We present a novel weighted $\ell_2$ projection method for estimating autocovariance sequences and spectral density functions from reversible Markov chains. Berg and Song (2023) introduced a least-squares shape-constrained estimation approach for the autocovariance function by projecting an initial estimate onto a shape-constrained space using an $\ell_2$ projection. While the least-squares objective is commonly used in shape-constrained regression, it can be suboptimal due to correlation and unequal variances in the input function. To address this, we propose a weighted least-squares method that defines a weighted norm on transformed data. Specifically, we transform an input autocovariance sequence into the Fourier domain and apply weights based on the asymptotic variance of the sample periodogram, leveraging the asymptotic independence of periodogram ordinates. Our proposal can equivalently be viewed as estimating a spectral density function by applying shape constraints to its Fourier series. We demonstrate that our weighted approach yields strongly consistent estimates for both the spectral density and the autocovariance sequence. Empirical studies show its effectiveness in uncertainty quantification for Markov chain Monte Carlo estimation, outperforming the unweighted moment LS estimator and other state-of-the-art methods.

翻译：我们提出了一种新颖的加权 $\ell_2$ 投影方法，用于估计可逆马尔可夫链的自协方差序列和谱密度函数。Berg 和 Song (2023) 提出了一种最小二乘形状约束估计方法，通过使用 $\ell_2$ 投影将初始估计值投影到形状约束空间中来估计自协方差函数。虽然最小二乘目标函数在形状约束回归中常用，但由于输入函数的相关性和不等方差性，它可能不是最优的。为了解决这个问题，我们提出了一种加权最小二乘法，该方法在变换后的数据上定义了一个加权范数。具体而言，我们将输入的自协方差序列变换到傅里叶域，并基于样本周期图的渐近方差应用权重，利用了周期图坐标的渐近独立性。我们的方法可以等价地视为通过对其傅里叶级数施加形状约束来估计谱密度函数。我们证明了我们的加权方法能为谱密度和自协方差序列产生强一致性估计。实证研究表明，该方法在马尔可夫链蒙特卡洛估计的不确定性量化方面表现有效，优于未加权的矩 LS 估计器及其他最先进方法。