# conjecture for general form of minimax estimator

I had previously posed an overly ambitious version of this conjecture here, Form of minimax estimator, which was quickly shot down by Václav Voráček (on twitter) and Iosif Pinelis (MO answer in the link).

The current conjecture is more modest. First I'll restate the problem.

Let $$\Delta$$ be the set of all probability distributions over $$\mathbb{N}=\{1,2,\ldots\}$$ and fix some $$\mathcal{P}\subseteq\Delta$$.

Assumptions on $$\mathcal{P}$$:

1. every $$\mathcal{P}$$ is permutation-invariant -- i.e., for every permutation $$\Pi$$ on $$\mathbb{N}$$ and every $$p\in\mathcal{P}$$, we have $$\Pi(p)\in\mathcal{P}$$.
2. $$\mathcal{P}$$ is a convex set.

Suppose additionally that $$\Delta$$ is endowed with some norm $$||\cdot||$$.

An estimator $$\hat p_n$$ for $$\mathcal{P}$$ is a mapping $$\mathbb{N}^n\to\Delta$$. We say that $$p_n^*$$ is a minimax optimal estimator if $$\inf_{\hat p^n}\sup_{P\in\mathcal{P}} \mathbb{E}||\hat p_n(X_1,\ldots,X_n)-P|| = \sup_{P\in\mathcal{P}} \mathbb{E}||p_n^*(X_1,\ldots,X_n)-P||,$$ where the infimum is over all estimators and the expectation is over $$n$$ independent copies of $$X\sim P$$.

I will now state the conjecture, in a strong form and a weak form.

Strong form. The minimax estimator $$p_n^*$$ can always be chosen to be of the following form: its $$i$$th coordinate is given by some function $$f:[0,1]\to[0,1]$$ of the empirical frequency of the $$i$$th symbol, $$f: n^{-1}\sum_{j=1}^n1[X_j=i] \mapsto [0,1],$$ where $$f$$ is fixed, deterministic, and entirely determined by $$\mathcal{P}$$ (and possibly $$n$$) and is further monotonically nondecreasing. (For instance, if $$\mathcal{P}=\Delta$$ and $$||\cdot||$$ is either $$\ell_1$$ or $$\ell_\infty$$, then minimax optimality is attained by the Maximum Likelihood Estimator, where $$f$$ is the identity function.)

Weak form. Same as the Strong form, with the additional conditions that $$||\cdot||=\ell_\infty$$ and $$\mathcal{P}$$ consists of all $$p\in\Delta$$ satisfying $$p_i\le v$$ for a given fixed $$v$$.

For both the strong and weak forms, I am also allowing improper estimators -- i.e., $$\hat p_n$$ whose range is not limited to $$\Delta$$.