Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0804.4738

Structural shrinkage of nonparametric spectral estimators for multivariate time series

Hilmar Böhm·Rainer von Sachs
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we investigate the performance of periodogram based estimators of the spectral density matrix of possibly high-dimensional time series. We suggest and study shrinkage as a remedy against numerical instabilities due to deteriorating condition numbers of (kernel) smoothed periodogram matrices. Moreover, shrinking the empirical eigenvalues in the frequency domain towards one another also improves at the same time the Mean Squared Error (MSE) of these widely used nonparametric spectral estimators. Compared to some existing time domain approaches, restricted to i.i.d. data, in the frequency domain it is necessary to take the size of the smoothing span as "effective or local sample size" into account. While Böhm and von Sachs (2007) proposes a multiple of the identity matrix as optimal shrinkage target in the absence of knowledge about the multidimensional structure of the data, here we consider "structural" shrinkage. We assume that the spectral structure of the data is induced by underlying factors. However, in contrast to actual factor modelling suffering from the need to choose the number of factors, we suggest a model-free approach. Our final estimator is the asymptotically MSE-optimal linear combination of the smoothed periodogram and the parametric estimator based on an underfitting (and hence deliberately misspecified) factor model. We complete our theoretical considerations by some extensive simulation studies. In the situation of data generated from a higher-order factor model, we compare all four types of involved estimators (including the one of Böhm and von Sachs (2007)).

Explore related subjects

math.STstat.TH

Keep this discovery

BibTeXRIS

Hilmar Böhm, Rainer von Sachs. 2008-08-13. Structural shrinkage of nonparametric spectral estimators for multivariate time series. https://doi.org/10.1214/08-ejs236

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Scale Invariance Property of PCA

The PCA algorithm is sensitive to changes in measurement scale. Measuring one variable of a system in inches rather than centimeters, say, alters both its principal axes and principal eigenvalues. Although this scale dependence is generally complicated, we show here that it nevertheless obeys a strict invariance property: under a continuous scale adjustment, the initial state's $k$-th largest principal component (ordered by eigenvalue) continuously evolves into the final state's $k$-th largest principal component, for each $k$. In this sense, we can say that the modes of PCA are "order-stable" with respect to changes in measurement scale. A special case occurs when scaling along directions that are orthogonal to some modes. Here, apparent eigenvalue crossings can occur. However, we show that we can interpret these apparent crossings as cases where the modes instantaneously swap their orientation, in this way maintaining the required order stability.

math.ST↗
arXiv2026

Small noise asymptotics for linear parabolic SPDEs in two space dimensions with unknown damping factors

We study parametric estimation for second order linear parabolic stochastic partial differential equations in two space dimensions with a small volatility parameter driven by a $Q$-Wiener process with an unknown damping parameter using high frequency spatio-temporal data. We first provide an estimator for the damping parameter of the $Q$-Wiener process utilizing realized quadratic variations based on spatial and temporal increments. We next propose minimum contrast estimators of the diffusive and advective parameters in the SPDE using a contrast function with the proposed estimator of the damping parameter. We then construct a quasi-maximum likelihood estimator of the reaction parameter in the SPDE using the approximate coordinate process derived from the estimators of the diffusive and advective parameters. We also provide simulation results of the proposed estimators.

math.ST↗
arXiv2026

Spike Estimation from Heteroscedastic Noise via Random Splitting

In this paper, we consider a spiked Wigner type matrix with a heteroscedastic and unknown variance profile. It is well known that in the supercritical regime of the BBP transition, strong spikes can create outliers in the spectrum. Unfortunately, in the heteroscedastic case, in general it is not possible to estimate the spike strength from these observed outlier consistently, as the latter is a solution to a Dyson equation with unknown parameters from the variance profile. In this paper, inspired by the work on sparse matrix completion \citep{BordenaveCosteNadakuditi2023}, we introduce an asymmetrized model by randomly splitting the spiked matrix into two parts, which transforms the noisy Wigner type matrix into a non Hermitian random matrix, while preserving the Hermitian spikes at the cost of a dilution. We establish a BBP type transition for the asymmetrized model, from which we can estimate the strength of the spikes precisely, even without knowing the variance profile of the noise part. We then further apply our approach to study the correlation between two correlated spiked models, where the spike/signal parts of the two models are correlated, and the noise parts are independent but may both be heteroscedastic. By applying our asymmetrization approach to the two models separately and also jointly, we are able to obtain a precise estimate of the correlation between the signal parts of the two models.

math.ST↗