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

Question #87366
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.
1
Expert's answer
2019-04-09T10:40:44-0400

Since


μ=EX1=kp\mu=EX_1=kp

and


EX12=Var(X1)(EX1)2=σ2+μ2EX_1^2=Var(X_1)-(EX_1)^2=\sigma^2+\mu^2




EX12=kp(1p)+k2p2EX_1^2=kp(1-p)+k^2p^2


Setting μ^1=μ\widehat{\mu}_1=\mu and μ^2=σ2+μ2\widehat{\mu}_2=\sigma^2+\mu^2 we obtain the moment estimators

θ^=(Xˉ,1ni=1n(XiXˉ)2)=(Xˉ,n1nS2)\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=1ni=1nXi=Xˉnp={1 \over n}\displaystyle\sum_{i=1}^nX_i=\bar{X}

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




S2=1n1i=1n(XiXˉ)2S^2 ={1 \over n-1}\displaystyle\sum_{i=1}^n(X_i-\bar{X})^2

1p=1ni=1n(XiXˉ)2Xˉ=n1nS2/Xˉ1-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}





p^=(μ^1+μ^12μ^2)/μ^1=1n1nS2/Xˉ\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


k^p^=Xˉk^=Xˉp^\widehat{k}\widehat{p}=\bar{X}\Rightarrow\widehat{k}={\bar{X} \over \widehat{p}}


k^=μ^12/(μ^1+μ^12μ^2)=Xˉ/(1n1nS2/Xˉ)\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 p^\widehat{p} is in the range of (0, 1).

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

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



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