Importance sampling algorithms for first passage time probabilities in the infinite server queue

This paper applies importance sampling simulation for estimating rare event probabilities of the first passage time in the infinite server queue with renewal arrivals and general service time distributions. We consider importance sampling algorithms which are based on large deviations results of the infinite server queue, and we consider an algorithm based on the cross-entropy method, where we allow light-tailed and heavy-tailed distributions for the interarrival times and the service times. Efficiency of the algorithms is discussed by simulation experiments.     

Importance sampling for integrated market and credit portfolio models

A sophisticated approach for computing the total economic capital needed for various stochastically dependent risk types is the bottom-up approach. In this approach, usually, market and credit risks of financial instruments are modeled simultaneously. As integrating market risk factors into standard credit portfolio models increases the computational burden of calculating risk measures, it is analyzed to which extent importance sampling techniques previously developed either for pure market portfolio models or for pure credit portfolio models can be successfully applied to integrated market and credit portfolio models. Specific problems which arise in this context are discussed. The effectiveness of these techniques is tested by numerical experiments for linear and non-linear portfolios.     

Alternative proof and interpretations for a recent state-dependent importance sampling scheme

Recently, a state-dependent change of measure for simulating overflows in the two-node tandem queue was proposed by Dupuis et al. (Ann. Appl. Probab. 17(4):1306–1346, 2007), together with a proof of its asymptotic optimality. In the present paper, we present an alternative, shorter and simpler proof. As a side result, we obtain interpretations for several of the quantities involved in the change of measure in terms of likelihood ratios. Part of this research has been funded by the Dutch BSIK/BRICKS project; part of this research was done while the first author was visiting INRIA/IRISA, Rennes, France.     

Subregion-Adaptive Integration of Functions Having a Dominant Peak

Abstract

Many statistical multiple integration problems involve integrands that have a dominant peak. In applying numerical methods to solve these problems, statisticians have paid relatively little attention to existing quadrature methods and available software developed in the numerical analysis literature. One reason these methods have been largely overlooked, even though they are known to be more efficient than Monte Carlo for well-behaved problems of low dimensionality, may be that when applied naively they are poorly suited for peaked-integrand problems. In this article we use transformations based on “split t” distributions to allow the integrals to be efficiently computed using a subregion-adaptive numerical integration algorithm. Our split t distributions are modifications of those suggested by Geweke and may also be used to define Monte Carlo importance functions. We then compare our approach to Monte Carlo. In the several examples we examine here, we find subregion-adaptive integration to be substantially more efficient than importance sampling.     

Non-local approximation of free-discontinuity functionals with linear growth: the one-dimensional case

We approximate, in the sense of Γ-convergence, one-dimensional free- discontinuity functionals with linear growth in the gradient by means of a sequence of non-local integral functionals depending on the averages of the gradient on small intervals.      

Importance sampling correction versus standard averages of reversible MCMCs in terms of the asymptotic variance

We establish an ordering criterion for the asymptotic variances of two consistent Markov chain Monte Carlo (MCMC) estimators: an importance sampling (IS) estimator, based on an approximate reversible chain and subsequent IS weighting, and a standard MCMC estimator, based on an exact reversible chain. Essentially, we relax the criterion of the Peskun type covariance ordering by considering two different invariant probabilities, and obtain, in place of a strict ordering of asymptotic variances, a bound of the asymptotic variance of IS by that of the direct MCMC. Simple examples show that IS can have arbitrarily better or worse asymptotic variance than Metropolis–Hastings and delayed-acceptance (DA) MCMC. Our ordering implies that IS is guaranteed to be competitive up to a factor depending on the supremum of the (marginal) IS weight. We elaborate upon the criterion in case of unbiased estimators as part of an auxiliary variable framework. We show how the criterion implies asymptotic variance guarantees for IS in terms of pseudo-marginal (PM) and DA corrections, essentially if the ratio of exact and approximate likelihoods is bounded. We also show that convergence of the IS chain can be less affected by unbounded high-variance unbiased estimators than PM and DA chains.     

