Matrix depth and covariance estimators

For $p\ge 1$, write $S^{p-1}:=\{u\in {\mathbb{R}}^p:\mathrm{\parallel }u{\mathrm{\parallel }}_2=1\}$. Let $U$ be a nonempty subset of this sphere, let $P$ be a probability law on ${\mathbb{R}}^p$, and let $\mathrm{\Gamma }⪰0$. Its matrix depth relative to $U$ is $D_U(\mathrm{\Gamma },P):={\inf }_{u\in U}\min \{P((u^{T}X{)}^2\le u^T\mathrm{\Gamma }u),P((u^{T}X{)}^2\ge u^T\mathrm{\Gamma }u)\},$ where $X$ has law $P$. Both weak inequalities include ties.

For observations $x_1,\dots ,x_n$, let $P_n:=\frac{1}{n}{\sum }_{i=1}^n{\delta }_{x_i}$ on the Borel sigma-algebra of ${\mathbb{R}}^p$ (The Borel sigma-algebra of a topological space). Each Dirac measure is a probability measure by The Dirac set function at a point and A Dirac set function is a probability measure. Their finite nonnegative weighted sum is a measure by Nonnegative scalar multiples and countable weighted sums of measures and Nonnegative scalar multiples and countable weighted sums of measures are measures, and its total mass is $n^{-1}{\sum }_{i}1=1$; hence $P_n$ is a probability law by Probability measures and probability spaces. Let $\beta >0$ be the constant of The Gaussian matrix-depth scale constant exists uniquely. The global dense depth correspondence is ${\mathrm{Argmax}}_{\mathrm{\Gamma }⪰0}{D}_{S^{p-1}}(\mathrm{\Gamma },P_n)$, over symmetric positive semidefinite matrices. For $1\le s\le p\mathrm{/}2$, define the global single-block class ${\mathcal{F}}_{p,s}:={\bigcup }_{c>0}{\mathcal{F}}_{p,s}(c)$ and the global single-block sparse correspondence ${\mathrm{Argmax}}_{\mathrm{\Gamma }\in {\mathcal{F}}_{p,s}}{D}_{U_s}(\mathrm{\Gamma },P_n)$. These are the unrestricted-spectrum versions; no known spectral bound is needed to define their selectors.

For known $M>0$, the dense bounded-class depth correspondence is ${\mathrm{Argmax}}_{\mathrm{\Gamma }\in {\mathcal{F}}_p(\beta M)}{D}_{S^{p-1}}(\mathrm{\Gamma },P_n)\mathrm{.}$ For known integer $1\le s\le p\mathrm{/}2$, the single-block sparse version is ${\mathrm{Argmax}}_{\mathrm{\Gamma }\in {\mathcal{F}}_{p,s}(\beta M)}{D}_{U_s}(\mathrm{\Gamma },P_n),$ where the classes and $U_s$ are defined in Bounded-spectrum covariance class and Single-block sparse covariance class and restricted directions. A measurable matrix-depth estimator is a Borel measurable selector $\widehat{\mathrm{\Gamma }}$ from the applicable argmax correspondence, with covariance output $\widehat{\mathrm{\Sigma }}=\widehat{\mathrm{\Gamma }}\mathrm{/}\beta$. Either global correspondence or the applicable bounded-class correspondence may be used. The compact search restriction contains $\beta \mathrm{\Sigma }$ whenever the target belongs to the corresponding class; it makes no computational feasibility claim.

Existence of a maximizer and existence of a measurable selector are separate obligations. Both are proved locally for the bounded classes, and the measurable-selection lemma also proves existence of Borel global maximizers by a data-dependent compact cap. Global argmax fibers are not asserted compact.