# Nonparametric Inference under B-bits Quantization

## TL;DR

This paper introduces a nonparametric testing method for quantized data, demonstrating its asymptotic properties and effectiveness through simulations and real data, especially when the number of bits exceeds a certain threshold.

## Contribution

It proposes a computationally efficient nonparametric testing procedure for B-bit quantized samples with theoretical guarantees and extensions to linearity and adaptive tests.

## Key findings

- Test statistic achieves classical minimax rate when B exceeds threshold
- Method is effective for spline models and nonparametric linearity testing
- Simulation and real-data studies confirm validity and effectiveness

## Abstract

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.