Large deviations and importance sampling for a tandem network with slow-down

We consider a variant of the two-node tandem Jackson network where the upstream server reduces its service rate when the downstream queue exceeds some prespecified threshold. The rare event of interest is the overflow of the downstream queue. Based on a game/subsolution approach, we rigorously identify the exponential decay rate of the rare event probabilities and construct asymptotically optimal importance sampling schemes. Research of P. Dupuis supported in part by the National Science Foundation (NSF-DMS-0404806 and NSF-DMS-0706003) and the Army Research Office (W911NF-05-1-0289). Research of K. Leder supported in part by the National Science Foundation (NSF-DMS-0404806 and NSF-DMS-0706003). Research of H. Wang supported in part by the National Science Foundation (NSF-DMS-0404806 and NSF-DMS-0706003).     

Numerical approximation using evolution PDE variational splines

This article deals with a numerical approximation method using an evolutionary partial differential equation (PDE) by discrete variational splines in a finite element space. To formulate the problem, we need an evolutionary PDE equation with respect to the time and the position, certain boundary conditions and a set of approximating points. We show the existence and uniqueness of the solution and we study a computational method to compute such a solution. Moreover, we established a convergence result with respect to the time and the position. We provided several numerical and graphic examples of approximation in order to show the validity and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 34: 5–18, 2018     

Penalization and regularization for multivalued pseudo-monotone variational inequalities with Mosco approximation on constraint sets

A coupling of penalization and regularization methods for a variational inequality with multi-valued pseudo-monotone operators is given. The regularization permits to include non-coercive operators. The effect of perturbation is also analyzed. This work is dedicated to the blessed memory of Professor Alex Rubinov.     

A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings

In this paper, we investigate the problem for finding the set of solutions for equilibrium problems, the set of solutions of the variational inequalities for k-Lipschitz continuous mappings and fixed point problems for nonexpansive mappings in a Hilbert space. We introduce a new viscosity extragradient approximation method which is based on the so-called viscosity approximation method and extragradient method. We show that the sequence converges strongly to a common element of the above three sets under some parameters controlling conditions. Finally, we utilize our results to study some convergence problems for finding the zeros of maximal monotone operators. Our results are generalization and extension of the results of Kumam [P. Kumam, Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space, Turk. J. Math. 33 (2009) 85–98], Wangkeeree [R. Wangkeeree, An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, Fixed Point Theory and Applications, 2008, Article ID 134148, 17 pages, doi:10.1155/2008/134148], Yao et al. [Y. Yao, Y.C. Liou, R. Chen, A general iterative method for an finite family of nonexpansive mappings, Nonlinear Analysis 69 (5–6) (2008) 1644–1654], Qin et al. [X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Analysis (69) (2008) 3897–3909], and many others.     

Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type

In many applications one seeks to recover an entire function of exponential type from its non-uniformly spaced samples. Whereas the mathematical theory usually addresses the question of when such a function in can be recovered, numerical methods operate with a finite-dimensional model. The numerical reconstruction or approximation of the original function amounts to the solution of a large linear system. We show that the solutions of a particularly efficient discrete model in which the data are fit by trigonometric polynomials converge to the solution of the original infinite-dimensional reconstruction problem. This legitimatizes the numerical computations and explains why the algorithms employed produce reasonable results. The main mathematical result is a new type of approximation theorem for entire functions of exponential type from a finite number of values. From another point of view our approach provides a new method for proving sampling theorems.

     

用重要性抽样法估计单调关联系统不可靠度

本文首先叙述了基于单调关联故障树模型成败型仿真的重要性抽样的基本原理，然后通过设计几种执行重要性抽样的试验方案，总结得到了在实践中执行重要性抽样若干有益的规则，最后的小结简述了似然比的概念在仿真研究中的重要应用。     

A relaxed projection method for variational inequalities

This paper presents a modification of the projection methods for solving variational inequality problems. Each iteration of the proposed algorithm consists of projection onto a halfspace containing the given closed convex set rather than the latter set itself. The algorithm can thus be implemented very easily and its global convergence to the solution can be established under suitable conditions.This work was supported in part by Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.     

