Asymptotic non-linear shrinkage formulas for weighted sample covariance

We compute asymptotic non-linear shrinkage formulas for covariance and precision matrix estimators for weighted sample covariances, in the spirit of Ledoit and P\'ech\'e. We detail explicitly the formulas for exponentially-weighted sample covariances. Those new tools pave a way for applying non-linear shrinkage methods on weighted sample covariance. We show experimentally the performance of the asymptotic shrinkage formulas. Finally, we test the robustness of the theory to a heavy-tailed distributions.

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

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.

Asymptotic spectrum of weighted sample covariance: a Marcenko-Pastur generalization

We propose an extension of the high dimensional spectrum analysis of sample covariance in the setting of the weighted sample covariance. We derive an asymptotic equation characterizing the limit density of the weighted sample eigenvalues generalizing for weighted sample covariance matrices the Marcenko-Pastur theorem.

Analysis of a multi-target linear shrinkage covariance estimator

Multi-target linear shrinkage is an extension of the standard single-target linear shrinkage for covariance estimation. We combine several constant matrices - the targets - with the sample covariance matrix. We derive the oracle and a \textit{bona fide} multi-target linear shrinkage estimator with exact and empirical mean. In both settings, we proved its convergence towards the oracle under Kolmogorov asymptotics. Finally, we show empirically that it outperforms other standard estimators in various situations.

Ledoit-Wolf linear shrinkage with unknown mean

This work addresses large dimensional covariance matrix estimation with unknown mean. The empirical covariance estimator fails when dimension and number of samples are proportional and tend to infinity, settings known as Kolmogorov asymptotics. When the mean is known, Ledoit and Wolf (2004) proposed a linear shrinkage estimator and proved its convergence under those asymptotics. To the best of our knowledge, no formal proof has been proposed when the mean is unknown. To address this issue, we propose a new estimator and prove its quadratic convergence under the Ledoit and Wolf assumptions. Finally, we show empirically that it outperforms other standard estimators.