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$\newcommand{\Beta}{\operatorname{Beta}}$I'm sampling a bunch of probabilities, $\theta_i \sim \Beta(a,b)$, from a common beta distribution, and then using each $\theta_i$ to sample a value $x_i$ out of $N_i$ possibilities. However, I am only able to see $x_i$ if $1 \leq x_i \leq N_i-1$, so there is 2-sided truncation of the data as well. Also, note that the family of beta distributions I'm interested in tend to have both $a$ and $b$ less than 1.

Given the situation, I would like to be able to estimate the beta parameters, $a$ and $b$ given only the observed $x_i$ and $N_i$ values. I have tried to incorporate the correct (I think) truncated binomial likelihood: $\left(\text{i.e. }\frac{\theta_i^{x_i}(1-\theta_i)^{N-x_i}}{1-\theta_i^N-(1-\theta_i)^N}\right) \, ,$ within a Laplace approximation to the marginal likelihood $\left(\text{marginal with respect to }\log\left(\frac{\theta_i}{1-\theta_i}\right)\right)$ to no avail (I always get overestimates of $a$ and $b$).

Does anyone have a good idea of how to do this, or is this simply a fool's errand without extremely large $N_i$ values?

Edit:
I'm trying this out simulating $\theta_i$ from a $\Beta(0.08,0.72)$ distribution, with all $N_i=1999$ for simplicity (though in general, they could be different).

Some example $(x_i, N_i)$ pairs are: $$[(442,1999),(1,1999),(22,1999),...,(5,1999),(601,1999),(737,1999)].$$ Note that any $x_i=0$ or $x_i=1999$ are excluded for this problem.

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  • $\begingroup$ Can you provide some actual values for the (x_i, N_i) pairs? $\endgroup$ – mef Dec 7 '15 at 23:09
  • $\begingroup$ You did not specify the distribution of $x_i \mid \theta_i$ so there is little to go o ... $\endgroup$ – kjetil b halvorsen Sep 16 at 8:12

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