# Nonlinear Shrinkage Estimation of Higher-Order Moments for Portfolio Optimization Under Uncertainty in Complex Financial Systems

## TL;DR

This paper introduces a new nonlinear shrinkage method to improve portfolio optimization by better estimating higher-order financial moments in complex systems.

## Contribution

The paper extends nonlinear shrinkage estimation to higher-order moments, improving accuracy in high-dimensional financial settings.

## Key findings

- Nonlinear shrinkage reduces mean squared errors and improves estimation of covariance and cokurtosis.
- Portfolios using nonlinear shrinkage show higher returns and lower risk in large asset universes.
- The method performs comparably to linear shrinkage in smaller asset universes.

## Abstract

This paper develops a nonlinear shrinkage estimation method for higher-order moment matrices within a multifactor model framework and establishes its asymptotic consistency under high-dimensional settings. The approach extends the nonlinear shrinkage methodology from covariance to higher-order moments, thereby mitigating the “curse of dimensionality” and alleviating estimation uncertainty in high-dimensional settings. Monte Carlo simulations demonstrate that, compared with linear shrinkage estimation, the proposed method substantially reduces mean squared errors (MSEs) and achieves greater Percentage Relative Improvement in Average Loss (PRIAL) for covariance and cokurtosis estimates; relative to sample estimation, it delivers significant gains in mitigating uncertainty for covariance, coskewness, and cokurtosis. An empirical portfolio analysis incorporating higher-order moments shows that, when the asset universe is large, portfolios based on the nonlinear shrinkage estimator outperform those constructed using linear shrinkage and sample estimators, achieving higher annualized return and Sharpe ratio with lower kurtosis and maximum drawdown, thus providing stronger resilience against uncertainty in complex financial systems. In smaller asset universes, nonlinear shrinkage portfolios perform on par with their linear shrinkage counterparts. These findings highlight the potential of nonlinear shrinkage techniques to reduce uncertainty in higher-order moment estimation and to improve portfolio performance across diverse and complex investment environments.