### 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 |
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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 1 |

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics