Hi! Can you help me with this:
we have X=(X1, X2, ... Xn) - iid, Xi ~ N(m,1), I need to prove that distribution of med(X)-X(j) is the same as distribution of X(n-j+1)-med(X), where med(X) is a sample median and X(j) is j-th order statistic.
well I tried something like this:
med(X)=X(k), then get joint distribution of (X(j), X(k)) and through linear transformation get distribution of (X(k)-X(j), X(j)) and then use a fact that X(k)-X(j) and X(j) are independent to get the distribution of X(k)-X(j) (but I dont know how to prove it that they are independent ) and then do this same with X(k) and X(n-j+1).
And it didn't work! (maybe I did something wrong or this is a bad idea)...
but I think it works only for n-odd...