Improved estimation in a non-Gaussian parametric regression

The paper considers the problem of estimating the parameters in a continuous time regression model with a non-Gaussian noise of pulse type. The vector of unknown parameters is assumed to belong to a compact set. The noise is specified by the Ornstein–Uhlenbeck process driven by the mixture of a Brownian motion and a compound Poisson process. Improved estimates for the unknown regression parameters, based on a special modification of the James–Stein procedure with smaller quadratic risk than the usual least squares estimates, are proposed. The developed estimation scheme is applied for the improved parameter estimation in the discrete time regression with the autoregressive noise depending on unknown nuisance parameters.     

On convergence of a discrete problem describing transport processes in the pressing section of a paper machine including dynamic capillary effects: One-dimensional case

This work presents a proof of convergence of a discrete solution to a continuous one. At first, the continuous problem is stated as a system of equations which describe the filtration process in the pressing section of a paper machine. Two flow regimes appear in the modeling of this problem. The model for the saturated flow is presented by the Darcy’s law and the mass conservation. The second regime is described by the Richards’ approach together with a dynamic capillary pressure model. The finite volume method is used to approximate the system of PDEs. Then, the existence of a discrete solution to the proposed finite difference scheme is proven. Compactness of the set of all discrete solutions for different mesh sizes is proven. The main theorem shows that the discrete solution converges to the solution of the continuous problem. At the end we present numerical studies for the rate of convergence.     

General model selection estimation of a periodic regression with a Gaussian noise

This paper considers the problem of estimating a periodic function in a continuous time regression model with an additive stationary Gaussian noise having unknown correlation function. A general model selection procedure on the basis of arbitrary projective estimates, which does not need the knowledge of the noise correlation function, is proposed. A non-asymptotic upper bound for L₂{\mathcal{L}_2} -risk (oracle inequality) has been derived under mild conditions on the noise. For the Ornstein–Uhlenbeck noise the risk upper bound is shown to be uniform in the nuisance parameter. In the case of Gaussian white noise the constructed procedure has some advantages as compared with the procedure based on the least squares estimates (LSE). The asymptotic minimaxity of the estimates has been proved. The proposed model selection scheme is extended also to the estimation problem based on the discrete data applicably to the situation when high frequency sampling can not be provided.     

Limiting Connection between Discrete and Continuous Time Forward Interest Rate Curve Models

We study discrete time Heath–Jarrow–Morton (HJM) type of interest rate curve models, where the forward interest rates – in contrast to the classical HJM models – are driven by a random field. Our main aim is to investigate the relationship between the discrete time forward interest rate curve model and its continuous time counterpart. We derive a general result on the convergence of discrete time models and we give special focus on the nearly unit root spatial autoregression model.     

Strong convergence for explicit space–time discrete numerical approximation methods for stochastic Burgers equations

In this paper we propose and analyze explicit space–time discrete numerical approximations for additive space–time white noise driven stochastic partial differential equations (SPDEs) with non-globally monotone nonlinearities such as the stochastic Burgers equation with space–time white noise. The main result of this paper proves that the proposed explicit space–time discrete approximation method converges strongly to the solution process of the stochastic Burgers equation with space–time white noise. To the best of our knowledge, the main result of this work is the first result in the literature which establishes strong convergence for a space–time discrete approximation method in the case of the stochastic Burgers equations with space–time white noise.     

Stochastic Differential Games With a Small Parameter

Much of stochastic game theory is concerned with diffusion models. Such models are often only idealizations of the actual physical process, which might be driven by a wide bandwidth process or be a discrete parameter system with correlated driving noises. For a two person zero-sum game, under quite general conditions, the optimal or nearly optimal strategies derived for the diffusion model are shown to be “nearly optimal” for the physical (say wideband noise driven) process. An approach based on occupation measures are used. We treat the problem of discounted cost, as well as the average cost per unit time problem. Weak convergence methods are utilized in the analysis     

一类带有空间时间白噪音随机弹性方程的全离散差分格式

随机弹性方程在结构工程中有许多应用.本文研究一类由空间时间白噪音扰动的随机弹性方程的全离散有限差分格式.通过引入新的函数,将随机弹性方程表示成一阶方程组的形式,然后对噪音项进行分片常数逼近,构造了带有空间时间白噪音随机弹性方程的全离散差分格式.基于对Gronwall不等式和Burkholder不等式的应用,证明了格式的L~p收敛性并得到了收敛阶.在数值实验中结合Monte-Carlo方法,所得实验结果与理论分析是一致的.     

