Bias-variancetradeoff2: Somestatisticalmethodsdiscussedin Chapters 7 and 8 use a shrinkage orsmoothingoftheMLestimator,suchthatastheamountofshrinkagein- creases, thebiasincreasesbutthevariancedecreases.Toillustratehowthiscanhappen,for Y binom(n, ), let

Bias-variancetradeoff2: Somestatisticalmethodsdiscussedin Chapters 7 and 8 use a

“shrinkage” or“smoothing”oftheMLestimator,suchthatastheamountofshrinkagein-

creases, thebiasincreasesbutthevariancedecreases.Toillustratehowthiscanhappen,for Y ∼ binom(n, π), let ˜π = (Y + c)~(n + 2c) beanestimatorof π for aconstant c ≥ 0.

(a) Showthat ˜π is aweightedaverageof ˆπ = Y ~n and 1/2,thusshrinkingtheMLestimator

ˆπ toward1/2,withgreatershrinkageas c increases.

(b) Findthebiasof ˜π and var(˜π). Showthatas c increases, the|bias|increasesandthe variancedecreases.

(c) Explainhowtheestimate ˜π = (y + c)~(n + 2c) results fromaBayesianapproachtoesti-

mation. Explaintheinfluenceof n on theextentofshrinkagetoward1/2.