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.

翻译：基于有损或不完全样本的统计推断常出现在信号/图像处理、医学图像存储、遥感、信号传输等研究领域。本文提出一种基于量化至B比特样本的非参数检验方法，通过计算高效的算法实现。在温和的技术条件下，我们建立了所提出的检验统计量的渐近性质，并研究了检验功效随B增加的变化规律。特别地，我们证明当B超过某一阈值时，所提出的非参数检验方法对于样条模型能达到经典极小极大检验速率（Shang and Cheng, 2015）。我们进一步将理论分析拓展至非参数线性检验和自适应非参数检验，扩大了所提方法的适用范围。通过大量模拟研究及实际数据分析验证了所提检验的有效性和可行性。