A well-conditioned estimator for large-dimensional covariance matrices

Ledoit, Olivier; Wolf, Michael

doi:10.1016/s0047-259x(03)00096-4

Cited by 2,318 publications

(2,374 citation statements)

References 23 publications

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“…These differ in the functions utilized and resulting properties, e.g., whether the original order of eigenvalues is maintained or whether the resulting estimates are guaranteed to be nonnegative; see Muirhead (1987) for a review of early work and Srivastava and Kubokawa (1999) for more recent references. Minimizing the squared error loss to determine an optimal amount of shrinkage, Ledoit and Wolf (2004) derived an estimator that regresses sample eigenvalues toward their mean and yields a weighted combination of the sample covariance matrix and an identity matrix. While this has seen diverse applications, including the analysis of high-dimensional genomic data (Schäfer and Strimmer 2005), Daniels and Kass (2001) reported overshrinkage of the smallest roots when eigenvalues were spread far apart and suggested shrinking the log sample eigenvalues toward their posterior mean as an alternative.…”

Section: Bias Due To Sampling Variancementioning

confidence: 99%

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Meyer

Kirkpatrick

2008

Genetics

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Eigenvalues and eigenvectors of covariance matrices are important statistics for multivariate problems in many applications, including quantitative genetics. Estimates of these quantities are subject to different types of bias. This article reviews and extends the existing theory on these biases, considering a balanced one-way classification and restricted maximum-likelihood estimation. Biases are due to the spread of sample roots and arise from ignoring selected principal components when imposing constraints on the parameter space, to ensure positive semidefinite estimates or to estimate covariance matrices of chosen, reduced rank. In addition, it is shown that reduced-rank estimators that consider only the leading eigenvalues and -vectors of the ''between-group'' covariance matrix may be biased due to selecting the wrong subset of principal components. In a genetic context, with groups representing families, this bias is inverse proportional to the degree of genetic relationship among family members, but is independent of sample size. Theoretical results are supplemented by a simulation study, demonstrating close agreement between predicted and observed bias for large samples. It is emphasized that the rank of the genetic covariance matrix should be chosen sufficiently large to accommodate all important genetic principal components, even though, paradoxically, this may require including a number of components with negligible eigenvalues. A strategy for rank selection in practical analyses is outlined.

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Section: Bias Due To Sampling Variancementioning

confidence: 99%

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Meyer

Kirkpatrick

2008

Genetics

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“…The original idea is due to Stein (1956), where it was applied to the estimation of the mean vector. Applications to variance matrix estimation include Jorion (1986), Muirhead (1987) and Ledoit and Wolf (2003, 2004a,b, 2012. Intuitively, the role of the shrinkage parameter is to balance the estimation error coming from the ill-conditioned variance matrix and the specification error associated with the target matrix.…”

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Design-free estimation of variance matrices

Abadir

Distaso

Žikeš

2014

Journal of Econometrics

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“…29 uses the inverse of this matrix, and thus cannot be used in combination with standardization based on empirical estimates of the mean and variance. To overcome this problem, we regularized the SCM via the well-known Ledoit-Wolf (LW) formula 31 . Briefly, given a SCM , the corresponding regularized covariance matrix is given by where is the trace of , is the number of variants, is the identity matrix and is the optimal shrinkage coefficient that minimizes the mean squared error between the estimated and true covariance matrix, whose determination is described by Ledoit and Wolf 31 .…”

Section: Comparison Of Map and Posterior Mean Estimatorsmentioning

confidence: 99%

Accurate liability estimation improves power in ascertained case-control studies

Weissbrod¹,

Lippert²,

Geiger³

et al. 2015

Nat Methods

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Linear mixed models (LMMs) have emerged as the method of choice for confounded genome-wide association studies. However, the performance of LMMs in nonrandomly ascertained case-control studies deteriorates with increasing sample size. We propose a framework called LEAP (Liability Estimator As a Phenotype; https://github.com/omerwe/LEAP) that tests for association with estimated latent values corresponding to severity of phenotype, and demonstrate that this can lead to a substantial power increase. Main TextIn recent years, genome-wide association studies (GWAS) have uncovered thousands of risk variants for genetic traits 1 . Only a small fraction of disease variance is explained by discovered variants, possibly because contemporary sample sizes are relatively small and that causal variants tend to have small effect sizes 2 . To identify such variants, future studies will need to include hundreds of thousands of individuals.Population structure and family relatedness 3 lead to spurious results and increased type I error rate. As sample sizes continue to increase, this difficulty becomes even 2 more severe, because larger samples are more likely to include individuals with a different genetic ancestry, or related individuals.Recently, LMMs have emerged as the method of choice for GWAS, due to their robustness to diverse sources of confounding 3 . LMMs gain resilience to confounding by testing for association conditioned on pairwise kinship coefficients between study subjects. Although designed for continuous phenotypes, LMMs have been successfully used in several large case-control GWAS 4-6 , because alternative methods cannot capture diverse sources of confounding 3 .However, LMMs in ascertained case-control studies, wherein cases are oversampled relative to the disease prevalence, lose power with increasing sample size compared to alternative methods 7 . This loss is due to several model violations: Dependence between tested causal variants and variants used to estimate kinship, dependence between genetic and environmental effects, and use of a non-continuous trait (Supplementary Note). Thus, the use of LMMs resolves the difficulty of sensitivity to confounding, but leads to a different difficulty instead.A possible remedy is to test for associations with a model that directly represents the case-control phenotype and takes the ascertainment scheme into account (Supplementary Note). Such models assume that observed case-control phenotypes are generated by an unobserved stochastic process with a well-defined distribution. One prominent example is the liability threshold model 8 , which associates individuals with a latent normally distributed variable called the liability, such that cases are individuals whose liability exceeds a given cutoff. Despite their elegance, such models are extremely computationally expensive, rendering whole genome association tests infeasible in most circumstances.As an alternative, we propose approximating such models by first estimating latent liability values and model param...

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A well-conditioned estimator for large-dimensional covariance matrices

Cited by 2,318 publications

References 23 publications

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Perils of Parsimony: Properties of Reduced-Rank Estimates of Genetic Covariance Matrices

Design-free estimation of variance matrices

Accurate liability estimation improves power in ascertained case-control studies

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