# What is the maximum entropy distribution given values for several quantiles of one sample?

Suppose I know the value of n specific quantiles from a large sample. What is the maximum entropy distribution over the real line given those values?

• Have you tried formulating and solving this as an optimization problem of maximizing entropy subject to equality constraints on the n known quantiles? I think you should be able to get the idea of things by Googling and looking at similar problems. Aug 11, 2016 at 1:47
• I'll poke about and see what I can find. But if g1ul10's response, below, is correct -- and I think it is -- then that approach will not work without further constraints. Aug 18, 2016 at 0:23
• Here is a similar problem with answers: stats.stackexchange.com/questions/361476/… Jul 1, 2023 at 1:08

Maximum entropy problems do not always admit a solution. The generic expression for the maximum entropy density $f(x)$ given a set of integral constraints $$\int dx \, h_i(x) \, f(x) = c_i$$ with $i=1\ldots N$ is $$f(x) = e^{\mu + \sum_{i=1}^N \lambda_i h_i(x)} \;.$$ The values of the parameter $\mu$ and $\lambda_i$ have to be found by imposing the fulfillment of the constraints and the fact the $f(x)$ integrates to one, that is it is a proper density. I left the boundaries of the integral in the constraints unspecified on purpose. The reason will become clear in a moment.
The quantile constraints $F(x_i)=q_i$ where $F$ is the distribution function (the integral of the $f(x)$), translate in having $h_i(x)=1-\theta_{x_i}(x)$ and $c_i=q_i$, where $\theta_z(x)$ is the Heaviside theta function, which is equal to $1$ if $x>z$ and zero otherwise. The problem is that now the expression for $f(x)$ given above is not an (improper) integrable function on the real line. Summarizing, your problem does not have a solution. If however you add further constraints, like a finite support or some specified moment, than the problem might become solvable.
• @andrewH No. The solution would have a constant behavior for large $x$ and the integral on the positive half-line would not exist. You could find a solution if you consider a finite domain, like $[0,1]$. In this case, however, the solution will be rather trivial, with a step-wise shape. Aug 29, 2016 at 8:13