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：我们引入三种多元中位数偏差的概念，即直线型、Tukey型与象限型中位数偏差。在适当的多元对称性概念下，这些中位数偏差均为零。我们研究 $B$ 个独立多元估计量的矩形凸包覆盖概率，特别关注保证最多 $\alpha$ 的误覆盖概率所需的估计量个数 $B$。证明表明：对于具有零象限型中位数偏差的估计量，需满足 $B\geq c\log_2(d/\alpha)$（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 $\theta_0$ 置信域，当且仅当存在一个具有渐近零象限型中位数偏差的（非平凡）估计量。