An Efficient Algorithm for Calculating the Likelihood and Likelihood Gradient of ARMA Models

David Burshtein

doi:10.1109/9.250487

An Efficient Algorithm for Calculating the Likelihood and Likelihood Gradient of ARMA Models

David Burshtein^*

^*Corresponding author for this work

School of Electrical Engineering

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

1 Scopus citations

Abstract

We obtain exact analytical expressions for the likelihood and likelihood gradient of stationary autoregressive moving average (ARMA) models. Let us denote the sample size by N, the autoregressive order by p, and the moving average order by q. The calculation of the likelihood requires (p + 2q + 1)N + o(N) multiply-add operations, and the calculation of the likelihood gradient requires (2p + 6q + 2)N + o(N) multiply-add operations. These expressions may be used to obtain an iterative, Newton-Raphson-type converging algorithm, with superlinear convergence rate, that computes the maximum-likelihood estimator in (2p + 6q + 2)N + o(N) multiply-add operations per iteration.

Original language	English
Pages (from-to)	336-340
Number of pages	5
Journal	IEEE Transactions on Automatic Control
Volume	38
Issue number	2
DOIs	https://doi.org/10.1109/9.250487
State	Published - Feb 1993

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title = "An Efficient Algorithm for Calculating the Likelihood and Likelihood Gradient of ARMA Models",

abstract = "We obtain exact analytical expressions for the likelihood and likelihood gradient of stationary autoregressive moving average (ARMA) models. Let us denote the sample size by N, the autoregressive order by p, and the moving average order by q. The calculation of the likelihood requires (p + 2q + 1)N + o(N) multiply-add operations, and the calculation of the likelihood gradient requires (2p + 6q + 2)N + o(N) multiply-add operations. These expressions may be used to obtain an iterative, Newton-Raphson-type converging algorithm, with superlinear convergence rate, that computes the maximum-likelihood estimator in (2p + 6q + 2)N + o(N) multiply-add operations per iteration.",

author = "David Burshtein",

note = "Funding Information: A thorough understanding of the characteristics of the single-part unreliable manufacturing system flow-control problem is important because it is an essential building block in the search Manuscript received July 5, 1990; revised June 16, 1991 and October 20, 1991. Paper recommended by A. Schwartz. This work was supported by the NSERC-Canada under Grant OGP0006820. The author is with Ecole Polytechnique de Montreal, Montreal, P.Q., H3C 3A7, Canada. IEEE Log Number 9203111.",

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N1 - Funding Information: A thorough understanding of the characteristics of the single-part unreliable manufacturing system flow-control problem is important because it is an essential building block in the search Manuscript received July 5, 1990; revised June 16, 1991 and October 20, 1991. Paper recommended by A. Schwartz. This work was supported by the NSERC-Canada under Grant OGP0006820. The author is with Ecole Polytechnique de Montreal, Montreal, P.Q., H3C 3A7, Canada. IEEE Log Number 9203111.

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N2 - We obtain exact analytical expressions for the likelihood and likelihood gradient of stationary autoregressive moving average (ARMA) models. Let us denote the sample size by N, the autoregressive order by p, and the moving average order by q. The calculation of the likelihood requires (p + 2q + 1)N + o(N) multiply-add operations, and the calculation of the likelihood gradient requires (2p + 6q + 2)N + o(N) multiply-add operations. These expressions may be used to obtain an iterative, Newton-Raphson-type converging algorithm, with superlinear convergence rate, that computes the maximum-likelihood estimator in (2p + 6q + 2)N + o(N) multiply-add operations per iteration.

AB - We obtain exact analytical expressions for the likelihood and likelihood gradient of stationary autoregressive moving average (ARMA) models. Let us denote the sample size by N, the autoregressive order by p, and the moving average order by q. The calculation of the likelihood requires (p + 2q + 1)N + o(N) multiply-add operations, and the calculation of the likelihood gradient requires (2p + 6q + 2)N + o(N) multiply-add operations. These expressions may be used to obtain an iterative, Newton-Raphson-type converging algorithm, with superlinear convergence rate, that computes the maximum-likelihood estimator in (2p + 6q + 2)N + o(N) multiply-add operations per iteration.

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An Efficient Algorithm for Calculating the Likelihood and Likelihood Gradient of ARMA Models

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