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Computer algebra derivation of the bias of linear estimators of autoregressive models. (English) Zbl 1115.62093
An AR(2) model is considered, \(z_t=\varpi_1z_{t-1}+\varphi_2 z_{t-2}+\varepsilon_t\), with i.i.d. Gaussian zero mean innovations \(\varepsilon_t\). For the Burg estimates \(\hat\varphi_i\) by the observations \(z_j\), \(j=1,\dots,n\), the principal part of the bias is derived, e.g., \[ \lim_{n\to\infty} n E(\hat\varphi_2-\varphi_2)=-(1+3\varphi_2)/n. \] Analogous results are obtained for estimates by observations of \(z_t+m\), where \(m\) is a constant to be estimated. For AR(3) the authors describe the computer derivation (via symbolic computations) of analogous formulas. E.g., for \(\varphi_3\) such formula includes 1,745,350 indivisible subexpressions in terms of roots of the characteristic equation. Numerical results for comparison of bias for the Burg, least squares and Yule-Walker estimates are presented. The authors’ conclusion is that Burg and LS estimates have bias much better than the Yule-Walker ones when the parameters are near the nonstationary boundary. Simulation results are presented.

MSC:
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
65C60 Computational problems in statistics (MSC2010)
62M09 Non-Markovian processes: estimation
68W30 Symbolic computation and algebraic computation
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