Viscosity approximation methods with weak contraction for L-Lipschitzian pseudocontractive self-mapping

In this paper, under the framework of real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^∗, we study the strong convergence of an implicit and an explicit composite viscosity approximation algorithm (I) and (II), respectively for a pseudocontractive mapping T by using the weakly contractive mapping f as follows:

(I)     

Importance Link Function Estimation for Markov Chain Monte Carlo Methods

Abstract

This article focuses on improving estimation for Markov chain Monte Carlo simulation. The proposed methodology is based upon the use of importance link functions. With the help of appropriate importance sampling weights, effective estimates of functionals are developed. The method is most easily applied to irreducible Markov chains, where application is typically immediate. An important conceptual point is the applicability of the method to reducible Markov chains through the use of many-to-many importance link functions. Applications discussed include estimation of marginal genotypic probabilities for pedigree data, estimation for models with and without influential observations, and importance sampling for a target distribution with thick tails.     

An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.      

Method of Moments Using Monte Carlo Simulation

Abstract

We present a computational approach to the method of moments using Monte Carlo simulation. Simple algebraic identities are used so that all computations can be performed directly using simulation draws and computation of the derivative of the log-likelihood. We present a simple implementation using the Newton-Raphson algorithm with the understanding that other optimization methods may be used in more complicated problems. The method can be applied to families of distributions with unknown normalizing constants and can be extended to least squares fitting in the case that the number of moments observed exceeds the number of parameters in the model. The method can be further generalized to allow “moments” that are any function of data and parameters, including as a special case maximum likelihood for models with unknown normalizing constants or missing data. In addition to being used for estimation, our method may be useful for setting the parameters of a Bayes prior distribution by specifying moments of a distribution using prior information. We present two examples—specification of a multivariate prior distribution in a constrained-parameter family and estimation of parameters in an image model. The former example, used for an application in pharmacokinetics, motivated this work. This work is similar to Ruppert's method in stochastic approximation, combines Monte Carlo simulation and the Newton-Raphson algorithm as in Penttinen, uses computational ideas and importance sampling identities of Gelfand and Carlin, Geyer, and Geyer and Thompson developed for Monte Carlo maximum likelihood, and has some similarities to the maximum likelihood methods of Wei and Tanner.     

Simulation-Based Optimal Design Using a Response Variance Criterion

Classical optimal design criteria suffer from two major flaws when applied to nonlinear problems. First, they are based on linearizing the model around a point estimate of the unknown parameter and therefore depend on the uncertain value of that parameter. Second, classical design methods are unavailable in ill-posed estimation situations, where previous data lack the information needed to properly construct the design criteria. Bayesian optimal design can, in principle, solve these problems. However, Bayesian design methods are not widely applied, mainly due to the fact that standard implementations for efficient and robust routine use are not available. In this article, we point out a concrete recipe for implementing Bayesian optimal design, based on the concept of simulation-based design introduced by Muller, Sanso, and De Iorio (2004 Muller, P., Sanso, B. and De Iorio, M. 2004. Optimal Bayesian Design by Inhomogeneous Markov Chain Simulation. Journal of the American Statistical Association, 99(467): 788–798. [Taylor & Francis Online] , [Google Scholar]). We develop further a predictive variance criterion and introduce an importance weighting mechanism for efficient computation of the variances. The simulation-based approach allows one to start the model-based optimization of experiments at an early stage of the parameter estimation process, in situations where the classical design criteria are not available. We demonstrate that the approach can significantly reduce the number of experiments needed to obtain a desired level of accuracy in the parameter estimates. A computer code package that implements the approach in a simple case is provided as supplemental material (available online).