Let Y denote the median and let ˉX denote the mean, of a random sample of size n=2k+1 from a distribution that is N(μ,σ2). How can I compute E(Y|ˉX=ˉx)?
Intuitively, because of the normality assumption, it makes sense to claim that E(Y|ˉX=ˉx)=ˉx and indeed that is the correct answer. Can that be shown rigorously though?
My initial thought was to approach this problem using the conditional normal distribution which is generally a known result. The problem there is that since I do not know the expected value and consequently the variance of the median, I would have to compute those using the k+1st order statistic. But that is very complicated and I would rather not go there unless I absolutely have to.
Let X denote the original sample and Z the random vector with entries Zk=Xk−ˉX. Then Z is normal centered (but its entries are not independent, as can be seen from the fact that their sum is zero with full probability). As a linear functional of X, the vector (Z,ˉX) is normal hence the computation of its covariance matrix suffices to show that Z is independent of ˉX.
Turning to Y, one sees that Y=ˉX+T where T is the median of Z. In particular, T depends on Z only hence T is independent of ˉX, and the distribution of Z is symmetric hence T is centered.