  We design a nonadaptive algorithm that, given oracle access to a function $f:
\{0,1\}^n \to \{0,1\}$ which is $\alpha$-far from monotone, makes poly$(n,
1/\alpha)$ queries and returns an estimate that, with high probability, is an
$\widetilde{O}(\sqrt{n})$-approximation to the distance of $f$ to monotonicity.
The analysis of our algorithm relies on an improvement to the directed
isoperimetric inequality of Khot, Minzer, and Safra (SIAM J. Comput., 2018).
Furthermore, we rule out a poly$(n, 1/\alpha)$-query nonadaptive algorithm that
approximates the distance to monotonicity significantly better by showing that,
for all constant $\kappa > 0,$ every nonadaptive $n^{1/2 -
\kappa}$-approximation algorithm for this problem requires $2^{n^\kappa}$
queries. This answers a question of Seshadhri (Property Testing Review, 2014)
for the case of nonadaptive algorithms. We obtain our lower bound by proving an
analogous bound for erasure-resilient (and tolerant) testers. Our method also
yields the same lower bounds for unateness and being a $k$-junta.
