2022-03-17 Answered
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Vasquez
Answered 2022-03-29 Author has 460 answers

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Given you have an independent random sample X1,X2,,Xn of a Bernoulli random variable with parameter p, estimate the variance of the maximum likelihood estimator of p using the Cramer-Rao lower bound for the variance
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since the probability function of X is binomial, we have:
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so: lnfX(X)=ln(ni=1nXi)+i=1nXiln(p)+(n-i=1nXi)ln(1-p)
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and: (lnfX(X)p)2=(nX¯p-(n-nX¯)(1-p))2=n2p2-2n2pX¯+n2X¯2p2(1-p)2
since E[X2]=μ2+σ2n, and for a Bernoulli random variable E[X]=μ=p=E[X] and Var[X]=σ2=p(1p):
E(lnfX(X)p)2=n2p2-2n2pE[X¯]+n2E[X¯2]p2(1-p)2=n2p2-2n2p2+n2(p2+p(1-p)n)p2(1-p)2=np(1-p)p2(1-p)2=np(1-p)
Therefore, Var[P^]=1nE[((/p) lnfx(X))2]=1nnp(1-p)=p(1-p)n2
However, I believe the true value I should have come up with is p(1p)n.

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