\bigr]^{-1}
E_\theta\bigl\{\dot{g}(\theta)\bigr\}$$ is a natural measure of the informativeness of the estimating equations \[@Heyde, Definition 2.1\]. For positive semidefinite matrices $A$ and $B$, write $A\succeq B$ if $A-B$ is positive semidefinite. For unbiased estimating equations $g(\theta)=0$ and $h(\theta)=0$, then we can say $g$ dominates $h$ if $\mathcal{E}\{g(\theta)\}
\succeq\mathcal{E}\{h(\theta)\}$. Under sufficient regularity conditions on the model and the estimating equations, the score equations are the optimal estimating equations \[@Bhapkar\]. Specifically, for the score equations, the Godambe information matrix equals the Fisher information matrix, ${{\mathcal}I}(\theta)$, so this optimality condition means ${{\mathcal}I}(\theta)\succeq\mathcal{E}\{g(\theta)\}$ for all unbiased estimating equations $g(\theta)=0$. Writing $M_{ij}$ for the $ij$th element of the matrix $M$, for the score equations in (\[score\]), ${{\mathcal}I}_{ij}(\theta) = \frac{1}{2}\operatorname{tr}(K^{-1}K_iK^{-1}K_j)$ \[@stein-book, page 179\]. For the approximate score equations (\[ascore\]), it is not difficult to show that $E_\theta\dot{g}(\theta,N)=-{{\mathcal}I}(\theta)$. Furthermore, writing $W^i$ for $K^{-1}K_i$ and defining the matrix ${{\mathcal}J}(\theta)$ by ${{\mathcal}J}_{ij}(\theta) =
\operatorname{cov}(U_1'W^iU_1,U_1'W^jU_1)$, we have $$\label{B}
\operatorname{cov}_\theta\bigl\{g(\theta,N)\bigr\} = {{\mathcal}I}(\theta) +
\frac{1}{4N}{{\mathcal}J}(\theta),$$ so that $\mathcal{E}\{g(\theta,N)\}={{\mathcal}I}(\theta) \{{{\mathcal}I}(\theta)
+\frac{1}{4N}{{\mathcal}J}(\theta) \}^{-1}{{\mathcal}I}(\theta)$, which, as $N\to\infty$, tends to ${{\mathcal}I}(\theta)$.

In fact, as also demonstrated empirically by @anitescu2012mfa, one may often not need $N$ to be that large to get estimating equations that are nearly as efficient as the exact score equations. Writing $U_{1j}$ for 