Lorenz ordering of means and medians (Q1072214)

The authors introduce a partial ordering \(''\leq_ L''\) on the set of nonnegative random variables, which they call the Lorenz partial order. They show that \(X\leq_ LY\) if and only if Y is a dilation of X (that is, Eh(X)\(\leq Eh(Y)\) for every convex function h for which the expectations exist). They show that \(\bar X_ n\leq_ L\bar X_{n-1}\) where, for each m, \(\bar X_ m\) denotes the sample mean of m exchangeable random variables \(X_ 1,...,X_ m.\) They also show that if \(X_ 1,...,X_{2n+3}\) are independent and identically distributed random variables with a common symmetric density then the sample medians are Lorenz ordered, that is, \(X_{n+2,2n+3}\leq_ LX_{n+1,2n+1}\) where, for each i and j (i\(\leq j)\), \(X_{i,j}\), denotes the i-th order statistic in a sample of size j of independent identically distributed random variables \(X_ 1,...,X_ j\). Finally they show that \(\bar X_{2n+1}\leq_ LX_{n+1,2n+1}\) if the parent distribution of the underlying independent identically distributed random variables is either exponential or Bernoulli or uniform.