Page 390 - Probability and Statistical Inference

7. Point Estimation 367

The expression on the rhs of the inequality in (7.5.1) is called the Cramér-Rao
lower bound (CRLB).
Proof Without any loss of generality, let us assume that 0 < V (T) < ∞. By
θ
the definition of expectation, we can write

which implies that

Now, observe that

so that by using the chain rule of differentiation, we get

Next, let us denote Y = . ] Note that Y is not an observable
random variable because it involves the unknown parameter θ. We now com-
bine (7.5.4) and (7.5.5) to rewrite

Also, one obviously has ∫ f(x; θ)dx = 1 so that one writes
χ

Hence, we have

for all θ ∈ Θ, since the X’s have identical distributions. Thus, (7.5.6) leads us
to conclude that