In this paper, we provide statistical guarantees for over-parameterized deep nonparametric regression in the presence of dependent data. By decomposing the error, we establish non-asymptotic error bounds for deep estimation, which is achieved by effectively balancing the approximation and generalization errors. We have derived an approximation result for Hölder functions with constrained weights. Additionally, the generalization error is bounded by the weight norm, allowing for a neural network parameter number that is much larger than the training sample size. Furthermore, we address the issue of the curse of dimensionality by assuming that the samples originate from distributions with low intrinsic dimensions. Under this assumption, we are able to overcome the challenges posed by high-dimensional spaces. By incorporating an additional error propagation mechanism, we derive oracle inequalities for the over-parameterized deep fitted $Q$-iteration.

UNLABELLED: In next generation sequencing (NGS)-based genetic studies, researchers typically perform genotype calling first and then apply standard genotype-based methods for association testing. However, such a two-step approach ignores genotype calling uncertainty in the association testing step and may incur power loss and/or inflated type-I error. In the recent literature, a few robust and efficient likelihood based methods including both likelihood ratio test (LRT) and score test have been proposed to carry out association testing without intermediate genotype calling. These methods take genotype calling uncertainty into account by directly incorporating genotype likelihood function (GLF) of NGS data into association analysis. However, existing LRT methods are computationally demanding or do not allow covariate adjustment; while existing score tests are not applicable to markers with low minor allele frequency (MAF). We provide an LRT allowing flexible covariate adjustment, develop a statistically more powerful score test and propose a combination strategy (UNC combo) to leverage the advantages of both tests. We have carried out extensive simulations to evaluate the performance of our proposed LRT and score test. Simulations and real data analysis demonstrate the advantages of our proposed combination strategy: it offers a satisfactory trade-off in terms of computational efficiency, applicability (accommodating both common variants and variants with low MAF) and statistical power, particularly for the analysis of quantitative trait where the power gain can be up to ∼60% when the causal variant is of low frequency (MAF < 0.01). AVAILABILITY AND IMPLEMENTATION: UNC combo and the associated R files, including documentation, examples, are available at http://www.unc.edu/∼yunmli/UNCcombo/ CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.