The Cramer-Rao lower bound is attainable if and only if we have equality in the Cauchy-Schwartz Inequality which implies that the score V(\textbf{X}, \theta)

V(X,) must have a representation of the form

V(\textbf{X}, \theta)=k_{n}(\theta )[W(\textbf{X})-h (\theta )],

V(X,)=kn

()[W(X)h()],

where the factor k_n(\theta)

kn

() is a proportionality factor, so it does not involve the data and the quantity W(\boldsymbol X)

W(X) is a statistic, i.e., it does not involve the parameter but just the data. Which form below indicates that the Cramer-Rao lower bound is NOT attainable for h(\theta) = \theta

h()=.