Non-regular estimation theory for piecewise continuous spectral densities
Non-regular estimation theory for piecewise continuous spectral densities
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DOI：
10.1016/j.spa.2007.04.001
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发表时间：
2008-02
影响因子：
1.4
通讯作者：
M. Taniguchi
中科院分区：
文献类型：
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作者：
M. Taniguchi
对于一类高斯平稳过程，谱密度[公式：见文本]被假设为一个分段连续函数，其中τ描述了不连续点，并且分段谱形式被η平滑地参数化。虽然参数θ的估计是一个非常基本的问题，但是对于这个问题还没有系统的渐近估计理论。本文提出了基于连续参数似然比的分段连续谱系统渐近估计理论。证明了对数似然比不是局部渐近正态（LAN）。引入了θ的两个估计量，即极大似然估计量θ³m和贝叶斯估计量θ³B。然后推导出θ³mand θ³Bare的渐近分布，并证明其为非正态分布。进一步地，我们观察到，θ π是渐近有效的，而θ π π不是渐近有效的。此外，还考虑了不同版本的阶跃光谱。
For a class of Gaussian stationary processes, the spectral density [Formula: see text] , 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 θ̂MLand the Bayes estimator θ̂B, are introduced. Then the asymptotic distributions of θ̂MLand θ̂Bare derived and shown to be non-normal. Furthermore we observe that θ̂Bis asymptotically efficient, but θ̂MLis not so. Also various versions of step spectra are considered.