On Certain Characteristics of the Distribution of the Latent Roots of a Symmetric Random Matrix Under General Conditions

Vaart, H. R. Van Der

doi:10.1214/aoms/1177704979

Cited by 15 publications

(7 citation statements)

References 2 publications

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“…The advantage of the solution λ0 is that it shrinks the highest and pushes up the lowest eigenvalue. Indeed, the general theorem of van der Vaart (1961) tells us that the highest eigenvalue i /n is upward biased and (obviously in this p > n case) the lowest eigenvalue downward biased. The disadvantage is that the shrinkage may be too extreme.…”

Section: The Proposed Estimatormentioning

confidence: 97%

An orthogonally equivariant estimator of the covariance matrix in high dimensions and for small sample sizes

Banerjee

Monni

2021

Journal of Statistical Planning and Inference

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Section: The Proposed Estimatormentioning

confidence: 97%

An orthogonally equivariant estimator of the covariance matrix in high dimensions and for small sample sizes

Banerjee

Monni

2021

Journal of Statistical Planning and Inference

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“…Jensen's inequality suggests that the largest (respectively, smallest) eigenvalue of the sample covariance matrix Σ is biased upward (respectively, downward), which implies that Σ tends to be ill-conditioned (van der Vaart 1961). This effect is most pronounced for Σ ≈ I.…”

Section: Background On Precision Matrix Estimationmentioning

confidence: 99%

Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator

Nguyên

Kühn

Esfahani

2022

Operations Research

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Note. The best result in each experiment is highlighted in bold.The optimal solutions of many decision problems such as the Markowitz portfolio allocation and the linear discriminant analysis depend on the inverse covariance matrix of a Gaussian random vector. In “Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator,” Nguyen, Kuhn, and Mohajerin Esfahani propose a distributionally robust inverse covariance estimator, obtained by robustifying the Gaussian maximum likelihood problem with a Wasserstein ambiguity set. In the absence of any prior structural information, the estimation problem has an analytical solution that is naturally interpreted as a nonlinear shrinkage estimator. Besides being invertible and well conditioned, the new shrinkage estimator is rotation equivariant and preserves the order of the eigenvalues of the sample covariance matrix. If there are sparsity constraints, which are typically encountered in Gaussian graphical models, the estimation problem can be solved using a sequential quadratic approximation algorithm.

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“…Indeed, note that while the full matrix,

\overset{boldB}{true}

, is an unbiased estimator of B , the spectrum of

\overset{boldB}{true}

is a biased estimate of the spectrum of B (van der Vaart, ; Takemura, ). Thus, the spectrum of

boldH \overset{boldB}{true} {boldH}^{normalT} {boldR}^{- 1}

, namely

{false{{\overset{σ}{true}}_{i}^{2} false}}_{i = 1}^{P}

, is also biased.…”

Section: More On the Marginal Inflation Estimatorsmentioning

confidence: 99%

Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures

Raanes¹,

Bocquet

Carrassi³

2019

Quart J Royal Meteoro Soc

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This paper studies multiplicative inflation: the complementary scaling of the state covariance in the ensemble Kalman filter (EnKF). Firstly, error sources in the EnKF are catalogued and discussed in relation to inflation; nonlinearity is given particular attention as a source of sampling error. In response, the “finite‐size” refinement known as the EnKF‐N is re‐derived via a Gaussian scale mixture, again demonstrating how it yields adaptive inflation. Existing methods for adaptive inflation estimation are reviewed, and several insights are gained from a comparative analysis. One such adaptive inflation method is selected to complement the EnKF‐N to make a hybrid that is suitable for contexts where model error is present and imperfectly parametrized. Benchmarks are obtained from experiments with the two‐scale Lorenz model and its slow‐scale truncation. The proposed hybrid EnKF‐N method of adaptive inflation is found to yield systematic accuracy improvements in comparison with the existing methods, albeit to a moderate degree.

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On Certain Characteristics of the Distribution of the Latent Roots of a Symmetric Random Matrix Under General Conditions

Cited by 15 publications

References 2 publications

An orthogonally equivariant estimator of the covariance matrix in high dimensions and for small sample sizes

An orthogonally equivariant estimator of the covariance matrix in high dimensions and for small sample sizes

Distributionally Robust Inverse Covariance Estimation: The Wasserstein Shrinkage Estimator

Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures

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