Complete $Q^{(m,n)}$ Rule Randomizations

Define

$$Q^{(m,n)}(f,Z) = \sum_{\vert{\bf p}\vert=m}w_{{\bf p}}f\{{\bf u}_{\bf p},Z\},$$

where $Z$ is an $n\times n$ orthogonal matrix, and $f\{{\bf u}_{\bf p},Z\}$ denotes the application of the transformation $Z$ to each of the points used in the original symmetric sum $f\{{\bf u}_{\bf p}\}$. All rule points for $Q^{(m,n)}(f,Z)$ are on the $n-$sphere surface and are linear transformations of the original points, so $Q^{(m,n)}(f,Z)$ also has polynomial degree $2m+1$. Denote the average of $N$ random copies of $Q^{(m,n)}(f,Z)$ by

$${\bar Q}_N^{(m,n)}(f) = \frac{1}{N} \sum_{i=1}^N Q^{(m,n)}(f,Z_i).$$

If random orthogonal matrices $Z_i$ are generated with Haar distribution from the set of all matrices in the orthogonal group (see Stewart [6], 1980), then the ``stochastic'' rule ${\bar Q}_N^{(m,n)}(f)$ is an unbiased degree $2m+1$ estimate for $I(f)$. A robust unbiased degree $2m+1$ error estimate ${\bar Q}_N^{(m,n)}(f)$ is provided by the Monte Carlo standard error

$$E_N(f)= \Big(\frac{1}{N(N-1)} \sum_{i=1}^N(Q^{(m,n)}(f,Z_i)-{\bar Q}_N^{(m,n)}(f))^2\Big)^\frac{1}{2}.$$