Nonparametric Inference under B-bits Quantization

Statistical inference based on lossy or incomplete samples is often needed in research armission. 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.