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
and
EX12=Var(X1)−(EX1)2=σ2+μ2
EX12=kp(1−p)+k2p2
Setting μ1=μ and μ2=σ2+μ2 we obtain the moment estimators
θ=(Xˉ,n1i=1∑n(Xi−Xˉ)2)=(Xˉ,nn−1S2)
np=n1i=1∑nXi=Xˉ
np(1−p)=n1i=1∑n(Xi−Xˉ)2
S2=n−11i=1∑n(Xi−Xˉ)2
1−p=Xˉn1i=1∑n(Xi−Xˉ)2=nn−1S2/Xˉ
p=(μ1+μ12−μ2)/μ1=1−nn−1S2/Xˉ
and
kp=Xˉ⇒k=pXˉ
k=μ12/(μ1+μ12−μ2)=Xˉ/(1−nn−1S2/Xˉ)
The estimator p is in the range of (0, 1).
But k may not be an integer.
It can be improved by an estimator that is k rounded to the nearest positive integer.
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