Here is another question about the median: is there a median that makes sense in two or more dimensions?
Yes, the maximium likelihood point estimate of the center of the Lapace distribution I gave in my previous email makes sense in any dimension, and with suitable (positive definite) choice of S, will satisfy sum_i | x_i - mu | / ( x_i - mu ) = 0
Here's an interesting question: suppose we have data X_1, ..., X_n drawn from a Gaussian distribution with unknown mean mu and known variance 1. We wish to estimate mu with a guess muhat. Virtually everyone uses the sample mean of the dataset as an estimate of mu, but note that mu is also the *median* of the distribution. Under what circumstances would we be justified in prefering the sample median of the data to estimate mu?
If the data was drawn from a Laplace rather than a Gaussian distribution.
Since the sample average is a sufficient statistic, the answer might be never, but I'm not sure. Might it be the case that that the sample median is preferable if we are using L1 loss, i.e., seeking to minimize E_mu |mu - muhat| ?
As others have pointed out, the posterior distribution of mu is symmetric about the average, so any reasonable loss function (including L1 loss) is going to be minimized at the average. -Thomas C