# Estimate unknown sum of iid random variables

Let $$X_1, X_2, \dots$$ be a sequence of independent and identically distributed discrete random variables with common mass function $$f_X(x)$$ defined for when $$x \in \{0,1,\dots,N\}$$ and $$N$$ is known.

I know the form of $$f_X(x)$$ (I can computationally compute this mass over the range).

Now define $$S = \sum_{i=1}^{M} X_i$$ to be the $$M$$-sum of these random variables on the state space $$\{0,1,\dots,NM\}$$.

The problem that I have is that I observe realisations of $$S$$, where $$M$$ is the unknown parameter. Does anyone have any ideas as to how I can estimate $$M$$?

The idea I had was to try to compute the mass function for $$S$$ using $$M$$ convolutions but am not sure if this blows up computationally as $$N,M$$ increase. From this, I think I can search to find the $$M$$ which gives the likelihood its highest value, but again I believe this will run into computational issues. To make this more precise, typically $$N \approx 10^4$$ and $$10^2 \leq M \leq 10^4$$.

Any help would be greatly appreciated! Thanks

• Can't you use $E[S]=M E[X]$? – user121049 Nov 1 '18 at 23:12
• Method of moments is plausible, however, if my sample size (of S) is small, then I reckon this will be too biased. Looking for a way to exploit the exact distribution of $X$ if possible, but thanks for your answer :) – user202654 Nov 2 '18 at 13:17
• With your $M$ being quite large can't you claim by Central Limit theorem that $S$ is distributed normally. – user121049 Nov 5 '18 at 8:25