  For nonparametric regression in the streaming setting, where data constantly
flow in and require real-time analysis, a main challenge is that data are
cleared from the computer system once processed due to limited computer memory
and storage. We tackle the challenge by proposing a novel one-pass estimator
based on penalized orthogonal basis expansions and developing a general
framework to study the interplay between statistical efficiency and memory
consumption of estimators. We show that, the proposed estimator is
statistically optimal under memory constraint, and has asymptotically minimal
memory footprints among all one-pass estimators of the same estimation quality.
Numerical studies demonstrate that the proposed one-pass estimator is nearly as
efficient as its non-streaming counterpart that has access to all historical
data.
