For instance, deep neural networks have led to impressive performance in fields such as computer vision, natural language processing, image/speech/audio recognition,social network filtering, machine translation, bioinformatics, drug design, medical image analysis, where they have demonstrated superior performance to human experts. The success of deep networks hinges on their rich expressiveness (see Delalleau and Bengio (2011), Raghu et al. (2017), Montufar et al. (2014), Bianchini and Scarselli (2014), Telgarsky (2016), Liang and Srikant (2017) and Yarotsky (2017, 2018)). Recently, deep networks have played an increasingly important role in statistics particularly in nonparametric curve fitting (see Kohler and Krzyżak (2005); Hamers and Kohler (2006); Kohler and Krzyżak (2017); Kohler and Mehnert (2011);Schmidt-Hieber (2017)). Applications of deep networks in other fields such as image processing or pattern recgnition include, to name a few, LeCun et al. (2015), Deng et al. (2013), Wan et al. (2014), Gal and Ghahramani (2016), etc. A fundamental problem in statistical applications of deep networks is how accurate they can estimate a nonparametric regression function. To describe the problem, let us consider i.i.d.

Statistical inference based on lossy or incomplete samples is of fundamental importance 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 quantized samples. In contrast to the classic nonparametric approach, our method lives on a coarse grid of sample information and are simple-to-use. Under mild technical conditions, we establish the asymptotic properties of the proposed procedures including asymptotic null distribution of the quantization test statistic as well as its minimax power optimality. Concrete quantizers are constructed for achieving the minimax optimality in practical use. Simulation results and a real data analysis are provided to demonstrate the validity and effectiveness of the proposed test. Our work bridges the classical nonparametric inference to modern lossy data setting.