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

Masanobu Taniguchi

doi:10.1016/0047-259X(86)90055-2

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

Masanobu Taniguchi^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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 language	English
Pages (from-to)	1-31
Number of pages	31
Journal	Journal of Multivariate Analysis
Volume	18
Issue number	1
DOIs	https://doi.org/10.1016/0047-259X(86)90055-2
Publication status	Published - 1986 Feb
Externally published	Yes

Keywords

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

ASJC Scopus subject areas

Statistics and Probability
Numerical Analysis
Statistics, Probability and Uncertainty

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10.1016/0047-259X(86)90055-2

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title = "Third order asymptotic properties of maximum likelihood estimators for Gaussian ARMA processes",

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.",

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

author = "Masanobu Taniguchi",

note = "Funding Information: Received March 15, 1984; revised February 20, 1985. AMS 1980 subject classifications. Primary 62Fl2, 62Ml5: secondary 62Ml0, 62EZO. Key words and phrases: Gaussian autoregressive moving average processes, spectral density, Toeplitz matrix, maximum likelihood estimator, third order asymptotic efficiency, Edgeworth expansion, residue theorem *This work was partially supported by Japan Society of Promotion of Science, United States-Japan Cooperative Science Program and Grant-in-Aid of the Ministry of Japan. Copyright: Copyright 2014 Elsevier B.V., All rights reserved.",

year = "1986",

month = feb,

doi = "10.1016/0047-259X(86)90055-2",

language = "English",

volume = "18",

pages = "1--31",

journal = "Journal of Multivariate Analysis",

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AU - Taniguchi, Masanobu

N1 - Funding Information: Received March 15, 1984; revised February 20, 1985. AMS 1980 subject classifications. Primary 62Fl2, 62Ml5: secondary 62Ml0, 62EZO. Key words and phrases: Gaussian autoregressive moving average processes, spectral density, Toeplitz matrix, maximum likelihood estimator, third order asymptotic efficiency, Edgeworth expansion, residue theorem *This work was partially supported by Japan Society of Promotion of Science, United States-Japan Cooperative Science Program and Grant-in-Aid of the Ministry of Japan. Copyright: Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1986/2

Y1 - 1986/2

N2 - 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.

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.

KW - Edgeworth expansion

KW - Gaussian autoregressive moving average processes

KW - Toeplitz matrix

KW - maximum likelihood estimator

KW - residue theorem

KW - spectral density

KW - third order asymptotic efficiency

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SN - 0047-259X

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EP - 31

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

IS - 1

ER -

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

Abstract

Keywords

ASJC Scopus subject areas

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