The trimmedmean calculates thesamplemeanafterexcludingasmallpercentageofthe lowestandhighestdatapoints,tolessentheimpactofoutliers.Forexample,withthefunction mean(y, 0.05) applied toavariable y, R finds the 10% trimmedmean, excluding5%ateach end. Foraheavy-tailedsymmetricdistribution,thetrimmedmeanisoftenmoreefficientthan the samplemeaninestimatingcenter.Toillustrate,randomlygenerate1000observationsfrom the standardCauchydistribution(4.11),forwhich E(Y ) doesnotexistbutthemedianand trimmed meanare0.

(a) Usethebootstraptofinda95%confidenceintervalusingthe10%trimmedmean.Com-

pare withtheordinary95%confidenceinterval(4.10)forthemean.

(b) Usethebootstraptofinda95%confidenceintervalforthemedian.Comparetothose for themeanandtrimmedmean.Whatdoesthissuggestabouttheseestimatorsfor heavy-taileddistributions?