Stat 312: Lecture 03Minimum Variance Unbiased EstimatorMoo K. Chung[email protected]September 9, 20041.Populationof interest is a collection of measur-able objects we are studying. LetX1,· · ·, Xnbea random sample from the population. Then sam-ple mean¯Xand sample varianceS2are unbiasedestimators of population meanμand populationvarianceσ2respectively.Proof.Note thatS2=1n-1Xi=1(Xi-¯X)2=1n-1hnXi=1X2i-n¯X2i.Then using the factE(¯X)2=V¯X+ (E¯X)2=σ2/n+μ2, it can be shown thatES2=σ2.2.There may be many unbiased estimators ofθ.Given two unbiased estimatorsˆθ1andˆθ2ofθ. Wechoose one that gives less variance. IfV(ˆθ1)≤V(ˆθ2),ˆθ1is called moreefficientthanˆθ2. An effi-cient estimator has less variability so we are morelikely to make an estimate close to the true param-eter value. The following coin flipping exampleclearly demonstrate this.