First of all I'd like to precise that I'm not an expert of the subject.

Suppose to have two random variables $X$ and $Y$ that are binomial, respectively $X\sim B(n_1,p)$ and $Y\sim B(n_2,p),$ note here that $p$ is the same. I know that $Z=X+Y \sim B(n_1+n_2,p).$

Let $\{x_1,\ldots,x_k\}$ be a sample for $X$ and $\{y_1,\ldots,y_k\}$ be a sample for $Y$, is there a standard method for estimating $n=n_1+n_2$ and $p$?

This is what I have done:

  1. take the "new sample" for $Z$ given by $\{x_1+y_1,\ldots, x_k+y_k\}$,
  2. using the Likelihood Estimator, obtain an estimation for $n$ and $p$,
  3. use he Fisher information in order to obtain the error over $n$ and $p$.

The method seems to work, but I have still some doubts. Let $S_k$ the group of permutation over $k$ elements. For every $\sigma\in S_k$ we can consider the "sample" given by $\{x_1+y_{\sigma(1)},\dots, x_k+y_{\sigma(k)}\}.$ Applying the Likelihood Estimator for each one of the "new samples" (there are $k!$ different sums) we obtain different estimation for $n$ and $p$.

What is the meaning of this? It can be used for calculating the error for $n$?

  • $\begingroup$ The real issue is that the errors in your estimates for $n$ and $p$ are highly (and negatively) correlated, so while you make a reasonable estimate $np$ as something close to $\frac1k(\sum x_i + \sum y_i)$, the potential relative errors on estimating $n$ and $p$ individually could be much larger. $\endgroup$ – Henry Nov 6 '14 at 14:27
  • $\begingroup$ @Henry: In the example that I have in mind $n$ is the number of birds in a given region and $p$ the visibility probability. I need to aggregate regions with similar $p$, otherwise the data are too small. In particular I need, if possible, an estimation only for $n$, where $p$ a priori is unknown. $\endgroup$ – amorvincomni Nov 6 '14 at 15:22
  • $\begingroup$ I recommend posting the question on stats/crossvalidated SE. $\endgroup$ – Nameless Nov 6 '14 at 23:25
  • $\begingroup$ @Nameless : I have posted the Q also here Note: in a preceding comment that I have deleted there was a wrong link. $\endgroup$ – amorvincomni Nov 11 '14 at 11:56
  • $\begingroup$ I suggest reading, Erdogan Günel & Daniel Chilko (1989) Estimation of Parameter n of the Binomial Distribution, Communications in Statistics - Simulation and Computation, 18:2, 537-551, DOI: 10.1080/03610918908812775 $\endgroup$ – confused_dragon Sep 1 '15 at 23:07

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