### Abstract

For a class of Gaussian stationary processes, the spectral density f_{θ} (λ), θ = (τ^{′}, η^{′})^{′}, is assumed to be a piecewise continuous function, where τ describes the discontinuity points, and the piecewise spectral forms are smoothly parameterized by η. Although estimating the parameter θ is a very fundamental problem, there has been no systematic asymptotic estimation theory for this problem. This paper develops the systematic asymptotic estimation theory for piecewise continuous spectra based on the likelihood ratio for contiguous parameters. It is shown that the log-likelihood ratio is not locally asymptotic normal (LAN). Two estimators for θ, i.e., the maximum likelihood estimator over(θ, ̂)_{ML} and the Bayes estimator over(θ, ̂)_{B}, are introduced. Then the asymptotic distributions of over(θ, ̂)_{ML} and over(θ, ̂)_{B} are derived and shown to be non-normal. Furthermore we observe that over(θ, ̂)_{B} is asymptotically efficient, but over(θ, ̂)_{ML} is not so. Also various versions of step spectra are considered.

Original language | English |
---|---|

Pages (from-to) | 153-170 |

Number of pages | 18 |

Journal | Stochastic Processes and their Applications |

Volume | 118 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2008 Feb |

### Fingerprint

### Keywords

- Asymptotic efficiency
- Bayes estimator
- Likelihood ratio
- Maximum likelihood estimator
- Non-regular estimation
- Piecewise continuous spectra

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Mathematics(all)
- Statistics and Probability
- Modelling and Simulation

### Cite this

**Non-regular estimation theory for piecewise continuous spectral densities.** / Taniguchi, Masanobu.

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 118, no. 2, pp. 153-170. https://doi.org/10.1016/j.spa.2007.04.001

}

TY - JOUR

T1 - Non-regular estimation theory for piecewise continuous spectral densities

AU - Taniguchi, Masanobu

PY - 2008/2

Y1 - 2008/2

N2 - For a class of Gaussian stationary processes, the spectral density fθ (λ), θ = (τ′, η′)′, is assumed to be a piecewise continuous function, where τ describes the discontinuity points, and the piecewise spectral forms are smoothly parameterized by η. Although estimating the parameter θ is a very fundamental problem, there has been no systematic asymptotic estimation theory for this problem. This paper develops the systematic asymptotic estimation theory for piecewise continuous spectra based on the likelihood ratio for contiguous parameters. It is shown that the log-likelihood ratio is not locally asymptotic normal (LAN). Two estimators for θ, i.e., the maximum likelihood estimator over(θ, ̂)ML and the Bayes estimator over(θ, ̂)B, are introduced. Then the asymptotic distributions of over(θ, ̂)ML and over(θ, ̂)B are derived and shown to be non-normal. Furthermore we observe that over(θ, ̂)B is asymptotically efficient, but over(θ, ̂)ML is not so. Also various versions of step spectra are considered.

AB - For a class of Gaussian stationary processes, the spectral density fθ (λ), θ = (τ′, η′)′, is assumed to be a piecewise continuous function, where τ describes the discontinuity points, and the piecewise spectral forms are smoothly parameterized by η. Although estimating the parameter θ is a very fundamental problem, there has been no systematic asymptotic estimation theory for this problem. This paper develops the systematic asymptotic estimation theory for piecewise continuous spectra based on the likelihood ratio for contiguous parameters. It is shown that the log-likelihood ratio is not locally asymptotic normal (LAN). Two estimators for θ, i.e., the maximum likelihood estimator over(θ, ̂)ML and the Bayes estimator over(θ, ̂)B, are introduced. Then the asymptotic distributions of over(θ, ̂)ML and over(θ, ̂)B are derived and shown to be non-normal. Furthermore we observe that over(θ, ̂)B is asymptotically efficient, but over(θ, ̂)ML is not so. Also various versions of step spectra are considered.

KW - Asymptotic efficiency

KW - Bayes estimator

KW - Likelihood ratio

KW - Maximum likelihood estimator

KW - Non-regular estimation

KW - Piecewise continuous spectra

UR - http://www.scopus.com/inward/record.url?scp=37249018463&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37249018463&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2007.04.001

DO - 10.1016/j.spa.2007.04.001

M3 - Article

AN - SCOPUS:37249018463

VL - 118

SP - 153

EP - 170

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 2

ER -