Montag, 14. Mrz 2022, 16:45 - 17:45 iCal

ISOR Colloquium

"Recent advances in large sample correlation matrices and their applications"

Speaker: Johannes Heiny (Ruhr University Bochum, Germany)

HS 7 OMP1 (#1.303)
Oskar-Morgenstern-Platz 1, 1090 Wien

Vortrag


Many fields of modern sciences are faced with high-dimensional data sets. In this talk, we investigate the spectral properties of large sample correlation matrices.

First, we consider a $p$-dimensional population with iid coordinates in the domain of attraction of a stable distribution with index $\alpha\in (0,2)$. Since the variance is infinite, the sample covariance matrix based on a sample of size $n$ from the population is not well behaved and it is of interest to use instead the sample correlation matrix $R$. We find the limiting distributions of the eigenvalues of $R$ when both the dimension $p$ and the sample size n grow to infinity such that $p/n\to \gamma$. The moments of the limiting distributions $H_{\alpha,\gamma}$ are fully identified as the sum of two contributions: the first from the classical Marchenko-Pastur law and a second due to heavy tails. Moreover, the family $\{H_{\alpha,\gamma}\}$ has continuous extensions at the boundaries $\alpha=2$ and $\alpha=0$ leading to the Marchenko-Pastur law and a modified Poisson distribution, respectively. A simulation study on these limiting distributions is also provided for comparison with the Marchenko-Pastur law.

In the second part of this talk, we assume that the coordinates of the $p$-dimensional population are dependent and $p/n \le 1$. Under a finite fourth moment condition on the entries we find that the log determinant of the sample correlation matrix $R$ satisfies a central limit theorem. In the iid case, it turns out the central limit theorem holds as long as the coordinates are in the domain of attraction of a stable distribution with index $\alpha>3$, from which we conjecture a promising and robust test statistic for heavy-tailed high-dimensional data. The findings are applied to independence testing and to the volume of random simplices.

Underlying papers:

J. Heiny, S. Johnston, J. Prochno: Thin-shell theory for rotationally invariant random simplices: projecteuclid.org/journals/electronic-journal-of-probability/volume-27/issue-none/Thin-shell-theory-for-rotationally-invariant-random-simplices/10.1214/21-EJP734.full

J. Heiny, J. Yao: Limiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations: arxiv.org/abs/2003.03857

 

The talk also can be joined online via our ZOOM MEETING: univienna.zoom.us/j/68724704896

Meeting room opens at: March 14, 2022 4.30 pm Vienna

Meeting ID: 687 2470 4896

Password: 613103

Zur Webseite der Veranstaltung


Veranstalter

Institut für Statistik und Operations Research


Kontakt

Sabine Sobotka-Tompits, BA
Fakultät für Wirtschaftswissenschaften
Institut für Statistik und Operations Research
+43 1 4277 38631
sabine.sobotka-tompits@univie.ac.at