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I repeatedly measure a value $S_n$ which is the sum of a set of $n$ hidden inputs. The goal is to identify the number of hidden inputs.

All of the hidden inputs are driven by an experimenter controlled stimulus. The stimulus is quantal (consists of all or none events) but multiple events can occur in each input. For each stimulus the probability density function of the input's activation level can be estimated. The random result of this activation is then scaled by a variable input specific weight $w_n$ before being summed into $S_n$.

If I use the central limit theorem then I can find $\sigma*{\sqrt n}$ by a least squares curve fit of the sample's cumulative distribution to the standard normal cdf. But I can't then solve for $n$ since the unknown set of $w_n$ makes $\sigma$ unknown.

My current thought is to use the Berry-Esseen theorem to try and get another handle on $n$ based on the convergence rate to normality. But I'm unsure how to proceed with the supremum in that function and would prefer to avoid the inequality (confidence intervals would be perfect).

How would you proceed?

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