Editing Order statistic (section)

== Mutual Information of Order Statistics ==
The [[mutual information]] and [[f-divergence]] between order statistics have also been considered.<ref> A. Dytso, M. Cardone and C. Rush, "Measuring Dependencies of Order Statistics: An Information Theoretic Perspective," in 2020 IEEE Information Theory Workshop, 2021, doi: 10.1109/ITW46852.2021.9457617.</ref> For example, if the parent distribution is continuous, then  for all <math>1 \le r, m\le n</math>
: <math>
I(X_{(r)}; X_{(m)}) =  I(U_{(r)}; U_{(m)}), 
</math>
In other words, mutual information is independent of the parent distribution.  For discrete random variables, the equality need not to hold and we only have 
: <math>
I(X_{(r)}; X_{(m)}) \le  I(U_{(r)}; U_{(m)}),
</math>

The mutual information between uniform order statistics is given by 
: <math>
 I(U_{(r)}; U_{(m)}) = T_{m-1} + T_{n-r} - T_{m-r+1} - T_n
</math>
where 
: <math>
T_k = \log(k!) - kH_k
</math>
where <math>H_k</math> is the <math>k</math>-th harmonic number.