Section 4.4.6 showedthat E(s2) = 2. Forindependentsamplingfroma N(, 2) distribution, 2 = [i(YiY )2]~n is theMLestimatorand 2

Section 4.4.6 showedthat E(s2) = σ2. Forindependentsamplingfroma N(μ, σ2) distribution,

ˆσ2 = [Σi(Yi−Y )2]~n is theMLestimatorand ˜σ2 = [Σi(Yi−Y )2]~(n+1) is theestimatorhaving minimum MSE.

(a) Showthat ˜σ2 is anasymptoticallyunbiasedestimatorof σ2.

(b) Showthat ˆσ2 is aconsistentestimatorof σ2. (Hint: Applythelawoflargenumbersto

[Σi(Yi − μ)2]~n and showthatitsdifferencefrom ˆσ2 goesto0as n increases.)