Answer in Statistics and Probability for Shivam Nishad #87366
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}displaystylesum_{i=1}^n(X_i-bar{X})^2)=(bar{X}, {n-1 over n}S^2)”
“np={1 over n}displaystylesum_{i=1}^nX_i=bar{X}”
“np(1-p)={1 over n}displaystylesum_{i=1}^n(X_i-bar{X})^2”
“S^2 ={1 over n-1}displaystylesum_{i=1}^n(X_i-bar{X})^2”
“1-p={{1 over n}displaystylesum_{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}Rightarrowwidehat{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.
