Third order asymptotic properties of maximum likelihood estimators for Gaussian ARMA processes

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23 Citations (Scopus)

Abstract

In this paper we investigate various third-order asymptotic properties of maximum likelihood estimators for Gaussian ARMA processes by the third-order Edgeworth expansions of the sampling distributions. We define a third-order asymptotic efficiency by the highest probability concentration around the true value with respect to the third-order Edgeworth expansion. Then we show that the maximum likelihood estimator is not always third-order asymptotically efficient in the class A3 of third-order asymptotically median unbiased estimators. But, if we confine our discussions to an appropriate class D (⊂ A3) of estimators, we can show that appropriately modified maximum likelihood estimator is always third-order asymptotically efficient in D.

Original languageEnglish
Pages (from-to)1-31
Number of pages31
JournalJournal of Multivariate Analysis
Volume18
Issue number1
DOIs
Publication statusPublished - 1986
Externally publishedYes

Fingerprint

ARMA Process
Gaussian Process
Maximum Likelihood Estimator
Maximum likelihood
Asymptotic Properties
Edgeworth Expansion
Sampling
Modified Maximum Likelihood
Asymptotic Efficiency
Sampling Distribution
Unbiased estimator
Maximum likelihood estimator
ARMA process
Asymptotic properties
Edgeworth expansion
Estimator

Keywords

  • Edgeworth expansion
  • Gaussian autoregressive moving average processes
  • maximum likelihood estimator
  • residue theorem
  • spectral density
  • third order asymptotic efficiency
  • Toeplitz matrix

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Numerical Analysis
  • Statistics and Probability

Cite this

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abstract = "In this paper we investigate various third-order asymptotic properties of maximum likelihood estimators for Gaussian ARMA processes by the third-order Edgeworth expansions of the sampling distributions. We define a third-order asymptotic efficiency by the highest probability concentration around the true value with respect to the third-order Edgeworth expansion. Then we show that the maximum likelihood estimator is not always third-order asymptotically efficient in the class A3 of third-order asymptotically median unbiased estimators. But, if we confine our discussions to an appropriate class D (⊂ A3) of estimators, we can show that appropriately modified maximum likelihood estimator is always third-order asymptotically efficient in D.",
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AB - In this paper we investigate various third-order asymptotic properties of maximum likelihood estimators for Gaussian ARMA processes by the third-order Edgeworth expansions of the sampling distributions. We define a third-order asymptotic efficiency by the highest probability concentration around the true value with respect to the third-order Edgeworth expansion. Then we show that the maximum likelihood estimator is not always third-order asymptotically efficient in the class A3 of third-order asymptotically median unbiased estimators. But, if we confine our discussions to an appropriate class D (⊂ A3) of estimators, we can show that appropriately modified maximum likelihood estimator is always third-order asymptotically efficient in D.

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KW - Gaussian autoregressive moving average processes

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KW - residue theorem

KW - spectral density

KW - third order asymptotic efficiency

KW - Toeplitz matrix

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