Adaptive multivariate trimmed mean

Let $d,N\ge 1$ be integers, let $0\le \eta \le 1\mathrm{/}10$, and let $0<\delta <1$ satisfy $N\ge 2560\log (2\mathrm{/}\delta )$. Given two observed halves ${\tilde{X}}_1,\dots ,{\tilde{X}}_N$ and ${\tilde{Y}}_1,\dots ,{\tilde{Y}}_N$ in ${\mathbb{R}}^d$, set

$$\epsilon =\max \{10\eta ,\frac{2560\log (2\mathrm{/}\delta )}{N}\}\in (0,1]\mathrm{.}$$

For $v\in S^{d-1}$, sort $⟨{\tilde{Y}}_i,v⟩$ nondecreasingly and let ${\alpha }_v$ and ${\beta }_v$ be its order statistics of ranks $\lceil \epsilon N\mathrm{/}2\rceil$ and $\lceil (1-\epsilon \mathrm{/}2)N\rceil$. Write ${\varphi }_{a,b}(t)=\min \{b,\max \{a,t\}\}$. For $Q>0$ define $U_Q(v)=\frac{1}{N}{\sum }_{i=1}^N{\varphi }_{{\alpha }_v-Q,{\beta }_v+Q}(⟨{\tilde{X}}_i,v⟩),\mathrm{\Gamma }(v,Q)=\{z\in {\mathbb{R}}^d:\mathrm{\mid }⟨z,v⟩-U_Q(v)\mathrm{\mid }\le 2\epsilon Q\}\mathrm{.}$ and $\mathrm{\Gamma }(Q)={\bigcap }_{v\in S^{d-1}}\mathrm{\Gamma }(v,Q)$. For each $j\in \mathbb{Z}$, put

$$K_j=\bigcap _{i\ge j}\mathrm{\Gamma }(2^i),S=\{j\in \mathbb{Z}:K_j\mathrm{\ne }\mathrm{\varnothing }\}\mathrm{.}$$

Return the unique minimum-norm point of $K_j$ if $S$ has least element $j$; if $S=\mathbb{Z}$, return the unique minimum-norm point of ${\bigcap }_{i\in \mathbb{Z}}\mathrm{\Gamma }(2^i)$; if $S=\mathrm{\varnothing }$, return the first-half sample mean $N^{-1}{\sum }_{i=1}^N{\tilde{X}}_i$. This three-branch selector is total on the displayed input domain; its existence and Borel measurability are proved in Totality and measurability of the repaired multivariate trimmed-mean selector ↗.