On sharp large deviations for sums of random vectors and multidimensional Laplace approximation

Barbe, Philippe; Broniatowski, Michel

doi:10.4213/tvp192

Cited by 3 publications

(1 citation statement)

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“…enables to reconstruct the whole sequence X 1 , ..., X n (rather than the sample up to a permutation) makes of P n a very flexible tool to represent complex and highly non-symmetric statistics. One can, for instance think of two-sample sequential rank statistics, V -statistics or fractional ARIMA processes as treated in Barbe and Broniatowski [1997] , [1998a] and [1998b]. In these references, sequential empirical measure representations allowed to obtain functional large deviation principles for classes of complex statistics as the ones evoked.…”

Section: Introductionmentioning

confidence: 99%

Limit Behavior of Sequential Empirical Measure Processes

El-Dakkak

2012

Stochastic Analysis and Applications

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Section: Introductionmentioning

confidence: 99%

Limit Behavior of Sequential Empirical Measure Processes

El-Dakkak

2012

Stochastic Analysis and Applications

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Small Variance Estimators for Rare Event Probabilities

Broniatowski

Caron

2013

ACM Trans. Model. Comput. Simul.

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Improving Importance Sampling estimators for rare event probabilities requires sharp approximations of conditional densities. This is achieved for events En := (u(X1) + ... + u(Xn)) ∈ An where the summands are i.i.d. and En is a large or moderate deviation event. The approximation of the conditional density of the vector (X1, ..., X kn ) with respect to En on long runs, when kn/n → 1, is handled. The maximal value of kn compatible with a given accuracy is discussed; simulated results are presented, which enlight the gain of the present approach over classical IS schemes. Detailed algorithms are proposed.

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Long runs under a conditional limit distribution

Broniatowski¹,

Caron²

2014

Ann. Appl. Probab.

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This paper presents a sharp approximation of the density of long runs of a random walk conditioned on its end value or by an average of a function of its summands as their number tends to infinity. In the large deviation range of the conditioning event it extends the Gibbs conditional principle in the sense that it provides a description of the distribution of the random walk on long subsequences. An approximation of the density of the runs is also obtained when the conditioning event states that the end value of the random walk belongs to a thin or a thick set with a nonempty interior. The approximations hold either in probability under the conditional distribution of the random walk, or in total variation norm between measures. An application of the approximation scheme to the evaluation of rare event probabilities through importance sampling is provided. When the conditioning event is in the range of the central limit theorem, it provides a tool for statistical inference in the sense that it produces an effective way to implement the Rao-Blackwell theorem for the improvement of estimators; it also leads to conditional inference procedures in models with nuisance parameters. An algorithm for the simulation of such long runs is presented, together with an algorithm determining the maximal length for which the approximation is valid up to a prescribed accuracy.Comment: Published in at http://dx.doi.org/10.1214/13-AAP975 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1010.361

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On sharp large deviations for sums of random vectors and multidimensional Laplace approximation

Cited by 3 publications

References 31 publications

Limit Behavior of Sequential Empirical Measure Processes

Limit Behavior of Sequential Empirical Measure Processes

Small Variance Estimators for Rare Event Probabilities

Long runs under a conditional limit distribution

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