Most likely event in a multinomial distribution setting

I'm looking at the following scenario:

$k$ categories, distributed by a multinomial ($p_1,\dots,p_k$) such that $p_1 \ge \dots \ge p_k$. Draw $n$ samples. I'm interested in estimators/lower bounds for $p_1$ in a scenario where I don't know which category has what probability under the multinomial distribution. For instance if you have 3 categories $A, B, C$, then $\mathrm{Pr(sample} \, \, A)$ could be $p_1$ or it could be $p_2$ or even $p_3$. You can think of this as trying to understand the a-priori 'dominant' event after actually observing $n$ samples, without actually knowing which one the dominant event was, atleast according to the distribution.

Any leads in this area would be helpful -- if you have suggestions (on how to approach this), keywords for Googling in this area or even research papers that look at this problem. I really appreciate your help!

• Is there any reason not to treat the most common category as the "dominant" one? Otherwise you'll be wrong with high probability the more data you have. – dsaxton Mar 4 '16 at 15:07
• Bechhofer, Elmaghraby, and Morse (1995) "A Single-Sample Multiple-Decision Procedure for Selecting the Multinomial Event Which Has the Highest Probability" projecteuclid.org/download/pdf_1/euclid.aoms/1177706362 – jan-glx Aug 10 '20 at 16:46

Let $$X \sim \mathcal{Multinom}(n,p_1,\dotsc,p_k)$$ and we want to find the index corresponding to the largest probability $$p_i$$, that is, finding $$\DeclareMathOperator*{\argmax}{arg\,max} j=\argmax_i p_i$$. As in the answer by @bdeonovic, the obvious estimator is $$\hat{j}=\argmax_i X_i/n$$, but we could also take the question as asking about additional inference, that is, some measure of uncertainty in that estimate.
The idea is to select a subset of indices such that the probability of that subset containing $$j$$ is at least some prespecified probability $$P^*$$, often chosen as $$P^*=0.95$$. So this is some kind of analogy with a confidence interval.
The rule Gupta proposes is to select all indices $$i$$ such that $$X_i \ge X_{\text{max}}-D$$ where $$D$$ is an integer chosen such that the $$P^*$$-condition is fulfilled. For that the coverage probability must be calculated for the full parameter space, and then the minimum found. He argues by numerical experiments that the worst configuration is $$p=(1/k, \dotsc, 1/k)$$. Then the paper gives some tables.
Lets say you have categories $$1,\ldots,k$$. Let $$x_1,\ldots,x_k$$ be the number observed in the corresponding categories. The MLE of the probabilities is $$x_i/n$$ and I think the best estimate you can get of $$p_1 > p_2 > ... > p_k$$ is if you ordered the MLE estimates.