The Maximum Deviation of Sample Spectral Densities

Woodroofe, Michael; Ness, John W. Van

doi:10.1214/aoms/1177698710

Cited by 40 publications

(33 citation statements)

References 7 publications

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“…A key step in the proof of Lemma 3 is to follow an example in page 315 of Zygmund (2002) by explicitly constructing a covariance matrix Σ, or equivalently the corresponding spectral density, for which the bias of the banding estimatorΣ B k is much larger than k −2β . Lemma 4 is an extension of a major result in Woodroofe and Van Ness (1967).…”

Section: Proof Of Theoremmentioning

confidence: 72%

“…Only a brief proof is given here. For more details, refer to Woodroofe and Van Ness (1967). According to (49), we have that…”

Section: Proof Of Lemmamentioning

confidence: 99%

“…Next we will truncate U p,i (λ) and V p,i (λ) as follow Before showing V p (λ) is negligible, we need some lemmas. For the proof of these lemmas, please refer to Woodroofe and Van Ness (1967). The first lemma is a standard result of trigonometric polynomial and the last two are based on Lemma (10) and Lemma (11), which are not hard to prove.…”

Section: Proof Of Lemmamentioning

confidence: 99%

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Optimal rates of convergence for estimating Toeplitz covariance matrices

Cai

Ren

Zhou

2012

Probab. Theory Relat. Fields

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Toeplitz covariance matrices are used in the analysis of stationary stochastic processes and a wide range of applications including radar imaging, target detection, speech recognition, and communications systems. In this paper, we consider optimal estimation of large Toeplitz covariance matrices and establish the minimax rate of convergence for two commonly used parameter spaces under the spectral norm. The properties of the tapering and banding estimators are studied in detail and are used to obtain the minimax upper bound. The results also reveal a fundamental difference between the tapering and banding estimators over certain parameter spaces. The minimax lower bound is derived through a novel construction of a more informative experiment for which the minimax lower bound is obtained through an equivalent Gaussian scale model and through a careful selection of a finite collection of least favorable parameters. In addition, optimal rate of convergence for estimating the inverse of a Toeplitz covariance matrix is also established.

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Section: Proof Of Theoremmentioning

confidence: 72%

“…Only a brief proof is given here. For more details, refer to Woodroofe and Van Ness (1967). According to (49), we have that…”

Section: Proof Of Lemmamentioning

confidence: 99%

Section: Proof Of Lemmamentioning

confidence: 99%

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Optimal rates of convergence for estimating Toeplitz covariance matrices

Cai

Ren

Zhou

2012

Probab. Theory Relat. Fields

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“…By the same argument as in Woodroofe and Van Ness [40] (see also [43, p. 118]), we have the inequality…”

Section: Proof Ifmentioning

confidence: 56%

Moderate deviations for quadratic forms in Gaussian stationary processes

Kakizawa

2007

Journal of Multivariate Analysis

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“…is continuously differentiable. We shall use results of Woodroofe and Van Ness (1967) to estimate the magnitude of 6*. However, their results are for a different type of estimator and so we must first approximate sN(,fj, h) by an estimator of the appropriate type.…”

mentioning

confidence: 99%

Determining the Bandwidth of a Kernel Spectrum Estimate

BeltraTo

Bloomfield

1987

Journal Time Series Analysis

129

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A cross-validated form of Whittle's frequency domain approximation to the likelihood function of a stationary Gaussian process is described, and proposed as a criterion for choosing the bandwidth in a kernel spectrum estimate. The criterion is shown to be equivalent, in large samples, to the mean integrated squared error. The statistical properties of the spectrum estimate whose bandwidth maximizes the criterion have been explored in a limited simulation.

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The Maximum Deviation of Sample Spectral Densities

Cited by 40 publications

References 7 publications

Optimal rates of convergence for estimating Toeplitz covariance matrices

Optimal rates of convergence for estimating Toeplitz covariance matrices

Moderate deviations for quadratic forms in Gaussian stationary processes

Determining the Bandwidth of a Kernel Spectrum Estimate

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