For continuous distributions on x>0 with known mean m, the exponential distribution f(x) = (1/m)exp(-x/m) is the maximum entropy distribution, with entropy H(f) = ln(m)+1. I have a problem where I know the P-th quantile Q and I want to know the maximum entropy distribution with that quantile.


The exponential distribution with P-th quantile Q has mean m = Q/ln(1-P). As stated this is the maximum entropy distribution for all distributions with mean m. Is it also the maximum entropy distribution over all continuous distributions with P-th quantile Q? If not, what is the maximum entropy distribution. Any help greatly appreciated. 


Thanks,


Trevor 



Maximum entropy probability distribution with known quantile
The quantile alone is insufficient to define a maximum entropy density. Intuitively this is because the quantile is a single point and is not enough to prescribe an entire density; you must specify additional moments. 


A related fact is that quantiles are not sufficient statistics for any distributions on $\mathbb{R}$, as noted [here](http://www.stats.ox.ac.uk/~steffen/teaching/grad/partial.pdf) on page 17.


