In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). This model is proposed by Johnstone to cope with empirical findings on various data sets. The question is to quantify the effect of the perturbation caused by the spike eigenvalues. A recent work by Baik and Silverstein establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. This paper establishes the limiting distributions of these extreme sample eigenvalues. As another important result of the paper, we provide a central limit theorem on random sesquilinear forms.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP118 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org

In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension is large compared to the sample size. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F-matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests with χ 2 approximation fails.Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.

In the spiked population model introduced by Johnstone [10], the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to quantify the effect of the perturbation caused by the spike eigenvalues. Baik and Silverstein [6] establishes the almost sure limits of the extreme sample eigenvalues associated to the spike eigenvalues when the population and the sample sizes become large. In a recent work [5], we have provided the limiting distributions for these extreme sample eigenvalues. In this paper, we extend this theory to a generalized spiked population model where the base population covariance matrix is arbitrary, instead of the identity matrix as in Johnstone's case. New mathematical tools are introduced for establishing the almost sure convergence of the sample eigenvalues generated by the spikes. 1991 Mathematics Subject Classification. Primary 60F15, 60F05; secondary 15A52, 62H25.

We have demonstrated a new strategy for enhancing solar evaporation by using floating light-absorbing materials. Floating Fe3O4/C magnetic particles with an average size of 500 nm were synthesized by carbonization of poly(furfuryl alcohol) (PFA) incorporated with Fe3O4 nanoparticles. The Fe3O4/C particles had a BET surface area of 429 m2 g−1, and a density of 1.44 g cm−3. Because of their hydrophobicity and a bulk packing density of 0.53 g cm−3, Fe3O4/C particles were floatable on water. Our results indicated that these Fe3O4/C particles enhanced the water evaporation rate by as much as a factor of 2.3 in the solar evaporation of 3.5% salt water. In addition, Fe3O4/C particles were easily recycled using a magnet, and stable after being recycled three times. Our work provides a low-cost and highly effective way for accelerating solar evaporation for industrial applications such as solar desalination, salt production, brine management and wastewater treatment

High-dimensional data appear in many fields, and their analysis has become increasingly important in modern statistics. However, it has long been observed that several well-known methods in multivariate analysis become inefficient, or even misleading, when the data dimension p is larger than, say, several tens. A seminal example is the well-known inefficiency of Hotelling's T2-test in such cases. This example shows that classical large sample limits may no longer hold for high-dimensional data; statisticians must seek new limiting theorems in these instances. Thus, the theory of random matrices (RMT) serves as a much-needed and welcome alternative framework. Based on the authors' own research, this book provides a firsthand introduction to new high-dimensional statistical methods derived from RMT. The book begins with a detailed introduction to useful tools from RMT, and then presents a series of high-dimensional problems with solutions provided by RMT methods.