# Thread: method of moments estimator for NBinom

1. ## method of moments estimator for NBinom

Derive the method of moments estimator for pi for a sample of size n from a NBinom(r, pi)-distribution. (Treat r as known, as it would be in a typical situation where you would be collecting data by repeating the Bernoulli trials.)

I have no idea where to start. Help!

2. Hello,

It's just basic use of the method of moments oO

Let X ~ NBinom(r,p). Then $\displaystyle E[X]=r\cdot\frac{p}{1-p}$ (you can compute that yourself, or find that on the wikipedia).

Then we approximate by taking the empirical mean instead of the expectation and consider the estimator of p instead of p. Which gives $\displaystyle \bar X_n=\frac{\sum_{i=1}^n X_i}{n}=r\cdot\frac{\hat p}{1-\hat p}$

Hence $\displaystyle \hat p=\frac{\bar X_n}{r+\bar X_n}$

and that's all...