The non‐asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probabilities in literature, the lower bounds on tail probabilities are relatively fewer. In this paper, we introduce systematic and user‐friendly schemes for developing non‐asymptotic lower bounds of tail probabilities. In addition, we develop sharp lower tail bounds for the sum of independent sub‐Gaussian and sub‐exponential random variables, which match the classic Hoeffding‐type and Bernstein‐type concentration inequalities, respectively. We also provide non‐asymptotic matching upper and lower tail bounds for a suite of distributions, including gamma, beta, (regular, weighted, and noncentral) chi‐square, binomial, Poisson, Irwin–Hall, etc. We apply the result to establish the matching upper and lower bounds for extreme value expectation of the sum of independent sub‐Gaussian and sub‐exponential random variables. A statistical application of signal identification from sparse heterogeneous mixtures is finally considered.

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In order to explore whether the experience within the role of movement influences information processing, this paper studies the visual search and prediction abilities of different women volleyball players and their brain neural mechanisms. A laboratory composing of a preparatory room, a master control room, and a test room was established. Subjects were required to perform experimental tasks in the test room alone. The results of multivariate analysis of variance showed that the main effect of the role was significant, indicating that different roles have different responses to different types of attacks. Volleyball players of different roles use different visual search strategies for visual search. The main attack team and the supporting attack team use short search durations more frequently for gaze duration.

In microbiome and genomic studies, the regression of compositional data has been a crucial tool for identifying microbial taxa or genes that are associated with clinical phenotypes. To account for the variation in sequencing depth, the classic log-contrast model is often used where read counts are normalized into compositions. However, zero read counts and the randomness in covariates remain critical issues. In this article, we introduce a surprisingly simple, interpretable, and efficient method for the estimation of compositional data regression through the lens of a novel high-dimensional log-error-in-variable regression model. The proposed method provides both corrections on sequencing data with possible overdispersion and simultaneously avoids any subjective imputation of zero read counts. We provide theoretical justifications with matching upper and lower bounds for the estimation error. The merit of the procedure is illustrated through real data analysis and simulation studies.

The non-asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper tail bounds in literature, the lower tail bound results are relatively fewer. In this partly expository paper, we introduce systematic and user-friendly schemes for developing non-asymptotic lower tail bounds with elementary proofs. In addition, we develop sharp lower tail bounds for the sum of independent sub-Gaussian and sub-exponential random variables, which matches the classic Hoeffding-type and Bernstein-type concentration inequalities, respectively. We also provide non-asymptotic matching upper and lower tail bounds for a suite of distributions, including gamma, beta, (regular, weighted, and noncentral) chi-squared, binomial, Poisson, Irwin-Hall, etc. We apply the result to establish the matching upper and lower bounds for extreme value expectation of the sum of independent sub-Gaussian and sub-exponential random variables. A statistical application of signal identification from sparse heterogeneous mixtures is finally studied.