Simulated minimum Hellinger distance estimation of stochastic volatility models

A simultaneously efficient and robust approach for distribution-free parametric inference, called the simulated minimum Hellinger distance (SMHD) estimator, is proposed. In the SMHD estimation, the Hellinger distance between the nonparametrically estimated density of the observed data and that of the simulated samples from the model is minimized. The method is applicable to the situation where the closed-form expression of the model density is intractable but simulating random variables from the model is possible. The robustness of the SMHD estimator is equivalent to the minimum Hellinger distance estimator. The finite sample efficiency of the proposed methodology is found to be comparable to the Bayesian Markov chain Monte Carlo and maximum likelihood Monte Carlo methods and outperform the efficient method of moments estimators. The robustness of the method to a stochastic volatility model is demonstrated by a simulation study. An empirical application to the weekly observations of foreign exchange rates is presented.     

Semiparametric regression for assessing agreement using tolerance bands

This article describes a Bayesian semiparametric approach for assessing agreement between two methods for measuring a continuous variable using tolerance bands. A tolerance band quantifies the extent of agreement in methods as a function of a covariate by estimating the range of their differences in a specified large proportion of population. The mean function of differences is modelled using a penalized spline through its mixed model representation. The covariance matrix of the errors may also depend on a covariate. The Bayesian approach is straightforward to implement using the Markov chain Monte Carlo methodology. It provides an alternative to the rather ad hoc frequentist likelihood-based approaches that do not work well in general. Simulation for two commonly used models and their special cases suggests that the proposed Bayesian method has reasonably good frequentist coverage. Two real data sets are used for illustration, and the Bayesian and the frequentist inferences are compared.     

Feedback and Weighting Mechanisms for Improving Jacobian Estimates in the Adaptive Simultaneous Perturbation Algorithm

It is known that a stochastic approximation (SA) analogue of the deterministic Newton-Raphson algorithm provides an asymptotically optimal or near-optimal form of stochastic search. However, directly determining the required Jacobian matrix (or Hessian matrix for optimization) has often been difficult or impossible in practice. This paper presents a general adaptive SA algorithm that is based on a simple method for estimating the Jacobian matrix while concurrently estimating the primary parameters of interest. Relative to prior methods for adaptively estimating the Jacobian matrix, the paper introduces two enhancements that generally improve the quality of the estimates for underlying Jacobian (Hessian) matrices, thereby improving the quality of the estimates for the primary parameters of interest. The first enhancement rests on a feedback process that uses previous Jacobian estimates to reduce the error in the current estimate. The second enhancement is based on an optimal weighting of per-iteration Jacobian estimates. From the use of simultaneous perturbations, the algorithm requires only a small number of loss function or gradient measurements per iteration—independent of the problem dimension—to adaptively estimate the Jacobian matrix and parameters of primary interest.      

A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability

In estimating the effect of a change in a random variable parameter on the (time-invariant) probability of structural failure estimated through Monte Carlo methods the usual approach is to carry out a duplicate simulation run for each parameter being varied. The associated computational cost may become prohibitive when many random variables are involved. Herein a procedure is proposed in which the numerical results from a Monte Carlo reliability estimation procedure are converted to a form that will allow the basic ideas of the first order reliability method to be employed. Using these allows sensitivity estimates of low computational cost to be made. Illustrative examples with sensitivities computed both by conventional Monte Carlo and the proposed procedure show good agreement over a range of probability distributions for the input random variables and for various complexities of the limit state function.     

Monte Carlo evaluation of methods for pulse transfer function estimation

This paper describes the results of a Monte Carlo evaluation made of the methods proposed in current literature for the estimation of the pulse transfer function of a linear, time-invariant dynamic system with feedback. Considered are two basic methods for estimating the coefficients of a pulse transfer function, given only the normal operating input and output of the system obscured by noise and over a limited period of time. The most commonly proposed method is a linear method in which a set of simultaneous linear equations is formed from the sampled data and the coefficients obtained by a matrix inversion. The other method is an eigenvector method proposed by Levin which uses the eigenvector associated with the smallest eigenvalue of a matrix formed from the sampled data. This paper presents a set of examples designed to compare linear and eigenvector estimation methods and to verify experimentally the theoretical results and approximations given by Levin. The comparison shows that the eigenvector method generally gives estimates with equal or smaller rms errorsqrt{Variance+(Bias)^2}than the linear method. The eigenvector estimates had bias magnitudes which were consistently less than their standard deviations; the linear estimates did not, and thus had rms errors which often consisted largely of the bias. The approximate covariance matrix given by Levin for the coefficients estimated with the eigenvector method is found to be reasonably accurate.     

Nonparametric probability density estimation: improvements to the histogram for laboratory data.

The histogram has long been used in the clinical laboratory for the depiction and manipulation of frequency data. We present recent results of refinements to the usual histogram procedures along with modern alternative methods of estimating frequency distributions, including the kernel and discrete maximum penalized likelihood estimation (DMPLE) approaches. We compared these nonparametric methods on 15 different types of simulated distributions, and on several sets (greater than 1000 subjects/set) of real data, including alanine aminotransferase, aspartate aminotransferase, and lactate dehydrogenase levels. Each frequency curve estimation technique was evaluated by measuring the integrated mean square error between each technique's prediction and the true underlying distribution, using Monte Carlo techniques on sample sets with size 49 and 119. The kernel method was the clear method of choice, both in performance (best in 22/36 cases) and in practical usage.     

