Joshua writes: << . . . Here is another question about the median: is there a median that makes sense in two or more dimensions? Suppose (X,Y) ~ f(x,y) where f(x,y) is the continuous joint pdf of the random variables X and Y. Is there a reasonable quantity to call the median?
For a *finite uniform* distribution, a reasonable way to generalize the 1D median (on R) to R^n is to use "convex peeling": do Take the convex hull of the data, then remove data on the boundary of its convex hull; while data remains. When eventually a removal leaves no remaining data, put these last points back and let their vector mean be the median. (For a continuous distribution, I suspect that by taking an arbitrarily large finite sample, the result of convex peeling should converge, with probability 1, to a point depending only on the original distribution. Better yet, there's probably some differential equation that implements this limit without resorting to samples. WPT ?) --Dan