WeSpeR: Population spectrum retrieval and spectral density estimation of weighted sample covariance

• URL: http://arxiv.org/abs/2410.14413v1
• Date: Fri, 18 Oct 2024 12:26:51 GMT
• Title: WeSpeR: Population spectrum retrieval and spectral density estimation of weighted sample covariance
• Authors: Benoit Oriol,
• Abstract summary: We prove that the spectral distribution $F$ of the weighted sample covariance has a continuous density on $mathbbR*$. We propose a procedure to compute it, to determine the support of $F$ and define an efficient grid on it. We use this procedure to design the $textitWeSpeR$ algorithm, which estimates the spectral density and retrieves the true spectral covariance spectrum.
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• Abstract: The spectrum of the weighted sample covariance shows a asymptotic non random behavior when the dimension grows with the number of samples. In this setting, we prove that the asymptotic spectral distribution $F$ of the weighted sample covariance has a continuous density on $\mathbb{R}^*$. We address then the practical problem of numerically finding this density. We propose a procedure to compute it, to determine the support of $F$ and define an efficient grid on it. We use this procedure to design the $\textit{WeSpeR}$ algorithm, which estimates the spectral density and retrieves the true spectral covariance spectrum. Empirical tests confirm the good properties of the $\textit{WeSpeR}$ algorithm.

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