Maximum entropy probability distribution with known quantile - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T06:53:27Z http://mathoverflow.net/feeds/question/91751 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91751/maximum-entropy-probability-distribution-with-known-quantile Maximum entropy probability distribution with known quantile Trevor Stewart 2012-03-20T20:42:21Z 2012-03-20T21:20:33Z <p>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.</p> <p>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. </p> <p>Thanks,</p> <p>Trevor </p> http://mathoverflow.net/questions/91751/maximum-entropy-probability-distribution-with-known-quantile/91754#91754 Answer by R Hahn for Maximum entropy probability distribution with known quantile R Hahn 2012-03-20T21:20:33Z 2012-03-20T21:20:33Z <p>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. </p> <p>A related fact is that quantiles are not sufficient statistics for any distributions on \$\mathbb{R}\$, as noted <a href="http://www.stats.ox.ac.uk/~steffen/teaching/grad/partial.pdf" rel="nofollow">here</a> on page 17.</p>