High-Dimensional Linear Models: A Random Matrix Perspective
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High-Dimensional Linear Models: A Random Matrix Perspective Jamshid Namdari and Debashis Paul University of California, Davis, USA
Lili Wang Zhejiang Gongshang University, Hangzhou, China Abstract Professor C.R.Rao’s Linear Statistical Inference is a classic that has motivated several generations of statisticians in their pursuit of theoretical research. This paper looks into some of the fundamental problems associated with linear models, but in a scenario where the dimensionality of the observations is comparable to the sample size. This perspective, largely driven by contemporary advancements in random matrix theory, brings new insights and results that can be helpful even for solving relatively low-dimensional problems. This overview also brings into focus the fundamental roles played by the eigenvalues of large covariance-type matrices in the theory of highdimensional multivariate statistics. AMS (2000) subject classification. Primary 62; Secondary 62H12, 62J05, 62J10. Keywords and phrases. Multivariate statistics, linear models, random matrix theory.
1 Introduction Professor C. R. Rao’s seminal contributions to all areas of statistics spanning over seven decades have inspired developments of many modern statistical techniques. Two of his books (Rao, 1952, 1965) are classics in statistical methods and their applications. Statistical inference techniques in linear models, and more broadly, multivariate statistical analysis from a linear models perspective, have been some of the most influential contributions of Rao. His works on testing hypotheses in multivariate analysis and in linear models (Rao, 1948, 1959), on statistical inference for factor analysis (Rao, 1955), on applications of principal components analysis (Rao, 1964), on estimation of variance components (Rao, 1972), and his 1975 Wald Memorial
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J. Namdari and et al.
Lectures on estimation theory for linear models (Rao, 1976) are some of the most well-cited works in the statistical literature. In view of this, in this paper we focus our attention to modern theoretical developments in these areas of multivariate statistics with the additional feature that the dimensions of the observation vectors are not fixed, but are allowed to increases with the sample size. One of the striking features of much of Rao’s work in the context of linear statistical inference is the broad generality of the conclusions, and hence their applicability to a multitude of practical problems. In this regard, modern high-dimensional statistics has evolved in broadly two distinct trajectories. One of these branches has utilized the idea of imposing meaningful and interpretable, though not always testable, structural assumptions on the parameters associated with a statistical model—such as sparsity, or low-rankedness—while the other branch attempted to generalize conclusions derived in classical multivariate analysis typically under the Gaussian distribution framework to broader classes, without imposing too many structural constraints on the parameter. The first branch has seen
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