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
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