6.2. Rao–Cram ́er Lower Bound and Efficiency 363(R4)The integral∫
f(x;θ)dxcan be differentiated twice under the integral sign as
afunctionofθ.
Note that conditions (R1)–(R4) mean that the parameterθdoes not appear
in the endpoints of the interval in whichf(x;θ)>0 and that we can interchange
integration and differentiation with respect toθ. Our derivation is for the continuous
case, but the discrete case can be handled in a similar manner. We begin with the
identity
1=∫∞−∞f(x;θ)dx.Taking the derivative with respect toθresults in
0=∫∞−∞∂f(x;θ)
∂θdx.The latter expression can be rewritten as0=∫∞−∞∂f(x;θ)/∂θ
f(x;θ)f(x;θ)dx,or, equivalently,
0=∫∞−∞∂logf(x;θ)
∂θf(x;θ)dx. (6.2.1)Writing this last equation as an expectation, we have established
E[
∂logf(X;θ)
∂θ]
= 0; (6.2.2)that is, the mean of the random variable∂log∂θf(X;θ)is 0. If we differentiate (6.2.1)
again, it follows that
0=∫∞−∞∂^2 logf(x;θ)
∂θ^2
f(x;θ)dx+∫∞−∞∂logf(x;θ)
∂θ∂logf(x;θ)
∂θ
f(x;θ)dx.(6.2.3)The second term of the right side of this equation can be written as an expectation,
which we callFisher informationand we denote it byI(θ); that is,
I(θ)=∫∞−∞∂logf(x;θ)
∂θ∂logf(x;θ)
∂θf(x;θ)dx=E[(
∂logf(X;θ)
∂θ) 2 ]. (6.2.4)
From equation (6.2.3), we see thatI(θ) can be computed fromI(θ)=−∫∞−∞∂^2 logf(x;θ)
∂θ^2f(x;θ)dx=−E[
∂^2 logf(X;θ)
∂θ^2]. (6.2.5)
Using equation (6.2.2), Fisher information is the variance of the random variable
∂logf(X;θ)
∂θ ; i.e.,
I(θ)=Var(
∂logf(X;θ)
∂θ). (6.2.6)
Usually, expression (6.2.5) is easier to compute than expression (6.2.4).