more sophisticated design for the $U_j$’s; this section describes such a design. Suppose that $n=Nm$ for some nonnegative integer $m$ and that $\beta_1,\ldots,\beta_N$ are fixed vectors of length $N$ with all entries $\pm1$ for which $\frac{1}{N}\sum_{j=1}^N \beta_j\beta'_j = I$. For example, if $N=2^q$ for a positive integer $q$, then the $\beta_j$’s can be chosen to be the design matrix for a saturated model of a $2^q$ factorial design in which the levels of the factors are set at $\pm1$ \[@BHH, Chapter 5\]. In addition, assume that $X_1,\ldots,X_m$ are random diagonal matrices of size $N$ and $Y_{jk}$, $j=1,\ldots,N; k=1,\ldots,m$ are random variables such that all the diagonal elements of the $X_j$’s and all the $Y_{jk}$’s are i.i.d. symmetric Bernoulli random variables. Then define $$\label{Uj}
U_j = \pmatrix{
Y_{j1}X_1
\cr
\vdots
\cr
Y_{jm}X_m}
\beta_j.$$ One can easily show that for any $Nm\times Nm$ matrix $M$, $E (\frac{1}{N}\sum_{j=1}^N U_j'MU_j ) = \operatorname{tr}(M)$. Thus, we can use this definition of the $U_j$’s in (\[ascore\]), and the resulting estimating equations are still unbiased.

This design is closely related to a class of designs introduced by @avron, who propose selecting the $U_j$’s as follows. Suppose $H$ is a Hadamard matrix, that is, an $n\times n$ orthogonal 