Uniform risk dominance of transformed Moore-Penrose estimators

For each $n\in\mathbb{N}$, let $p_n\in\mathbb{N}$ satisfy $p_n/n\to c$ for some constant $c\in(1,\infty)$. Observe independent random vectors $X_{1,n},\dots,X_{n,n}\in\mathbb{R}^{p_n}$ of the form $X_{i,n}=\Sigma_n^{1/2}Z_{i,n}$, where $\Sigma_n\in\mathbb{R}^{p_n\times p_n}$ is symmetric positive definite, $\mathbb{E}[Z_{i,n}]=0$, $\mathbb{E}[Z_{i,n}Z_{i,n}^\top]=I_{p_n}$, and moments are uniformly bounded (for example, $\sup_{n,i,j}\mathbb{E}|(Z_{i,n})_j|^{4+\delta}<\infty$ for some $\delta>0$). Define

If $p_n>n$, then $\operatorname{rank}(S_n)\le n<p_n$ deterministically, so $S_n$ is singular for every sample realization; let $S_n^\dagger$ denote its Moore-Penrose pseudoinverse.

This setup follows Bodnar & Parolya (2024).

Let $\mathcal{C}$ be a prescribed spectral class of covariance sequences $\Sigma=(\Sigma_n)_{n\ge1}$, e.g. eigenvalues uniformly bounded away from $0$ and $\infty$: there exist constants $0<m<M<\infty$ such that $m\le\lambda_{\min}(\Sigma_n)\le\lambda_{\max}(\Sigma_n)\le M$ for all $n$. For any estimator $\widehat\Theta_n$ of $\Sigma_n^{-1}$, define Frobenius risk

Fix a benchmark class $\mathcal{B}$ of measurable estimators $\widehat\Theta_n^{(b)}$, $b\in\mathcal{B}$ (e.g. ridge/linear-shrinkage families such as $(S_n+\lambda_n I_{p_n})^{-1}$ or $\alpha_n S_n^\dagger+\beta_n I_{p_n}$, with deterministic or data-driven tuning). A transformed Moore-Penrose estimator is of the form

with measurable data-driven $T_n$.

Source-established result (Bodnar--Parolya, arXiv:2403.15792v2): asymptotic trace-moment formulas are derived and used to construct specific fully data-driven shrinkage estimators with asymptotic quadratic-loss optimality for those constructions.