Let $X = (x_1, \ldots, x_n)$ be a sample of i.i.d values. There are several robust estimators of sample location, most notably sample median and Hodges-Lehmann estimator.
Now let $W = (w_1, \ldots, w_n), w_i >= 0$ be (non-random) reliability weights. Weighted mean is an example of estimator that takes weights into account, but it is non-robust.
Is there a statistically (like HL-estimator) and computationally efficient ($O(n log(n))$ at least) estimator of sample location that accounts for weights?