We develop a theory of nonlinear shrinkage covariance estimation for nonparanormal (Gaussian-copula) models, in which each observed coordinate is an unknown strictly increasing transformation of a latent Gaussian vector. This model accommodates arbitrary marginal skewness and heavy marginal tails while retaining a Gaussian dependence structure, and it is the natural semiparametric setting for heavy-tailed, asymmetric financial returns. Our estimator, marginal-free nonlinear shrinkage (MENS), applies an oracle nonlinear shrinkage function to the eigenvalues of the normal-scores rank-covariance matrix. We give the almost-sure convergence of the empirical spectral distribution of the normal-scores covariance to the generalized Marchenko-Pastur law of Sigma, and asymptotic optimality of MENS among rotation-equivariant estimators under Frobenius loss. We establish a Baik-Ben Arous-Peche phase transition for spiked latent correlations. The MENS attains the robustness of rank-based estimation and the efficiency of nonlinear shrinkage at once within this class. We corroborate the theory with a simulation study that isolates the marginal-invariance property and the spiked transition. In an out-of-sample minimum-variance backtest on S&P 500 stocks, MENS delivers a better-conditioned covariance estimate, lower realized portfolio volatility, and lower turnover than linear shrinkage, illustrating its practical value for high-dimensional allocation and decision-making.
H. Karamikabir, Mohammad Arashi Department of Statistics, Faculty of Intelligent Systems Engineering et al.· 0 citations
We study estimation of the p*p residual scatter (shape) matrix in a high-dimensional multivariate linear regression, where p and n grow proportionally. When the coefficient matrix obeys a known linear restriction of rank q<d, as in multivariate analysis of variance, growth-curve models, and reduced-rank regression, the restricted fit leaves additional residual degrees of freedom that sharpen estimation of the shape matrix. To accommodate heavy-tailed errors, we work with independent elliptically distributed rows under a mild scale condition, a finite second moment on the radii, which is far weaker than the usual sub-Gaussian assumptions and covers every multivariate-t law with more than two degrees of freedom. Shrinking the restricted residual sample covariance directly is unsound here, since its limiting spectrum depends on the radial distribution. We instead shrink a scale-invariant scatter of the restricted residuals, whose spectrum is distribution-free over the elliptical family and obeys the same limiting law as under Gaussian errors, at a smaller effective aspect ratio. The resulting estimator attains the rotation-equivariant oracle and is asymptotically optimal within that class, and a Stein-type combination with the unrestricted estimator dominates it while remaining safe under misspecification. We further correct for the case in which the restriction is itself selected from the data. Simulations, a growth-curve experiment, and two real-data analyses illustrate the results.
H. Karamikabir, Mohammad Arashi Department of Statistics, Faculty of Intelligent Systems Engineering et al.· 0 citations
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