  Designing scalable estimation algorithms is a core challenge in modern
statistics. Here we introduce a framework to address this challenge based on
parallel approximants, which yields estimators with provable properties that
operate on the entirety of very large, distributed data sets. We first
formalize the class of statistics which admit straightforward calculation in
distributed environments through independent parallelization. We then show how
to use such statistics to approximate arbitrary functional operators in
appropriate spaces, yielding a general estimation framework that does not
require data to reside entirely in memory. We characterize the $L^2$
approximation properties of our approach and provide fully implemented examples
of sample quantile calculation and local polynomial regression in a distributed
computing environment. A variety of avenues and extensions remain open for
future work.
