  In adaptive importance sampling, and other contexts, we have $K>1$ unbiased
and uncorrelated estimates $\hat\mu_k$ of a common quantity $\mu$. The optimal
unbiased linear combination weights them inversely to their variances but those
weights are unknown and hard to estimate. A simple deterministic square root
rule based on a working model that $\mathrm{Var}(\hat\mu_k)\propto k^{-1/2}$
gives an unbisaed estimate of $\mu$ that is nearly optimal under a wide range
of alternative variance patterns. We show that if
$\mathrm{Var}(\hat\mu_k)\propto k^{-y}$ for an unknown rate parameter $y\in
[0,1]$ then the square root rule yields the optimal variance rate with a
constant that is too large by at most $9/8$ for any $0\le y\le 1$ and any
number $K$ of estimates. Numerical work shows that rule is similarly robust to
some other patterns with mildly decreasing variance as $k$ increases.
