Cramér type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cramér type moderate deviation theorem for general self-normalized processes which include many well-known Studentized nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized U-statistics.

Stein's method is a powerful tool for proving central limit theorems along with explicit error bounds in probability theory, where uniform and non-uniform Berry-Esseen bounds spark general interest. Nonlinear statistics, typified by Hoeffding's class of U-statistics, L-statistics, random sums and functions of nonlinear statistics, are building blocks in various statistical inference problems. However, because the standardized statistics often involve unknown nuisance parameters, the Studentized analogues are most commonly used in practice. This paper begins with a brief review of some standard techniques in Stein's method, and their applications in deriving Berry-Esseen bounds and Cramér moderate deviations for nonlinear statistics, and then using the concentration inequality approach, establishes Berry-Esseen bounds for Studentized nonlinear statistics in a general framework. As direct applications, sharp Berry-Esseen bounds for Studentized U- and L- statistics are obtained.

Over the last two decades, many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool. Can the discoveries by such data mining approaches be spurious due to high dimensionality and limited sample size? Can our fundamental assumptions on exogeneity of covariates needed for such variable selection be validated with the data? To answer these questions, we need to derive the distributions of the maximum spurious correlations given certain number of predictors, namely, the distribution of the correlation of a response variable Y with the best s linear combinations of p covariates X, even when X and Y are independent. When the covariance matrix of X possesses the restricted eigenvalue property, we derive such distributions for both finite s and diverging s, using Gaussian approximation and empirical process techniques. However, such a distribution depends on the unknown covariance matrix of X. Hence, we use the multiplier bootstrap procedure to approximate the unknown distributions and establish the consistency of such a simple bootstrap approach. The results are further extended to the situation where residuals are from regularized fits. Our approach is then applied to construct the upper confidence limit for the maximum spurious correlation and testing exogeneity of covariates. The former provides a baseline for guarding against false discoveries due to data mining and the latter tests whether our fundamental assumptions for high-dimensional model selection are statistically valid. Our techniques and results are illustrated by both numerical examples and real data analysis.

Two-sample U-statistics are widely used in a broad range of applications, including those in the fields of biostatistics and econometrics. In this paper, we establish sharp Cramér-type moderate deviation theorems for Studentized two-sample U-statistics in a general framework, including the two-sample t-statistic and Studentized Mann-Whitney test statistic as prototypical examples. In particular, a refined moderate deviation theorem with second-order accuracy is established for the two-sample t-statistic. These results extend the applicability of the existing statistical methodologies from the one-sample t-statistic to more general nonlinear statistics. Applications to two-sample large-scale multiple testing problems with false discovery rate control and the regularized bootstrap method are also discussed.