Maximizing the uniformity of density function subject to moment constraints

Question

Background

I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below:

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Definition: Maximally Uniform Probability Measure

Let $X$ be a continuous random variable with probability measure defined on $\mathcal{D}_X=\mathcal{B}([a,b])$, and let $L_m=\{B \in \mathcal{D}_X: \lambda(B)=m\}$, where $\lambda(B)$ is the Lebesgue measure of Borel set $B$ and $0\leq m \leq \lambda([a,b])$. Let $C$ be a set of constraints on the probability measure of $X$, and $\mathcal{P}_C$ be the class of all continuous probability measures on $\mathcal{D}_X$ that satisfy $C$.

For given constraints $C$, a maximally uniform probability measure $P^*_X \in \mathcal{P}_C$ achieves the minimum maximum probability measure for every $L_m$, where minimization is over $\mathcal{P}_C$.

Specifically, for a set of constraints $C$ and associated probability measures $\mathcal{P}_C$, the maximally uniform probability measure $P^*_X \in \mathcal{P}_C$ satisfies the following condition:

$\left\{\max\limits_{B\in L_m}{P^*_X(B)}\leq \max\limits_{B\in L_m}{P_X(B)}\;\; \forall m\right\}\forall P_X \in \mathcal{P}_C$

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For example, if I specify the class of Borel sets of Lebesgue measure 3 (i.e., $L_3$), and pick a probability measure from $P \in \mathcal{P}_C$ then I would proceed to find the interval in $L_3$ that has the highest probability measure wrt $P$. If $P$ is maximally uniform, then when I do this exercise for any other $P' \in \mathcal{P}_C$, I will get a Borel set of Lebesgue measure 3 that contains more probability than $P$. This will be also be the case for any other class of Borel sets of given Lebesgue measure (i.e., there is nothing special about 3).

My thinking is that I can solve this problem as a variational optimization problem (at least for $-\infty < a\leq b,< \infty)$, by minimizing the path length of the distribution function $F$ associated with each $P_X \in \mathcal{P}_C$.

Below is the formulation of my problem as a variational-calculus problem Note: $f(x)=F'(x)=$ target density function associated with the maximally uniform probability measure.

$\min\limits_{f(x)} \int\limits_a^b \sqrt{1+f(x)^2} dx$

$s.t.$

$\int\limits_a^b f(x) dx = 1$

$\int\limits_a^b xf(x) dx = 0$

$\int\limits_a^b x^if(x) dx = a_i \;\;i\in \{S \subset \mathbb{N^+}/1\}$

$f(x) \geq 0$

Using the euler-lagrange euqations, I have solved the general formulation of the above to get the following implicit solution:

$\frac{f(x)}{\sqrt{{f(x)}^{2}+1}}= \sum\limits_{i\in S \cup \{0,1\}}\mu_i{x}^{i}$ where the $\mu_i$ are the langrange multipliers (one per constraint) that must be solved for to make this implicit solution satisfy the constraints.

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What I need help with

In order of priority (I will accept any answer that addresses either of these):

1. Is there an explicit solution of the above? (even if it uses infinite series)
2. Does the concept of uniformity via path length minimization admit pathological cases (in the continuous measure case only)?

Answer 1

I was able to solve this explicitly:

Given a set of moment constraints $\mathcal{G}_I(a,b,m_I)$, the maximally uniform density function, $S^*(t)$, will take the form:

$$S^*_I(t) = \frac{\sum\limits_{i \in I} \lambda_i t^i}{\sqrt{1-\left[\sum\limits_{i \in I} \lambda_i t^i\right]^2}}\mathbf{1}_{a\leq t\leq b}(t)$$

Where the Lagrange multipliers $\mathbf{\lambda}:=(\lambda_i)_I$ are chosen such that:

$$ S^*(t|\mathbf{\lambda}) \in \mathcal{M}_I(a,b,m_I)$$