Asymptotic Efficiency Bounds

Any consistent and asymptotically normal estimator with a variance-covariance matrix of the stabilizing transformation attaining the Cramér-Rao efficiency bound is said to be asymptotically efficient (cf. Amemiya, 1985, p. 124). It is well known that the Cramér-Rao bound is given by the inverse of the information matrix. Throughout this chapter, let J(θ₀) denote the information matrix for a single observation, evaluated at the true parameter vector, defined as 5.1.1$$ J\left( {\theta _0 } \right) \equiv - E\left[ {\frac{{\partial ^2 \ln f\left( {Z|\theta _0 } \right)}} {{\partial \theta \partial \theta '}}} \right], $$ where ∂2 lnf (z | θ )/∂θ∂θ′ is the Hessian matrix for a single observation containing the second derivatives of its loglikelihood contribution ln f(z | θ). Let S(θ) ≡ ∂ln f (z | θ)/∂θ denote the vector of first derivatives of the loglikelihood contribution of a single observation, henceforth referred to as the score. Using the information matrix equality at the individual level, (5.1.1) can be rewritten as 5.1.2$$ J\left( {\theta _0 } \right) = E\left[ {S(\theta _0 )S(\theta _0 )'} \right] = V\left[ {S(\theta _0 )} \right], $$ which will be more convenient for the results stated in the following two sections.