amount of data. For example, if $\kappa(K)=5$, a bound of 0.01 is obtained with $N=180$ and a bound of 0.1 with $N=18$.

It is possible to obtain unbiased estimating equations similar to (\[ascore\]) whose statistical efficiency does not depend on $\kappa(K)$. Specifically, if we write $\operatorname{tr}(K^{-1}K_i)$ as $\operatorname{tr}((G')^{-1}K_iG^{-1})$, where $G$ is any matrix satisfying $G'G=K$, we then have that $$\label{symscore}
h_i(\theta,N) = \frac{1}{2}Z'K^{-1}K_iK^{-1}Z
-\frac{1}{2N}\sum_{j=1}^N
U_j'\bigl(G'\bigr)^{-1}K_iG^{-1}U_j
= 0$$ for $i=1,\ldots,p$ are also unbiased estimating equations for $\theta$. In this case, $\operatorname{cov}_\theta\{h(\theta,N)\}\preceq
( 1+\frac{1}{N} ){{\mathcal}I}(\theta)$, whose proof is similar to that of Theorem \[tmain\] but exploits the symmetry of $(G')^{-1}K_iG^{-1}$. This bound is less than or equal to the bound in (\[Bbound\]) on $\operatorname{cov}_\theta\{g(\theta,N)\}$. Whether it is preferable to use (\[symscore\]) rather than (\[ascore\]) depends on a number of factors, including the sharpness of the bound in (\[Bbound\]) and how much more work it takes to compute $G^{-1}U_j$ than to compute $K^{-1}U_j$. An example of how the action of such a matrix square root can be approximated efficiently using only $O(n)$ storage is presented by @chen2011computing.

Dependent designs {#sec3}
=================

\[secdependent\] Choosing the $U_j$’s independently is simple and convenient, but one can reduce the variation in the stochastic approximation by using a 