Feynman-Kac Particle Integration with Geometric Interacting Jumps

This article is concerned with the design and analysis of discrete time Feynman-Kac particle integration models with geometric interacting jump processes. We analyze two general types of model, corresponding to whether the reference process is in continuous or discrete time. For the former, we consider discrete generation particle models defined by arbitrarily fine time mesh approximations of the Feynman-Kac models with continuous time path integrals. For the latter, we assume that the discrete process is observed at integer times and we design new approximation models with geometric interacting jumps in terms of a sequence of intermediate time steps between the integers. In both situations, we provide nonasymptotic bias and variance theorems w.r.t. the time step and the size of the system, yielding what appear to be the first results of this type for this class of Feynman-Kac particle integration models. We also discuss uniform convergence estimates w.r.t. the time horizon. Our approach is based on an original semigroup analysis with first order decompositions of the fluctuation errors.     

Modeling clustered binary data with nonparametric unobserved heterogeneity: An application to stock crash analysis

Various random effects models have been developed for clustered binary data; however, traditional approaches to these models generally rely heavily on the specification of a continuous random effect distribution such as Gaussian or beta distribution. In this article, we introduce a new model that incorporates nonparametric unobserved random effects on unit interval (0,1) into logistic regression multiplicatively with fixed effects. This new multiplicative model setup facilitates prediction of our nonparametric random effects and corresponding model interpretations. A distinctive feature of our approach is that a closed-form expression has been derived for the predictor of nonparametric random effects on unit interval (0,1) in terms of known covariates and responses. A quasi-likelihood approach has been developed in the estimation of our model. Our results are robust against random effects distributions from very discrete binary to continuous beta distributions. We illustrate our method by analyzing recent large stock crash data in China. The performance of our method is also evaluated through simulation studies.     

Multi-Label Markov Random Fields as an Efficient and Effective Tool for Image Segmentation,Total Variations and Regularization

One of the classical optimization models for image segmentation is the well known Markov Random Fields (MRF) model. This model is a discrete optimization problem, which is shown here to formulate many continuous models used in image segmentation. In spite of the presence of MRF in the literature, the dominant perception has been that the model is not effective for image segmentation. We show here that the reason for the non-effectiveness is due to the lack of access to the optimal solution. Instead of solving optimally, heuristics have been engaged. Those heuristic methods cannot guarantee the quality of the solution nor the running time of the algorithm. Worse still, heuristics do not link directly the input functions and parameters to the output thus obscuring what would be ideal choices of parameters and functions which are to be selected by users in each particular application context.We describe here how MRF can model and solve efficiently several known continuous models for image segmentation and describe briefly a very efficient polynomial time algorithm, which is provably fastest possible, to solve optimally the MRF problem. The MRF algorithm is enhanced here compared to the algorithm in Hochbaum (2001) by allowing the set of assigned labels to be any discrete set. Other enhancements include dynamic features that permit adjustments to the input parameters and solves optimally for these changes with minimal computation time. Several new theoretical results on the properties of the algorithm are proved here and are demonstrated for images in the context of medical and biological imaging. An interactive implementation tool for MRF is described, and its performance and flexibility in practice are demonstrated via computational experiments.We conclude that many continuous models common in image segmentation have discrete analogs to various special cases of MRF and as such are solved optimally and efficiently, rather than with the use of continuous techniques, such as PDE methods, which restrict the type of functions used and furthermore, can only guarantee convergence to a local minimum.     

Decomposition of unreliable assembly/disassembly networks with limited buffer capacity and random processing times

An assembly/disassembly (A/D) network is a manufacturing system in which machines perform assembly and/or disassembly operations. We consider tree-structured systems of unreliable machines that produce discrete parts. Processing times, times to failure and times to repair in the inhomogeneous system are assumed to be stochastic and machine-dependent. Machines are separated by buffers of limited capacity. We develop Markov process models for discrete time and continuous time systems and derive approximate decomposition equations to determine performance measures such as production rate and average buffer levels in an iterative algorithm. An improved parameter updating procedure leads to a dramatic improvement with respect to convergence reliability. Numerical results demonstrate that the methods are quite accurate.     

A lower bound on convergence rates of nonadaptive algorithms for univariate optimization with noise

This paper considers complexity bounds for the problem of approximating the global minimum of a univariate function when the function evaluations are corrupted by random noise. We take an average-case point of view, where the objective function is taken to be a sample function of a Wiener process and the noise is independent Gaussian. Previous papers have bounded the convergence rates of some nonadaptive algorithms. We establish a lower bound on the convergence rate of any nonadaptive algorithm.     

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise II. Fully discrete schemes

We present an abstract framework for analyzing the weak error of fully discrete approximation schemes for linear evolution equations driven by additive Gaussian noise. First, an abstract representation formula is derived for sufficiently smooth test functions. The formula is then applied to the wave equation, where the spatial approximation is done via the standard continuous finite element method and the time discretization via an I-stable rational approximation to the exponential function. It is found that the rate of weak convergence is twice that of strong convergence. Furthermore, in contrast to the parabolic case, higher order schemes in time, such as the Crank-Nicolson scheme, are worthwhile to use if the solution is not very regular. Finally we apply the theory to parabolic equations and detail a weak error estimate for the linearized Cahn-Hilliard-Cook equation as well as comment on the stochastic heat equation.     

