The standardestimator in thenormaldistributionisdistributedas S2n 22(n 1)/(n 1) and thus unbiased forthevariance 2, butnotfortheprecision

The standardestimator

in thenormaldistributionisdistributedas S2n ∼ σ2χ2(n − 1)/(n − 1) and thus unbiased forthevariance σ2, butnotfortheprecision ρ2 = 1/σ2, norforthestandard deviation σ, asdiscussedinExample4.3.

a) Findandcomparethemeansquareerrorsof Sn and theMLE ˆσn = Sn p (n − 1) as estimatesofthestandarddeviation σ

b) Determineaconstant an such that

is anunbiasedestimatoroftheprecision ρ2 for n > 3.

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n i=1 (X; -X)