ exponential power distribution (EPD) family, which covers wide range of tail behavior like Gaussian ($\kappa=2$) or Laplace ($\kappa=1$) distribution. It is also convenient for such adaptive estimation of scale parameter $\sigma$ as its standard ML estimation is $\sigma^\kappa$ being average $\|x-\mu\|^\kappa$. By just replacing average with exponential moving average: $(\sigma_{T+1})^\kappa=\eta(\sigma_T)^\kappa +(1-\eta)|x_T-\mu|^\kappa$ 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 $\kappa$ 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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