On rates of convergence in functional linear regression

This paper investigates the rate of convergence of estimating the regression weight function in a functional linear regression model. It is assumed that the predictor as well as the weight function are smooth and periodic in the sense that the derivatives are equal at the boundary points. Assuming that the functional data are observed at discrete points with measurement error, the complex Fourier basis is adopted in estimating the true data and the regression weight function based on the penalized least-squares criterion. The rate of convergence is then derived for both estimators. A simulation study is also provided to illustrate the numerical performance of our approach, and to make a comparison with the principal component regression approach.     

Robust state estimation and fault diagnosis for uncertain hybrid nonlinear systems

Robust state estimation and fault diagnosis are challenging problems in the research of hybrid systems. In this paper, a novel robust hybrid observer is proposed for a class of uncertain hybrid nonlinear systems with unknown mode transition functions, model uncertainties and unknown disturbances. The observer consists of a mode observer for discrete mode estimation and a continuous observer for continuous state estimation. It is shown that the mode can be identified correctly and the continuous state estimation error is exponentially uniformly bounded. Robustness to unknown transition functions, model uncertainties and disturbances can be guaranteed by disturbance decoupling and selecting proper thresholds. The transition detectability and mode identifiability conditions are rigorously analyzed. Based on the robust hybrid observer, a robust fault diagnosis scheme is presented for faults modeled as discrete modes with unknown transition functions, and the analytical properties are investigated. Simulations of a hybrid three-tank system demonstrate that the proposed approach is effective.     

Discrete schemes for processes in random media

We present discrete schemes for processes in random media. We prove two results. The first one is the convergence of Sinai's random walks in random environments to the Brox model. The second one is the convergence of random walks in media with random “gates” to a continuous process in a Poisson potential. The proofs are based on the following idea: we consider the discrete media as random potentials for continuous models. Received: 6 May 1999 / Revised version: 18 October 1999 / Published online: 20 October 2000     

Ruin probabilities in the discrete time renewal risk model

In this paper, we study the discrete time renewal risk model, an extension to Gerber’s compound binomial model. Under the framework of this extension, we study the aggregate claim amount process and both finite-time and infinite-time ruin probabilities. For completeness, we derive an upper bound and an asymptotic expression for the infinite-time ruin probabilities in this risk model. Also, we demonstrate that the proposed extension can be used to approximate the continuous time renewal risk model (also known as the Sparre Andersen risk model) as Gerber’s compound binomial model has been proposed as a discrete-time version of the classical compound Poisson risk model. This allows us to derive both numerical upper and lower bounds for the infinite-time ruin probabilities defined in the continuous time risk model from their equivalents under the discrete time renewal risk model. Finally, the numerical algorithm proposed to compute infinite-time ruin probabilities in the discrete time renewal risk model is also applied in some of its extensions.     

Convergence properties of ordinal comparison in the simulation of discrete event dynamic systems

Recent research has demonstrated that ordinal comparison has fast convergence despite the possible presence of large estimation noise in the design of discrete event dynamic systems. In this paper, we address the fundamental problem of characterizing the convergence of ordinal comparison. To achieve this goal, an indicator process is formulated and its properties are examined. For several performance measures frequently used in simulation, the rate of convergence for the indicator process is proven to be exponential for regenerative simulations. Therefore, the fast convergence of ordinal comparison is supported and explained in a rigorous framework. Many performance measures of averaging type have asymptotic normal distributions. The results of this paper show that ordinal comparison converges monotonically in the case of averaging normal random variables. Such monotonicity is useful in simulation planning.The author would like to thank C. G. Cassandras, X. Chao, S. G. Strickland, X. Xie, and the reviewers for their helpful suggestions.     

Robust tensor completion using transformed tensor singular value decomposition

In this article, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by using other unitary transform matrices than that by using discrete Fourier transform matrix. This would be more effective for robust tensor completion. Experimental results for hyperspectral, video and face datasets have shown that the recovery performance for the robust tensor completion problem by using transformed tensor SVD is better in peak signal‐to‐noise ratio than that by using Fourier transform and other robust tensor completion methods.     

The detection and estimation of the change point in a disccrete-time stochastic system

A piecewise-constant process containing a single jump is observed under noise in the context of discrete time. The conditional density and maximum a posteriori (MAP) estimator of the jump time as well as the Bayes detector of the jump itself are determined using the powerful measure transformation approach. The Bayes detector provides a convenient sequential detection rule for practical on-line implementation. An asymptotic result for the distribution of the MAP estimator's estimation error and the corresponding convergence rate are derived. This result provides a reference measure of optimal performance for jump-time estimators in discrete-time stochastic systems that does not depend on the jump time's prior distribution