计量经济分析 = Econometric analysis

It is now over a decade since the pioneering contribution of Gordon (1993), which is commonly regarded as the first instance of modern sequential Monte Carlo (SMC) approaches. Initially focussed on applications to tracking and vision, these techniques are now very widespread and have had a significant impact in virtually all areas of signal and image processing concerned with Bayesian dynamical models. This paper is intended to serve both as an introduction to SMC algorithms for nonspecialists and as a reference to recent contributions in domains where the techniques are still under significant development, including smoothing, estimation of fixed parameters and use of SMC methods beyond the standard filtering contexts.

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An Overview of Existing Methods and Recent Advances in Sequential Monte Carlo

On generalise l'estimateur bien connu de Hill de l'indice d'une fonction de reparatition avec queue de variation reguliere a une estimation de l'indice d'une loi de valeurs extremes. On demontre la convergence et la normalite asymptotique. On utilise l'estimateur pour certaines estimations comme celle d'une quantile elevee et d'un point d'extremite

A moment estimator for the index of an extreme-value distribution

This paper discusses a novel strategy for simulating rare events and an associated Monte Carlo estimation of tail probabilities. Our method uses a system of interacting particles and exploits a Feynman-Kac representation of that system to analyze their fluctuations. Our precise analysis of the variance of a standard multilevel splitting algorithm reveals an opportunity for improvement. This leads to a novel method that relies on adaptive levels and produces, in the limit of an idealized version of the algorithm, estimates with optimal variance. The motivation for this theoretical work comes from problems occurring in watermarking and fingerprinting of digital contents, which represents a new field of applications of rare event simulation techniques. Some numerical results show performance close to the idealized version of our technique for these practical applications.

/pdf/sequential-monte-carlo-for-rare-event-estimation-2d78yakm4r.pdf

Sequential Monte Carlo for rare event estimation

Throughout this chapter, we suppose that E is a Banach space with norm ‖ • ‖, and (Σ, e) is a measurable space. Further, we suppose that S is a C0-semigroup of continuous linear operators acting on E and that Q: e → ℒ(E) is a spectral measure, so that 

$$ X = \left( {\Omega ,{{\left\langle {{S_t}} \right\rangle }_{t \geqslant 0}},{{\left\langle {{M_t}} \right\rangle }_{t \geqslant 0}};{{\left\langle {{X_t}} \right\rangle }_{t \geqslant 0}}} \right)$$

is a σ-additive (S, Q)-process. Recall that this means that for each t ≥ 0, M t : S t → ℒ(E) is a σ-additive set function defined on a σ-algebra S t of subsets of Ω containing the collection e t {X} of all basic events before time t. To ensure that St is not too large, we suppose that it is contained in the completion with respect to the measure M t of the σ-algebra σ(e t {X}) generated by e t {X}, that is, the σ-algebra produced by augmenting σ(e t {X}) with all subsets of M t -null sets belonging to σ(e t {X}).

Feynman-Kac Formulae

We present in this article a genetic type interacting particle systems algorithm and a genealogical model for estimating a class of rare events arising in physics and network analysis. We represent the distribution of a Markov process hitting a rare target in terms of a Feynman-Kac model in path space. We show how these branching particle models described in previous works can be used to estimate the probability of the corresponding rare events as well as the distribution of the process in this regime.

https://hal.inria.fr/inria-00071391/document

Genetic genealogical models in rare event analysis

In this article, a genetic-type algorithm based on interacting particle systems is presented, together with a genealogical model, for estimating a class of rare events arising for instance in telecommunication networks, nuclear engineering, etc. The distribution of a Markov process hitting a rare but critical set is represented in terms of a Feynman-Kac model in path space. Approximation results obtained previously for these models are applied here to estimate the probability of the rare events as well as the probability distribution of the critical trajectories.

/pdf/limit-theorems-for-the-multilevel-splitting-algorithm-in-the-30kngbepj8.pdf

Limit theorems for the multilevel splitting algorithm in the simulation of rare events

Splitting techniques in rare event simulation

We provide an overview of the probability and statistical tools underlying the extreme value theory, which aims to predict occurrence of rare events. Firstly, we explain that the asymptotic distribution of extreme values belongs, in some sense, to the family of the generalised extreme value distributions which depend on a real parameter, called the extreme value index. Secondly, we discuss statistical tail estimation methods based on estimators of the extreme value index.

/pdf/extreme-value-analysis-an-introduction-3eoeqzywyq.pdf

Extreme Value Analysis: an Introduction

This article focuses on branching particle interpretations of rare events. We connect importance sampling techniques with interacting particle algorithms, and multi-splitting branching models. These Monte Carlo methods are illustrated with a variety of examples arising in particle trapping analysis, as well as in ruin type estimation problems. We also provide a rather detailed presentation of the asymptotic theory of these particle algorithms, including exponential extinction probabilities, Lp-mean error bounds, central limit theorem, and fluctuation variance comparaisons.

Pascal Lezaud

Papers

Genetic genealogical models in rare event analysis

Limit theorems for the multilevel splitting algorithm in the simulation of rare events

Splitting techniques in rare event simulation

Extreme Value Analysis: an Introduction

Branching and interacting particle interpretations of rare event probabilities