Third order asymptotic properties of maximum likelihood estimators for Gaussian ARMA processes. (English) Zbl 0595.62091
The k-th order asymptotic efficiency is defined by the highest probability concentration around the true value by the k-th order Edgeworth expansion. The evaluation of various third-order asymptotic properties of MLE’s for Gaussian ARMA processes is based on asymptotic moments of some statistics corresponding to the first three derivatives of the likelihood. The author proves that even in smooth cases the MLE is not always asymptotically efficient in the class \(A_ 3\) of third-order asymptotically median unbiased estimators. However, an appropriately modified MLE is always third-order asymptotically efficient in a subclass \(D\subset A_ 3\).
Reviewer: J.Anděl
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62F12 | Asymptotic properties of parametric estimators |
| 62M15 | Inference from stochastic processes and spectral analysis |
| 62E20 | Asymptotic distribution theory in statistics |
Keywords:
Gaussian autoregressive moving average processes; spectral density; Toeplitz matrix; maximum likelihood estimator; residue theorem; asymptotic efficiency; highest probability concentration; Edgeworth expansion; Gaussian ARMA processes; asymptotically median unbiased estimatorsReferences:
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