Form of minimax estimator

Let $$\Delta$$ be the set of all probability distributions over $$\mathbb{N}=\{1,2,\ldots\}$$ and fix some $$\mathcal{P}\subseteq\Delta$$. 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 conjecture that $$p_n^*$$ can always be chosen to be of the following form: its $$i$$th coordinate is given by some function $$f_i:[0,1]\to[0,1]$$ of the empirical frequency of the $$i$$th symbol, $$f_i: n^{-1}\sum_{j=1}^n1[X_j=i] \mapsto [0,1],$$ where the $$f_i, i\in\mathbb{N}$$ are fixed, deterministic, and entirely determined by $$\mathcal{P}$$ and are 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 the $$f_i$$ are the identity function.)

Question: Is this true? Known?

Update: Václav Voráček provided what appears to be a counterexample: $$P(1)>0\implies P(2)=c$$ for some constant $$c$$, for all $$P\in\mathcal{P}$$, and $$P(1)=0\implies P(2)=0$$.

Hence I additionally assume that $$\mathcal{P}$$ is permutation-invariant, in which case I conjecture that all of the $$f_i$$ can be taken to be identical.

Update II. I don't mind if $$f$$ is also a function of $$n$$ (in addition to the first argument, which is the empirical frequency) -- as long as it is monotone in the first argument.

Update III. Following some discussion (mostly with Václav), I'm making the conjecture more modest yet. Let's restrict $$\mathcal{P}$$ to those distributions $$P$$ for which $$\sup_{i\in\mathbb{N}}P(i)(1-P(i))\le v$$ and the norm $$||\cdot||$$ to be $$\ell_\infty$$.

$$\newcommand\P{\mathcal P}\newcommand\N{\mathbb N}\newcommand\de{\delta}$$You wrote:

Hence I additionally assume that $$\mathcal{P}$$ is permutation-invariant, in which case I conjecture that all of the $$f_i$$ can be taken to be identical.

I do not think that this is true. For instance, suppose that $$\P$$ is the set of uniform distributions on subsets of $$\N$$ of cardinality $$2$$. That is, $$P\in\P$$ iff $$P=P_{a,b}:=(\de_a+\de_b)/2$$ for some distinct natural $$a$$ and $$b$$, where $$\de_a$$ is the Dirac probability measure supported on $$\{a\}$$.

Then I think a minimax estimator (which you referred to as "minimax optimal estimator") of $$P$$ does not exist. If we let $$\hat p_n(a):=\frac1n\,\sum_{i=1}^n 1(X_i=a)$$ and replace $$\N$$ by $$[N]:=\{1,\dots,N\}$$ for a natural $$N\ge2$$, then I think a (randomized) minimax estimator of $$P$$ will be

• $$P_{\hat a_n,\hat b_n}$$ if $$\{\hat a_n,\hat b_n\}=\hat A_n:=\{a\in[N]\colon \hat p_n(a)>0\}$$ and the cardinality $$|\hat A_n|$$ of the set $$\hat A_n$$ is $$2$$ and
• $$P_{\hat a_n,\hat c_n}$$ if $$A_n$$ is a singleton set $$\{\hat a_n\}$$ and, given, $$\hat a_n$$, the point $$\hat c_n$$ is uniformly distributed over the set $$[N]\setminus\{\hat a_n\}$$.

Finally, I think the minimax risk will be attained in the limit as $$N\to\infty$$.

Hopefully, it is not exceedingly hard to verify the above claims.