# DELTA METHOD IN LARGE DEVIATIONS AND MODERATE DEVIATIONS FOR ESTIMATORS

@article{Gao2011DELTAMI, title={DELTA METHOD IN LARGE DEVIATIONS AND MODERATE DEVIATIONS FOR ESTIMATORS}, author={Fuqing Gao and Xingqiu Zhao}, journal={Annals of Statistics}, year={2011}, volume={39}, pages={1211-1240} }

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…

## 68 Citations

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## References

SHOWING 1-10 OF 71 REFERENCES

Moderate deviations of empirical periodogram and non-linear functionals of moving average processes

- Mathematics
- 2006

A moderate deviation principle for non-linear functionals, with at most quadratic growth, of moving average processes (or linear processes) is established. The main assumptions on the moving average…

Large deviations for M-estimators

- Mathematics
- 2006

We study the large deviation principle for M-estimators (and maximum likelihood estimators in particular). We obtain the rate function of the large deviation principle for M-estimators. For…

Moderate deviations of minimum contrast estimators under contamination

- Mathematics
- 2001

Since statistical models are simplifications of reality, it is important in estimation theory to study the behavior of estimators also under distributions (slightly) different from the proposed…

Moderate deviations for M-estimators

- Mathematics
- 2002

General sufficient conditions for the moderate deviations of M-estimators are presented. These results are applied to many different types of M-estimators such as thep-th quantile, the spatial…

Large Deviations Limit Theorems for the Kernel Density Estimator

- Mathematics
- 1998

We establish pointwise and uniform large deviations limit theorems of Chern- off-type for the non-parametric kernel density estimator based on a sequence of independent and identically distributed…

A Large Deviation Result for Parameter Estimators and its Application to Nonlinear Regression Analysis

- Mathematics
- 1986

Elaborating on the work of Ibragimov and Has'minskii (1981) we prove a law of large deviations (LLD) for $M$-estimators, i.e., those estimators which maximize a functional, continuous in the…

Moderate Deviations and Large Deviations for Kernel Density Estimators

- Mathematics
- 2003

AbstractLet fn be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in ℝd. It is proved…

A central limit theorem for M-estimators by the Von Mises method

- Mathematics
- 1987

Asymptotic normality of M- or maximum likelihood type estimators was established in a classic paper by Huber (1967). Reeds (1976) argued that this could have been obtained simply as an application of…

On Probabilities of Excessive Deviations for Kolmogorov-Smirnov, Cramer-von Mises and Chi-Square Statistics

- Mathematics
- 1990

Let α n be the classical empirical process and T: D[0, 1]→R. Assume T satisfies the Lipschitz condition. Using the Komlos-Major-Tusnady inequality, bounds for P(T(α n )≥x n √n) are obtained for every…

Moderate and Cramer-type large deviation theorems for M-estimators

- Mathematics
- 1988

The known central limit result for broad classes of M-estimators is refined to moderate and large deviation behaviour. The results are applied in relating the local inaccuracy rate and the asymptotic…