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The coordinatewise median of vectors applies the univariate sample median separately to each coordinate. It is easy to compute but depends on the chosen coordinate axes.

Let $m=F^{-1}(1/2)$. Since $F$ has an everywhere positive density, it is continuous and strictly increasing, so $F(m)=1/2$. Membership in the Kolmogorov neighborhood of a distribution givesBy the symmetry of the standard normal distribution,The stated asymptotic-bias formula for the sample median therefore yields

Write $y_i=|x_i|$. If $n$ is odd, the sample median $t=T_n$ equals one observation, with equally many $y_i$ below and above it; the corresponding terms $\mathrm{sign}(y_i/t-1)$ cancel and the median term is zero. If $n$ is even, the conventional median lies strictly between the two middle observations when the $y_i$ are distinct, so exactly half the terms are $-1$ and half are $+1$. In either caseHence $T_n$ is a Scale M-estimator.

As $b\downarrow 0$, the Huber estimating equation approaches the sign equationwhose solution is the sample median. Hence the most B-robust location M-estimator is the median, with minimum gross-error sensitivity