UsetheMarkovinequality(Exercise2.50)toshowthatas n : (a) If is mean-squared-errorconsistent in thesensethat E( )2 0, then

UsetheMarkovinequality(Exercise2.50)toshowthatas n → ∞:

(a) If ˆθ is mean-squared-errorconsistent in thesensethat E(ˆθ − θ)2 → 0, then ˆθ

p→

θ.

(b) If E(ˆθ) → θ and var(ˆθ) → 0, then ˆθ

p→

θ. (Here,useMSE(ˆθ) = var(ˆθ) + (bias)2.)