Ledoit-Wolf linear shrinkage with unknown mean

Paper: arXiv 2304.07045

Authors: Benoit Oriol, Alexandre Miot

Abstract

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 to extend the linear shrinkage and its convergence properties to translation-invariant estimators. We expose four estimators respecting those conditions, proving their properties. Finally, we show empirically that a new estimator we propose outperforms other standard estimators.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 5.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper fills a notable theoretical gap by extending Ledoit-Wolf linear shrinkage convergence proofs to setting with an unknown mean under Kolmogorov asymptotics. While the mathematical contribution is rigorous and well-formalized, the empirical section focuses mainly on basic performance validation rather than extensive real-world portfolio backtesting.

Research Flowchart

  flowchart TD
    A[Goal: Estimate covariance matrix with unknown mean under Kolmogorov asymptotics] --> B[Input: Multi-Asset High-Dimensional Financial Data]
    B --> C[Method: Extend Ledoit-Wolf linear shrinkage to translation-invariant estimators]
    C --> D[Develop 4 translation-invariant shrinkage estimators]
    D --> E[Prove theoretical convergence & empirical performance]
    E --> F[Outcome: New proposed estimator outperforms standard covariance estimators]