Adaptive truncated covariance estimator

Input a finite sample $X_1,\dots ,X_m\in {\mathbb{R}}^p$, confidence parameter $\beta >1$, and positive finite scale bounds $0<{\sigma }_{\min }\le {\sigma }_{\max }$. The computational domain requires $k=\lfloor 3.5\beta \rfloor +1\le m\mathrm{/}2$. Statistical guarantees are stated separately: for a clean iid finite-fourth-moment law, use the matrix variance of Matrix variance for covariance estimation and a bracket containing ${\sigma }_0$. The statistical conditions are not needed merely to evaluate this finite-sample procedure.

Call Geometric median of block means on $(X_1,\dots ,X_m)$ with block count $k$ to obtain the pilot $\widehat{\mu }$. Use its consecutive blocks in index order, of size $\lfloor m\mathrm{/}k\rfloor +1$ for the first $m-k\lfloor m\mathrm{/}k\rfloor$ blocks and $\lfloor m\mathrm{/}k\rfloor$ for the rest. This uses the same sample and is not a sample split.

For a real symmetric matrix $A=Q\mathrm{diag}({\lambda }_i)Q^T$, define $\psi (A)=Q\mathrm{diag}(\psi ({\lambda }_i))Q^T$, where $\psi (x)=\mathrm{sign}(x)\min (\mathrm{\mid }x\mathrm{\mid },1)$. The real spectral theorem makes this definition independent of the chosen orthonormal eigenbasis. Define the finite grid and truncation thresholds ${\sigma }_j=2^j{\sigma }_{\min },J=\{j\in \mathbb{Z}:{\sigma }_{\min }\le {\sigma }_j<2{\sigma }_{\max }\},{\theta }_j=\frac{1}{{\sigma }_j}\sqrt{\frac{\beta }{m}}\mathrm{.}$ For every $j\in J$, form ${\widehat{\mathrm{\Sigma }}}_{m,j}=\frac{1}{m{\theta }_j}{\sum }_{i=1}^m\psi \text{\hspace{0.17em}⁣}({\theta }_j(X_i-\widehat{\mu })(X_i-\widehat{\mu }{)}^T)\mathrm{.}$ Choose the smallest $j^{\ast }\in J$ such that $\mathrm{\parallel }{\widehat{\mathrm{\Sigma }}}_{m,k^{\mathrm{\prime }}}-{\widehat{\mathrm{\Sigma }}}_{m,j^{\ast }}{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le 6{\sigma }_{k^{\mathrm{\prime }}}\sqrt{\frac{\beta }{m}}\text{for every }{k}^{\mathrm{\prime }}\in J,k^{\mathrm{\prime }}>j^{\ast },$ and output ${\widehat{\mathrm{\Sigma }}}^{\ast }={\widehat{\mathrm{\Sigma }}}_{m,j^{\ast }}$.

This is the clean heavy-tail moment procedure. It makes no Huber contamination or sample-replacement guarantee.