非零均值平稳球不变随机过程的参数与半参数Fisher信息矩阵
On the parametric and semiparametric Fisher information matrix for non-zero mean stationary spherical invariant random processes
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中文总结 AI 辅助
本文将Whittle公式推广至非零均值平稳复合高斯过程，在统一框架下推导出参数与半参数Fisher信息矩阵的闭式谱域表达式，支持重尾信号处理性能分析。
中文摘要 AI 辅助
经典的Whittle公式为多维、实值、零均值、纯非确定平稳高斯过程（GPs）的渐近Fisher信息矩阵（FIM）速率提供了闭式表达式，该表达式在最大似然框架下用其参数化谱表示。然而，该结果所依据的高斯假设限制了其对许多表现出重尾或非高斯行为的真实世界信号的适用性。在本文中，我们将Whittle的结果推广到具有任意均值的多维、实值平稳复合高斯过程（CGPs），在一个涵盖完全已知、参数化和完全未知密度生成器的统一框架下进行。基于n个连续观测的Slepian-Bangs公式，并将其扩展到半参数设置，我们利用块Toeplitz矩阵的渐近性质，获得了渐近FIM速率的闭式谱域表达式。所得表达式对所有密度生成器族通用，通过纳入与非零均值相关的分布依赖贡献，以及一个对非高斯分布选择不变的附加协方差项，推广了标准的零均值高斯公式。这一扩展使得对重尾信号处理应用进行高效且理论上有依据的性能分析成为可能。
英文摘要
The classical Whittle formula provides a closed-form expression for the asymptotic Fisher information matrix (FIM) rate of multidimensional, real-valued, zero-mean, purely nondeterministic stationary Gaussian processes (GPs), expressed in terms of their parameterized spectra within a maximum-likelihood framework. However, the Gaussian assumption underlying this result restricts its applicability to many real-world signals exhibiting heavy-tailed or non-Gaussian behavior. In this paper, we extend Whittle's result to multidimensional, real-valued stationary compound Gaussian processes (CGPs) with arbitrary mean, under a unified framework encompassing fully known, parameterized, and completely unknown density generators. Building upon the Slepian-Bangs formula for $n$ consecutive observations and extending it to the semiparametric setting, we leverage the asymptotic properties of block Toeplitz matrices to obtain a closed-form spectral-domain expression for the asymptotic FIM rate. The resulting expression, common to all density generator families, generalizes the standard zero-mean Gaussian formula by incorporating a distribution-dependent contribution associated with the nonzero mean, and an additional covariance term that is invariant to the choice of non-Gaussian distribution. This extension enables efficient and theoretically grounded performance analysis for heavy-tailed signal processing applications.
发表机构
- Samovar laboratory, Telecom SudParis, Institut Polytechnique de Paris（萨莫瓦尔实验室，电信巴黎理工学院，巴黎理工学院）
- Taif University, College of Engineering, Dept. of Electrical Engineering（塔伊夫大学，工程学院，电气工程系）
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