Special Session
Wishart matrices
Organized by: Dietrich von Rosen
Description:
Contributions are expected to connect to the Wishart matrix, at least in a broad sense. Topics of interest can be estimation of the parameters in the Wishart distribution including robust estimation and estimation under sparsity assumptions, deriving moments (in particular inverse moments), extending results from the real Wishart distribution to the complex and quaternion Wishart distribution, studying the spectral density of a Wishart matrix under different type of assumption, studying the extreme eigenvalues (including the condition number) and many others.
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The Wishart distribution bares its name after John Wishart who in 1928 derived the joint distribution of the estimators of all second order moments in a normal distribution (Wishart, 1928).
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Pearson (1896) introduced the joint normal distribution of the elements in a vector of
size p, i.e., a p-dimensional multivariate normal distribution. The distribution includes the
variances and covariances of all pairs of elements of a p-sized vector. Almost twenty years later Fisher (1915) was the first to derive the joint distribution of the sample variances and the sample product moment (covariance) in a bivariate (p=2) normal distribution. After thirteen more years Wishart (1928) entered the scene. He presented the joint distribution of all 0.5p(p+1) sample variances and sample covariances in his seminal work. The name Wishart distribution was first used 1933 by Wilks.
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The Wishart distribution appears, besides in multivariate statistical analysis, in many scientific fields that involves random vectors or matrices. For example, one can mention finance (Okhrin & Schmidt, 2008; Glombek 2014), physics (Castillo & Metz, 2018), climatology and meteorology (Thonfelt & Nielsen, 2008; Akbari et. al., 2010), image processing (Fukunga, 1990), wireless antenna systems (Zanella & Chiani, 2009), biology (Alter et al., 2000, Holter et al., 2000), neurology (Mengucci et al., 2021), psychology (Levy & Mislevy 2017; Hecht et al., 2021), genetics (Patterson et al., 2006; Novembre & Stephens, 2008), stochastic processes (Gourieroux & Sufana, 2010; Yu, et al., 2017; Golosny et al., 2012), nuclear physics (Fyodorov & Sommers, 1997), quantum gravity (Ambjörn et al., 1994), entangled systems in quantum computation (Majumdar, 2011) as well as in many more areas.
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A relatively young field in statistics is random matrix theory. The area connects physics, mathematics and statistics. In particular in the 50ties Wigner's semi-circle law appeared (Wigner, 1958) but several authors had touched the problem before, in particular Wigner (1955). The research questions focused on eigenvalues of symmetric matrices which size increase to infinity, i.e., there is a sequence of "growing" matrices. These matrices consisted from the beginning of independent random elements and the main interest was to understand what happens to the eigenvalues, e.g., the empirical spectral density, under some special asymptotics (asymptotics of growing dimension, Kolmogorov asymptotics) was of interest. Arharov (1971) discussed explicitly a Wishart matrix (sample covariance matrix), i.e., an underlying normal distribution is assumed but the elements in the matrix do not have to be independent, and relayed on results of Marčhenko & Pastur (1967). Jonsson (1982) followed up Archarov's (1971) work in detail. For many results in this direction see the book by Bai & Silverstein (2009), where also many additional references can be found.
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Contributions are expected to connect to the Wishart matrix, at least in a broad sense. Topics of interest can be estimation of the parameters in the Wishart distribution including robust estimation and estimation under sparsity assumptions, deriving moments (in particular inverse moments), extending results from the real Wishart distribution to the complex and quaternion Wishart distribution, studying the spectral density of a Wishart matrix under different type of assumption, studying the extreme eigenvalues (including the condition number) and many others.
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