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

Covariance 

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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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