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Asymptotic Maximum Likelihood

  • Robert Azencott
  • Didier Dacunha-Castelle
Part of the Applied Probability book series (APPLIEDPROB, volume 2)

Abstract

Let X be a stationary, centered, nonzero gaussian process, with spectral density f. Let μn be the law of X1Xn, hn the density of μn on ℝn and L (f, X1Xn) = log hn (X1Xn) the log-likelihood of X. We have seen (Chapter 12, Section 1.2) that
$$- 2{\mathcal{L}_n}\left( {f,{X_1} \ldots {X_n}} \right) = n\log 2\pi + \log \,\det {T_n}\left( {2\pi f} \right) + X(n) * {\left[ {{T_n}\left( {2\pi f} \right)} \right]^{ - 1}}X(n)$$
(1)
where Tn is the Toeplitz matrix and X(n)* = (X1Xn).

Keywords

Spectral Density Maximum Likelihood Estimator Toeplitz Operator Asymptotic Normality Toeplitz Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • Robert Azencott
    • 1
  • Didier Dacunha-Castelle
    • 1
  1. 1.Equipe de Recerche Associée au C.N.R.S. 532 Statistique Appliquée MathématiqueUniversité de Paris-SudOrsay CedexFrance

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