Linear operator over a simplex space in a multinomial distribution parameter estimation problem

Question

This is actually a variant of a well-known problem of how the parameters of a multinomial distribution can be estimated by maximum likelihood, and this arises from a final year project I undertook about single-cell RNA-seq. But I don't know how (or whether) it can be solved.

Settings

Define $S_{D}=\{\mathbf{x_{i}}\in\mathbb{R}^{D}:0\leq x_{ij}\leq1\:\forall j\:,\sum_{j}x_{ij}=1\}$ to be a simplex space of dimension $D-1$. Let $L(S_D)$ denote the set of all linear operators over $S_D$.

Let $\mathbf{Y}_i \sim \text{Multinom}(N,P\mathbf{a})$, where $\mathbf{a}$ is an unknown fixed parameter in $S^D$, with the unknown fixed operator $P\in L(S_D)$. $N$ is a known positive integer.

Problem Statement

Given the data $\{\mathbf{y}\}_{i\leq n}$,

1. how can/is it possible that $\mathbf{a}$ and $P$ (under the canonical basis) be estimated by maximum likelihood?
2. In fact, is there an estimator $\hat{\mathbf{a}}$ of $\mathbf{a}$ whose asymptotic distribution $\hat{\mathbf{a}}-\mathbf{a}$ can be computed?
3. Furthermore, if we are allowed to have exactly one estimate of $P$ that can be arbitrarily close to $P$ under some matrix norm, does there exist an algorithm that can compute for each iteration an estimator of $\mathbf{a}$ that converges in probability to $\mathbf{a}$?
4. Finally, suppose we define the estimator $$\hat{\mathbf{a}}_1 = \frac{\sum_{i}^n\hat{P}^{-1}y_i}{n}$$ is the asymptotic distribution of $\hat{\mathbf{a}}_1$ converging in distribution to some distribution indepedent of $P$ and $\mathbf{a}$ if $\|\hat{P}^{-1}P-I\| \leq \delta$ for some $\delta>0$, and $\hat{P} \in L(S_D)$ is symmetric?

Answer 1

This seems to be a very complicated problem. May I know the exact scientific question that leads to this problem?

However, I hope you are aware that $S_D$, if it inherits the operations of the D-dimensional Euclidean space $\mathbb{R}^D$, then it is not a vector space at all, and there can be no such $L(S_D)$. However, if $S_D$ is conferred the Aitchison geometry, it will be a vector space.

Some observations: if we allow $S_D$ not to be a vector space and to inherit all structures from $\mathbb{R}^D$, let $C_D$ be the set of all linear operators over $\mathbb{R}^D$ satisfying

1. matrix representations under the standard basis contain only column vectors from $S_D$.
2. it maps vectors in $S_D$ to $S_D$

It can then be observed that for a vector $y\in S_D$, if $Q\in C_D$ contains only $y$ as its column vectors, then $Qy=y$.

This implies that there is no hope in solving the minimisation problem to get a good answer, that is the following problem

$$\min_{P , x\in S_D} \|Px -y \|$$

because $P = I, x=y$ and $P = Q, x=y$ would be two solutions.