Delta method in large deviations and moderate deviations for estimators

The delta method is a popular and elementary tool for deriving limiting distributions of transformed statistics, while applications of asymptotic distributions do not allow one to obtain desirable accuracy of approximation for tail probabilities. The large and moderate deviation theory can achieve this goal. Motivated by the delta method in weak convergence, a general delta method in large deviations is proposed. The new method can be widely applied to driving the moderate deviations of estimators and is illustrated by examples including the Wilcoxon statistic, the Kaplan--Meier estimator, the empirical quantile processes and the empirical copula function. We also improve the existing moderate deviations results for $M$-estimators and $L$-statistics by the new method. Some applications of moderate deviations to statistical hypothesis testing are provided.Comment: Published in at http://dx.doi.org/10.1214/10-AOS865 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bernstein type's concentration inequalities for symmetric Markov processes

Using the method of transportation-information inequality introduced in \cite{GLWY}, we establish Bernstein type's concentration inequalities for empirical means $\frac 1t \int_0^t g(X_s)ds$ where $g$ is a unbounded observable of the symmetric Markov process $(X_t)$. Three approaches are proposed : functional inequalities approach ; Lyapunov function method ; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied

The silicon isotope composition of Ethmodiscus rexlaminated diatom mats from the tropical West Pacific: Implications for silicate cycling during the Last Glacial Maximum

The cause of massive blooms of Ethmodiscus rex laminated diatom mats (LDMs) in the eastern Philippine Sea (EPS) during the Last Glacial Maximum (LGM) remains uncertain. In order to better understand the mechanism of formation of E. rex LDMs from the perspective of dissolved silicon (DSi) utilization, we determined the silicon isotopic composition of single E. rex diatom frustules (δ30SiE. rex) from two sediment cores in the Parece Vela Basin of the EPS. In the study cores, δ30SiE. rex varies from −1.23‰ to −0.83‰ (average −1.04‰), a range that is atypical of marine diatom δ30Si and that corresponds to the lower limit of reported diatom δ30Si values of any age. A binary mixing model (upwelled silicon versus eolian silicon) accounting for silicon isotopic fractionation during DSi uptake by diatoms was constructed. The binary mixing model demonstrates that E. rex dominantly utilized DSi from eolian sources (i.e., Asian dust) with only minor contributions from upwelled seawater sources (i.e., advected from Subantarctic Mode Water, Antarctic Intermediate Water, or North Pacific Intermediate Water). E. rex utilized only ~24% of available DSi, indicating that surface waters of the EPS were eutrophic with respect to silicon during the LGM. Our results suggest that giant diatoms did not always use a buoyancy strategy to obtain nutrients from the deep nutrient pool, thus revising previously proposed models for the formation of E. rex LDMs