Consider $n$ independent and identically distributed random variables $ \{X_i\}_{i=1,...n} $ with support on some interval $[a,b]$ and its $n$'th order statistic $\max_{i \in \{1,...n\}} X_i$ . The entropy of the maximum is
$ - \int_a^b F^n(x) \ln F^n(x) dx ,$ where $F(x)= \Pr (X \le x) $. It seems natural that the entropy should be decreasing in $n$ (just think about $n$ very large). Is this a known result?
I did in fact prove that the entropy is monotone, but the proof turned out to be lengthy and messy. I would expect that there is a simple argument. Does anyone know?