Answer to Question #87366 in Statistics and Probability for Shivam Nishad

Question

Let X1, X2, ..., Xn be a random sample from a Binomial distribution with parameters n
and p, both unkown. Obtain estimators of n and p, using method of moments.

Answer 1

Since

$$\mu=EX_1=kp$$

and

$$EX_1^2=Var(X_1)-(EX_1)^2=\sigma^2+\mu^2$$

$$EX_1^2=kp(1-p)+k^2p^2$$

Setting $\widehat{\mu}_1=\mu$ and $\widehat{\mu}_2=\sigma^2+\mu^2$ we obtain the moment estimators

$$\widehat{\theta}=(\bar{X}, {1 \over n}\displaystyle\sum_{i=1}^n(X_i-\bar{X})^2)=(\bar{X}, {n-1 \over n}S^2)$$

$$np={1 \over n}\displaystyle\sum_{i=1}^nX_i=\bar{X}$$

$$np(1-p)={1 \over n}\displaystyle\sum_{i=1}^n(X_i-\bar{X})^2$$

$$S^2 ={1 \over n-1}\displaystyle\sum_{i=1}^n(X_i-\bar{X})^2$$

$$1-p={{1 \over n}\displaystyle\sum_{i=1}^n(X_i-\bar{X})^2 \over \bar{X}}={n-1 \over n}S^2/\bar{X}$$

$$\widehat{p}=(\widehat{\mu}_1+\widehat{\mu}_1^2-\widehat{\mu}_2)/\widehat{\mu}_1=1-{n-1 \over n}S^2/\bar{X}$$

and

$$\widehat{k}\widehat{p}=\bar{X}\Rightarrow\widehat{k}={\bar{X} \over \widehat{p}}$$

$$\widehat{k}=\widehat{\mu}_1^2/(\widehat{\mu}_1+\widehat{\mu}_1^2-\widehat{\mu}_2)=\bar{X}/(1-{n-1 \over n}S^2/\bar{X})$$

The estimator $\widehat{p}$ is in the range of (0, 1).

But $\widehat{k}$ may not be an integer.

It can be improved by an estimator that is $\widehat{k}$ rounded to the nearest positive integer.