标题： 长记忆高斯时间序列光谱密度的贝叶斯非参数估计 显示英文标题

标题： Bayesian nonparametric estimation of the spectral density of a long memory Gaussian time series

摘要： 设$\mathbf {X}=\{X_t, t=1,2,... \}$为一个平稳的高斯随机过程，均值为$EX_t=\mu$，协方差函数为$\gamma(\tau)=E(X_t-\mu)(X_{t+\tau}-\mu)$。令$f(\lambda)$为其对应的谱密度；若谱密度$f(\lambda)$可以写成一个缓慢变化函数$\tilde{f}(\lambda)$和量$\lambda ^{-2d}$的乘积，则称该平稳高斯过程具有长程依赖性。 本文提出了一种新的贝叶斯非参数方法来估计$\mathbf {X}$的谱密度。 我们证明了，在先验分布满足某些特定假设的情况下，当$f(\cdot)$和$d$是感兴趣的对象时，我们的方法能够保证后验一致性。 后验序列的收敛速度在很大程度上依赖于先验的结构；我们提供了一些一般性结果，并且考虑了分数指数（FEXP）先验族（详见下文）。 由于在长记忆设定下缺乏合理的依据，我们避免使用威特尔（Whittle）逼近似然函数的方法，而是选择使用真实的高斯似然函数。 显示英文摘要

摘要： Let $\mathbf {X}=\{X_t, t=1,2,... \}$ be a stationary Gaussian random process, with mean $EX_t=\mu$ and covariance function $\gamma(\tau)=E(X_t-\mu)(X_{t+\tau}-\mu)$. Let $f(\lambda)$ be the corresponding spectral density; a stationary Gaussian process is said to be long-range dependent, if the spectral density $f(\lambda)$ can be written as the product of a slowly varying function $\tilde{f}(\lambda)$ and the quantity $\lambda ^{-2d}$. In this paper we propose a novel Bayesian nonparametric approach to the estimation of the spectral density of $\mathbf {X}$. We prove that, under some specific assumptions on the prior distribution, our approach assures posterior consistency both when $f(\cdot)$ and $d$ are the objects of interest. The rate of convergence of the posterior sequence depends in a significant way on the structure of the prior; we provide some general results and also consider the fractionally exponential (FEXP) family of priors (see below). Since it has not a well founded justification in the long memory set-up, we avoid using the Whittle approximation to the likelihood function and prefer to use the true Gaussian likelihood.