We introduce three notions of multivariate median bias, namely, rectilinear, Tukey, and orthant median bias. Each of these median biases is zero under a suitable notion of multivariate symmetry. We study the coverage probabilities of rectangular hull of $B$ independent multivariate estimators, with special attention to the number of estimators $B$ needed to ensure a miscoverage of at most $\alpha$. It is proved that for estimators with zero orthant median bias, we need $B\geq c\log_2(d/\alpha)$ for some constant $c > 0$. Finally, we show that there exists an asymptotically valid (non-trivial) confidence region for a multivariate parameter $\theta_0$ if and only if there exists a (non-trivial) estimator with an asymptotic orthant median bias of zero.

翻译：本文引入了三种多元中位数偏差的概念，即直线型、Tukey 和象限中位数偏差。在适当的多元对称性概念下，每种中位数偏差均为零。我们研究了$B$个独立多元估计量的矩形壳覆盖概率，特别关注确保误覆盖率不超过$\alpha$所需的估计量个数$B$。证明了对于具有零象限中位数偏差的估计量，需要满足$B\geq c\log_2(d/\alpha)$，其中常数$c>0$。最后，我们证明存在一个渐近有效（非平凡）的多元参数$\theta_0$置信区域，当且仅当存在一个（非平凡）估计量其渐近象限中位数偏差为零。