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I have some observations are recorded as Proportion, from a binomial distribution with unknown success numbers. Typically in standard case, we observe only number of successes say $X$ out of trials say $N.$

But in my case, I just see the proportion of successes $P_i$ without knowing $X_i$ and $N_i$. So given I have sequence of data $P_1, P_2, \ldots$, what can be the theoretical distribution, given that the legitimate range for those observation is $0, 1$ inclusive?

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    $\begingroup$ The distribution of $P_i$ depends on $N_i$, so if the $N_i$ are different for different $i$, there isn't a single distribution in the first place, as the $P_i$ are not identically distributed. Furthermore, as long as the $N_i$ are not regularly repeating in some sense, distributions can change all the time even in directions where they hadn't been before. $\endgroup$
    – Christian Hennig
    Commented Jun 17, 2022 at 20:06
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    $\begingroup$ The distribution of $P_i$ given $N_i$ is known, so if you have a prior distribution for $N_i$, you can compute an overall distribution for $P_i$. But without assuming anything about the $N_i$, there is no way. (This is the Bayesian answer, the first one was frequentist assuming the $N_i$ to be fixed.) $\endgroup$
    – Christian Hennig
    Commented Jun 17, 2022 at 20:08
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    $\begingroup$ Scaled binomial. But if you don't know $X$ or $N$, you can only make guesses as to what they might be. $\endgroup$
    – AdamO
    Commented Jun 17, 2022 at 20:38
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    $\begingroup$ When you have an upper bound for each $N_i$ and the $P_i$ are recorded with sufficient precision, often you can recover the $N_i$ and the $P_iN_i$ from the mere fact that both must be non-negative integers. I have done this to reverse-engineer data in a few questions here on CV (but, alas, it's not possible to search for them...) The method is a brute-force search: multiply $P_i$ by all values of $N_i$ from $1$ through the upper bound and identify any results that are unusually close to integers. Usually, choosing the smallest such $N_i$ leads to a conservative analysis. $\endgroup$
    – whuber ♦
    Commented Jun 17, 2022 at 21:25
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    $\begingroup$ My post at stats.stackexchange.com/a/309058/919 provides details of how to do such a search along with example code. $\endgroup$
    – whuber ♦
    Commented Jun 20, 2022 at 12:40

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