Baxter’s inequality and sieve bootstrap for random fields

Meyer, Marco; Jentsch, Carsten; Kreiß, Jens-Peter

doi:10.3150/16-bej835

Cited by 16 publications

(7 citation statements)

References 40 publications

(53 reference statements)

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“…The conditions in Assumption 4.1 are analogous to those used in the analysis of stationary time series, where conditions on the rate of decay of the autocovariances coefficients are given. In the lemma below we obtain a bound between the rows of D n and D n (the proof is based on methods developed in Meyer et al (2017)).…”

Section: Finite Dimension Approximationmentioning

confidence: 99%

“…PROOF The proof is based on the innovative technique developed in Meyer et al (2017) (who used the method to obtain Baxter bounds for stationary spatial processes). We start by deriving the normal equations corresponding to (89) and ( 90) for 1 ≤ s ≤ n and c = 1, .…”

Section: Proof Of Lemma 23mentioning

confidence: 99%

“…To bound the difference between the rows of D n = C −1 n and D n (the submatrix of D), we use that the entries of D n and D n are the entries of coefficients in a regression. This allows us to use the Baxter inequality methods developed in Meyer et al (2017) to bound the difference between projections on finite dimensional spaces and infinite dimensional spaces. The infinite and finite dimensional spaces we will use are H = sp(X…”

Section: Example: Locally Stationary Time-varying Var(1)mentioning

confidence: 99%

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Graphical models for nonstationary time series

Basu¹,

Rao²

2021

Preprint

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We propose NonStGGM, a general nonparametric graphical modeling framework for studying dynamic associations among the components of a nonstationary multivariate time series. It builds on the framework of Gaussian Graphical Models (GGM) and stationary time series Gaussian Graphical model (StGGM), and complements existing works on parametric graphical models based on change point vector autoregressions (VAR). Analogous to StGGM, the proposed framework captures conditional noncorrelations (both intertemporal and contemporaneous) in the form of an undirected graph. In addition, to describe the more nuanced nonstationary relationships among the components of the time series, we introduce the new notion of conditional nonstationarity/stationarity and incorporate it within the graph architecture. This allows one to distinguish between direct and indirect nonstationary relationships among system components, and can be used to search for small subnetworks that serve as the "source" of nonstationarity in a large system. Together, the two concepts of conditional noncorrelation and nonstationarity/stationarity provide a parsimonious description of the dependence structure of the time series.In GGM, the graphical model structure is encoded in the sparsity pattern of the inverse covariance matrix. Analogously, we explicitly connect conditional noncorrelation and stationarity between and within components of the multivariate time series to zero and Toeplitz embeddings of an infinite-dimensional inverse covariance operator. In order to learn the graph, we move to the Fourier domain. We show that in the Fourier domain, conditional stationarity and noncorrelation relationships in the inverse covariance operator are encoded with a specific sparsity structure of its integral kernel operator. Within the local stationary framework we show that these sparsity patterns can be recovered from finite-length time series by node-wise regression of discrete Fourier Transforms (DFT) across different Fourier frequencies. We illustrate the features of our general framework under the special case of timevarying Vector Autoregressive models. We demonstrate the feasibility of learning NonStGGM structure from data using simulation studies.

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Section: Finite Dimension Approximationmentioning

confidence: 99%

Section: Proof Of Lemma 23mentioning

confidence: 99%

Section: Example: Locally Stationary Time-varying Var(1)mentioning

confidence: 99%

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Graphical models for nonstationary time series

Basu¹,

Rao²

2021

Preprint

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“…See [26] for a comprehensive review. Recently, [32] proposed an AR sieve bootstrap for linear random fields. To the best of our knowledge, the development of spatial bootstrap methods focuses mainly on the block bootstrap type methods, and a frequency domain bootstrap method for possibly nonlinear random fields remains absent from the literature.…”

Section: Introductionmentioning

confidence: 99%

Frequency domain bootstrap methods for random fields

Yau

Chen

2021

Electron. J. Statist.

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This paper develops a frequency domain bootstrap method for random fields on Z 2 . Three frequency domain bootstrap schemes are proposed to bootstrap Fourier coefficients of observations. Then, inversetransformations are applied to obtain resamples in the spatial domain. As a main result, we establish the invariance principle of the bootstrap samples, from which it follows that the bootstrap samples preserve the correct second-order moment structure for a large class of random fields. The frequency domain bootstrap method is simple to apply and is demonstrated to be effective in various applications including constructing confidence intervals of correlograms for linear random fields, testing for signal presence using scan statistics, and testing for spatial isotropy in Gaussian random fields. Simulation studies are conducted to illustrate the finite sample performance of the proposed method and to compare with the existing spatial block bootstrap and subsampling methods.

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“…It has been used by [3] in proving the consistency of the autoregressive model fitting process and the corresponding autoregressive spectral density estimator, and in proving the validity of autoregressive sieve bootstrap for a stationary time series in [9,10,31]. Due to the widespread applicability of Baxter's inequality in these areas and others, there has been a great deal of activities in extending it to the setups of multivariate stationary processes in [17,11], random fields in [33], and rectangular arrays in [34]. In these extensions, the boundedness of the spectral density function of the underlying process appears to be an absolutely essential and indispensable part of proving Baxter's inequality.…”

Section: Introductionmentioning

confidence: 99%

Baxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processes

Inoue¹,

Kasahara²,

Pourahmadi³

2018

Bernoulli

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For a multivariate stationary process, we develop explicit representations for the finite predictor coefficient matrices, the finite prediction error covariance matrices and the partial autocorrelation function (PACF) in terms of the Fourier coefficients of its phase function in the spectral domain. The derivation is based on a novel alternating projection technique and the use of the forward and backward innovations corresponding to predictions based on the infinite past and future, respectively. We show that such representations are ideal for studying the rates of convergence of the finite predictor coefficients, prediction error covariances, and the PACF as well as for proving a multivariate version of Baxter's inequality for a multivariate FARIMA process with a common fractional differencing order for all components of the process.2010 Mathematics Subject Classification. Primary 60G25; secondary 62M20, 62M10.

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Baxter’s inequality and sieve bootstrap for random fields

Cited by 16 publications

References 40 publications

Graphical models for nonstationary time series

Graphical models for nonstationary time series

Frequency domain bootstrap methods for random fields

Baxter’s inequality for finite predictor coefficients of multivariate long-memory stationary processes

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