On semiparametric statistical inferences in the moderate deviation zone

Ermakov, Mikhail

doi:10.1007/s10958-008-9104-5

“…Bahadur and Zabell (1979) developed a subadditive method to study the large deviations for general sample means. For some developments of large deviations and moderate deviations in statistics, see Fu (1982), Kester and Kallenberg (1986), Sieders and Dzhaparidze (1987), Inglot and Ledwina (1990), Borovkov and Mogul ′ skii (1992), Puhalskii and Spokoiny (1998), Bercu (2001), Joutard (2004) and Arcones (2006) for large deviations of estimators; Kallenberg (1983), Gao (2001), Arcones (2002), Inglot and Kallenberg (2003), Djellout, Guillin and Wu (2006) and Ermakov (2008) for moderate deviations of estimators; Louani (1998), Worms (2001), Gao (2003), Lei and Wu (2005) for large deviations and moderate deviations of kernel density estimators, and references therein. On the other hand, large deviations of estimators can be applied to Bahadur efficiency to determine the Bahadur slope [Bahadur (1967), Nikitin (1995), He and Shao (1996)] and hypothesis testing [see Dembo and Zeitouni (1998), Sections 3.5 and 7.1].…”

mentioning

confidence: 99%

Delta method in large deviations and moderate deviations for estimators

Gao

¹

,

Zhao²

2011

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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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“…For the semiparametric estimation the local asymptotic minimax lower bounds in the zone of moderate deviation probabilities have been established in [12]. In [12] the statistical functionals take the values in R 1 . The results were based on the assumptions that (2.1-2.3) hold uniformly for the families of "least-favourable"distributions.…”

Section: Resultsmentioning

confidence: 99%

The Sharp Lower Bound of Asymptotic Efficiency of Estimators in the Zone of Moderate Deviation Probabilities

Ermakov¹

2012

Preprint

0

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“…For the semiparametric estimation the local asymptotic minimax lower bounds in the zone of moderate deviation probabilities have been established in [12]. In [12] the statistical functionals take the values in R 1 .…”

Section: Ermakovmentioning

confidence: 99%

“…For the semiparametric estimation the local asymptotic minimax lower bounds in the zone of moderate deviation probabilities have been established in [12]. In [12] the statistical functionals take the values in R 1 . The results were based on the assumptions that (2.2)-(2.4) hold uniformly for the families of "leastfavourable" distributions.…”

Section: Ermakovmentioning

confidence: 99%