Weighted shape-constrained estimation for the autocovariance sequence from a reversible Markov chain

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.