  Statistical inference based on lossy or incomplete samples is often needed in
research areas such as signal/image processing, medical image storage, remote
sensing, signal transmission. In this paper, we propose a nonparametric testing
procedure based on samples quantized to $B$ bits through a computationally
efficient algorithm. Under mild technical conditions, we establish the
asymptotic properties of the proposed test statistic and investigate how the
testing power changes as $B$ increases. In particular, we show that if $B$
exceeds a certain threshold, the proposed nonparametric testing procedure
achieves the classical minimax rate of testing (Shang and Cheng, 2015) for
spline models. We further extend our theoretical investigations to a
nonparametric linearity test and an adaptive nonparametric test, expanding the
applicability of the proposed methods. Extensive simulation studies {together
with a real-data analysis} are used to demonstrate the validity and
effectiveness of the proposed tests.
