Bias-variance tradeoff : Forabinomialparameter , considertheBayesianestimator (Y + 1)~(n + 2) that occurswithauniformpriordistribution. (a) Deriveitsbiasanditsvariance.ComparethesewiththebiasandvarianceoftheML estimator = Y ~n. (b) ShowthatitsMSE = [n(1)+(12)2]~(n+2)2. For

Bias-variance tradeoff : Forabinomialparameter π, considertheBayesianestimator (Y +

1)~(n + 2) that occurswithauniformpriordistribution.

(a) Deriveitsbiasanditsvariance.ComparethesewiththebiasandvarianceoftheML estimator ˆπ = Y ~n.

(b) ShowthatitsMSE = [nπ(1−π)+(1−2π)2]~(n+2)2. For n = 10 and for n = 1000, plotthis and theMSEoftheMLestimatorasafunctionof π. Describethe π valuesforwhichthe MSE issmallerfortheBayesestimator.Explainhowthisreflectsitsshrinkagetoward 1/2 andthebias/variancetradeoff.