Simplified Estimators Possessing “Nice” Asymptotic Properties
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The examples considered in Sections 4 and 5 of the preceding Chapter indicate that the asymptotic m.l. estimators \(\mathop \theta \limits^ \sim \) of the parameters θ appearing in the expression for spectral density fθ of a Gaussian random process Xt, t = …,-1,0,1, … while they are simpler than the exact m.l. estimators \(\mathop \theta \limits^\_ \), they are nevertheless most often roots of rather complex nonlinear equations so that their determination also requires a substantial amount of time and effort. Only the problem of estimating the parameters ⌊1, …, ⌊q and σ2 in the autoregressive process with spectral density (II.4.3) was an exception. In Subsection 4.1 of the preceding Chapter it was shown that for this problem the asymptotic m.l. estimators \(\mathop L\limits^ \sim \)1, …, \(\mathop L\limits^ \sim \)q are roots of a simple system of linear equations (II.4.6) with respect to the variables ⌊1 …, ⌊q and that the estimator \(\mathop \sigma \limits^ \sim \)2 of the parameter σ2 is given by a relatively simple formula (II.4.7).
KeywordsSpectral Density Unknown Parameter Random Vector Asymptotic Property Consistent Estimator
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