On stochastic orders of absolute value of order statistics in symmetric distributions

Let Y₁, ..., Y_n be the order statistics of a simple random sample from a finite or infinite population, having median = M. We compare the variables | Y_j - M | and | Y_m - M |, where Y_m is the sample median, that is, m = frac(n + 1, 2) for odd n. The comparison is in terms of the likelihood ratio order, which implies stochastic order as well as other orders. The results were motivated by the study of best invariant and minimax estimators for the k / N quantile of a finite population of size N, with a natural loss function of the type g (| F_N (t) - frac(k, N) |), where F_N is the population distribution function, t is an estimate, and g is an increasing function.