Spectrum Estimation: A Unified Framework for Covariance Matrix Estimation and PCA in Large Dimensions

Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or even larger. In such settings, there is a common remedy for both statistical problems: nonlinear shrinkage of the eigenvalues of the sample covariance matrix. The optimal nonlinear shrinkage formula depends on unknown population quantities and is thus not available. It is, however, possible to consistently estimate an oracle nonlinear shrinkage, which is motivated on asymptotic grounds. A key tool to this end is consistent estimation of the set of eigenvalues of the population covariance matrix (also known as the spectrum), an interesting and challenging problem in its own right. Extensive Monte Carlo simulations demonstrate that our methods have desirable finite-sample properties and outperform previous proposals.Comment: 40 pages, 8 figures, 5 tables, University of Zurich, Department of Economics, Working Paper No. 105, Revised version, July 201

On the Generalization of the Hébraud-Lequeux Model to Multidimensional Flows

In this article we build a model for multidimensional flows based on the idea of Hébraud and Lequeux for soft glassy materials. Care is taken to build a frame indifferent multi-dimensional model. The main goal of this article is to prove that the methodology we have developed to study the well-posedness and the glass transition for the original Hébraud-Lequeux model can be successfully generalized. Thus this work may be used as a starting point for more sophisticated studies in the modeling of general flows of glassy materials

Nonlinear shrinkage estimation of large-dimensional covariance matrices

Many statistical applications require an estimate of a covariance matrix and/or its inverse. When the matrix dimension is large compared to the sample size, which happens frequently, the sample covariance matrix is known to perform poorly and may suffer from ill-conditioning. There already exists an extensive literature concerning improved estimators in such situations. In the absence of further knowledge about the structure of the true covariance matrix, the most successful approach so far, arguably, has been shrinkage estimation. Shrinking the sample covariance matrix to a multiple of the identity, by taking a weighted average of the two, turns out to be equivalent to linearly shrinking the sample eigenvalues to their grand mean, while retaining the sample eigenvectors. Our paper extends this approach by considering nonlinear transformations of the sample eigenvalues. We show how to construct an estimator that is asymptotically equivalent to an oracle estimator suggested in previous work. As demonstrated in extensive Monte Carlo simulations, the resulting bona fide estimator can result in sizeable improvements over the sample covariance matrix and also over linear shrinkage.Comment: Published in at http://dx.doi.org/10.1214/12-AOS989 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Accelerated forgetting of contextual details due to focal medio-dorsal thalamic lesion

Effects of thalamic nuclei damage and related white matter tracts on memory performance are still debated. This is particularly evident for the medio-dorsal thalamus which has been less clear in predicting amnesia than anterior thalamus changes. The current study addresses this issue by assessing 7 thalamic stroke patients with consistent unilateral lesions focal to the left medio-dorsal nuclei for immediate and delayed memory performance on standard visual and verbal tests of anterograde memory, and over the long-term (>24 h) on an object-location associative memory task. Thalamic patients showed selective impairment to delayed recall, but intact recognition memory. Patients also showed accelerated forgetting of contextual details after a 24 h delay, compared to controls. Importantly, the mammillothalamic tract was intact in all patients, which suggests a role for the medio-dorsal nuclei in recall and early consolidation memory processes

Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size

This paper analyzes whether standard covariance matrix tests work when dimensionality is large, and in particular larger than sample size. In the latter case, the singularity of the sample covariance matrix makes likelihood ratio tests degenerate, but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well-defined. We study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together, with their ratio converging to a finite non-zero limit. We find that the existing test for sphericity is robust against high dimensionality, but not the test for equality of the covariance matrix to a given matrix. For the latter test, we develop a new correction to the existing test statistic that makes it robust against high dimensionality.Concentration asymptotics, equality test, sphericity test

The Economic Impacts of Climate Change: Evidence from Agricultural Profits and Random Fluctuations in Weather

This paper measures the economic impact of climate change on US agricultural land by estimating the effect of the presumably random year-to-year variation in temperature and precipitation on agricultural profits. Using long-run climate change predictions from the Hadley 2 Model, the preferred estimates indicate that climate change will lead to a $1.1 billion (2002$) or 3.4% increase in annual profits. The 95% confidence interval ranges from -$1.8 billion to$4.0 billion and the impact is robust to a wide variety of specification checks, so large negative or positive effects are unlikely. There is considerable heterogeneity in the effect across the country with California’s predicted impact equal to -$2.4 billion (or nearly 50% of state agricultural profits). Further, the analysis indicates that the predicted increases in temperature and precipitation will have virtually no effect on yields among the most important crops. These crop yield findings suggest that the small effect on profits is not due to short-run price increases. The paper also implements the hedonic approach that is predominant in the previous literature. We conclude that this approach may be unreliable, because it produces estimates of the effect of climate change that are very sensitive to seemingly minor decisions about the appropriate control variables, sample and weighting. Overall, the findings contradict the popular view that climate change will have substantial negative welfare consequences for the US agricultural sector.Cost of climate change, Hedonics, Agricultural profits, Agricultural production, Crop yields