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Analysis of a multi-target linear shrinkage covariance estimator

Benoit Oriol

TL;DR

This work addresses covariance estimation in high dimensions by extending shrinkage to multiple fixed targets. It derives an explicit oracle estimator for projecting the population covariance onto the span of the sample covariance and multiple targets, and provides a bona fide estimator that converges quadratically in mean to the oracle, under Kolmogorov-type asymptotics with $p_n/n \rightarrow c>0$. Theoretical results are complemented by experiments showing improved MSE over single-target methods, robustness to heavy tails, and applicability to nonstationary time series such as GMV portfolio optimization. Practically, the paper offers guidance on incorporating multiple targets to substantially improve covariance estimation in challenging settings.

Abstract

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.

Analysis of a multi-target linear shrinkage covariance estimator

TL;DR

This work addresses covariance estimation in high dimensions by extending shrinkage to multiple fixed targets. It derives an explicit oracle estimator for projecting the population covariance onto the span of the sample covariance and multiple targets, and provides a bona fide estimator that converges quadratically in mean to the oracle, under Kolmogorov-type asymptotics with . Theoretical results are complemented by experiments showing improved MSE over single-target methods, robustness to heavy tails, and applicability to nonstationary time series such as GMV portfolio optimization. Practically, the paper offers guidance on incorporating multiple targets to substantially improve covariance estimation in challenging settings.

Abstract

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.
Paper Structure (37 sections, 5 theorems, 76 equations, 3 figures, 1 table)

This paper contains 37 sections, 5 theorems, 76 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction and related work
  2. Notation, definitions and hypotheses
  3. Oracle and proposed estimator
  4. Experimental results
  5. Target set impact: Usefulness of a target
  6. Target set impact: Adding useless targets to a good target set
  7. Heavy-tails robustness
  8. Application to non-stationary time series
  9. Limitations
  10. Conclusion
  11. Unknown mean formulas
  12. Experimental settings and extensions
  13. Experiment 1: Target set impact
  14. Experiment 3: Heavy-tail robustness
  15. Theoretical proofs
  16. ...and 22 more sections

Key Result

Proposition 1

The oracle coefficients solving the minimization problem min_or are, for $i \in \llbracket 1, N_n \rrbracket$, with $\langle \cdot \rangle := \langle \cdot \rangle_n$: Thus, we define the oracle covariance estimator $\Sigma_n^{*} = \mathcal{P}_{\mathcal{S}_n^+} \left(c_0^* S_n + \sum_{k=1}^{N_n} c_{k}^* T_n^{(k)}\right)$, where $\mathcal{P}_{\mathcal{S}_n^+}$ is the orthogonal projection on the c

Figures (3)

  • Figure 1: PRIAL of the MTSE, and the oracle, with target sets 1 and 2, and the Ledoit-Wolf STSE as reference.
  • Figure 2: PRIAL difference between the oracle MTSE and Ledoit Wolf estimator (MTSE_o - LW), and bona fide MTSE (MTSE - LW).
  • Figure 3: Convergence of $\lVert S_n^* - \Sigma_n^{*}\rVert_n^2$ for different values of $\nu$.

Theorems & Definitions (13)

  • Definition 1: Empirical covariance
  • Definition 2: Set of targets
  • Proposition 1: Oracle estimator
  • Definition 3: Known mean - Variance estimators
  • Theorem 1: Variance estimators
  • Definition 4: Bona fide estimator
  • Theorem 2: Loss convergence
  • Definition 5: Unknown mean - Variance estimators
  • Remark 1: Implementation trick
  • Remark 2
  • ...and 3 more