Define the ei as in (2.3), and let f(e) be the density function of e. Then note that for every real θ,                        θ = IE(e + θ) = Z ∞ −∞ (e + θ)f(e)de = Z ∞ −∞ ef(e − θ)de, and that by...

Define the ei as in (2.3), and let f(e) be the density function of e. Then note that for every real θ,

                       θ = IE(e + θ) = Z ∞ −∞ (e + θ)f(e)de = Z ∞ −∞ ef(e − θ)de,

and that by differentiating with respect to θ we get

                          1 = Z ∞ −∞ e{−f ′ (e − θ)/f(e − θ)}f(e − θ)de, ∀θ.

Under θ = 0, the Cauchy–Schwartz inequalityimplies that

                         1 ≤ ³ Z ∞ −∞ e 2 dF(e) ´³ Z ∞ −∞ {−f ′ (e)/f(e)} 2 (e) = σ 2 I(f).