On confidence intervals and tests for autocorrelations

We propose a statistical procedure for estimating the asymptotic variances and covariances of sample autocorrelations from a stationary time series so that confidence regions and tests on a finite subset of autocorrelations can be implemented. The corresponding algorithm is described. The accuracy of the asymptotic confidence intervals for finite samples is studied by Monte Carlo simulations. Further, our method is illustrated with examples from the literature.     

Estimating the probability of misclassification and variate selection

This paper considers the problem of estimating the probability of misclassifying normal variates using the usual discriminant function when the parameters are unknown. The probability of misclassification is estimated, by Monte Carlo simulation, as a function of n₁ and n₂ (sample sizes), p (number of variates) and α (measure of separation between the two populations). The probability of misclassification is used to determine, for a given situation, the best number and subset of variates for various sample sizes. An example using real data is given.     

Space–time adaptive finite difference method for European multi-asset options

The multi-dimensional Black–Scholes equation is solved numerically for a European call basket option using a priori–a posteriori error estimates. The equation is discretized by a finite difference method on a Cartesian grid. The grid is adjusted dynamically in space and time to satisfy a bound on the global error. The discretization errors in each time step are estimated and weighted by the solution of the adjoint problem. Bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations. Comparisons are made with Monte Carlo and quasi-Monte Carlo methods in one dimension, and the performance of the method is illustrated by examples in one, two, and three dimensions.     

Integrating subset simulation with probabilistic re-analysis to estimate reliability of dynamic systems

It is often expensive to estimate the failure probability of highly reliable systems by Monte Carlo simulation. Subset Simulation breaks the original problem of estimating a small probability into the estimation of a sequence of large conditional probabilities, which is more efficient. The conditional probabilities are estimated by Markov Chain simulation. Uncertainty in the power spectral density of the excitation makes it necessary to re-evaluate the reliability for many power spectral densities that are consistent with the evidence about the system excitation. Subset Simulation is more efficient than Monte Carlo simulation, but still requires a new simulation for each admissible power spectral density. This paper presents an efficient method to re-evaluate the reliability of a dynamic system under stationary Gaussian stochastic excitation for different load spectra. We accomplish that by re-weighting the results of a single Subset Simulation. This method is applicable to both linear and nonlinear systems provided that all of the spectra contain the same amount of energy. The authors are currently working on an extension of the method to nonlinear systems, even when the sampling and true power spectral density functions contain different amounts of energy.     

不确定控制系统概率鲁棒性分析——自适应重要抽样法

将自适应重要抽样(AIS)法应用于不确定控制系统的概率鲁棒性分析问题,克服了标准MonteCarlo仿真(MCS)方法不能有效解决小概率事件的困难.给出了一种新的AIS方案.首先采用了一种递归的估计条件众数算法来产生一组使得系统不稳定或性能不可接受的不确定参数向量样本.然后利用这组样本来估计初始高斯型重要抽样密度函数的参数,并执行随后的迭代仿真过程.仿真结果验证了该方法的有效性.     

Efficient estimation of distance‐dependent metrics in edge‐failing networks

This article introduces a method for estimating performability metrics built upon non‐binary network states, determined by the hop distances between distinguished nodes. The estimation is performed by a Monte Carlo simulation where the sampling space is reduced using edge sets known as d‐pathsets and d‐cutsets. Numerical experiments over two mesh‐like networks are presented. They show significant efficiency improvements relative to the crude Monte Carlo method, in particular as link failures become rarer events, which is usually the case in most real communication networks.     

Convergence of parameter sensitivity estimates in a stochastic experiment

To reduce the error in estimating the gradient (parameter sensitivity) of an unknown function is of great importance in stochastic optimization problems. Three kinds of parameter sensitivity estimates using the Monte Carlo method are discussed in this paper. The estimates depend on the number of replications,N, and the change in parameter,Delta d. The convergence properties asN rightarrow inftyandDelta d rightarrow 0for these estimates are obtained. The result explains many theoretical and practical issues in the study of discrete event dynamic systems, as well as continuous dynamic systems, by the Monte Carlo method. It is proved that an estimate based on averaging the gradients calculated along each sample path by a perturbation of the path is much better than the other estimates if the output functions are uniformly differentiable with probability one (w.p.1). It is also concluded that in computer simulations one should always choose the same seed for bothdandd + Delta din estimating the parameter sensitivity. Combining the results in this paper with existing stochastic approximation algorithms may yield algorithms with faster convergence.     

Monte Carlo methods for matrix computations on the grid

Many scientific and engineering applications involve inverting large matrices or solving systems of linear algebraic equations. Solving these problems with proven algorithms for direct methods can take very long to compute, as they depend on the size of the matrix. The computational complexity of the stochastic Monte Carlo methods depends only on the number of chains and the length of those chains. The computing power needed by inherently parallel Monte Carlo methods can be satisfied very efficiently by distributed computing technologies such as Grid computing. In this paper we show how a load balanced Monte Carlo method for computing the inverse of a dense matrix can be constructed, show how the method can be implemented on the Grid, and demonstrate how efficiently the method scales on multiple processors.     

A COMMENT ON FAST MONTE CARLO INTEGRATION OF P.D.F. ESTIMATORS

In their recent paper Batchelor and Hand (1) proposed a method of estimating marginal probability density functions based on a fast Monte Carlo integration procedure devised by the authors. In this note it is shown that the required probability density functions can be estimated directly, thus obviating the computationally involved numerical integration altogether.     

A Bayesian propensity score adjustment for latent variable modeling and MCMC algorithm

The estimation of the differences among groups in observational studies is frequently inaccurate owing to a bias caused by differences in the distributions of covariates. In order to estimate the average treatment effects when the treatment variable is binary, Rosenbaum and Rubin [1983. The central role of the propensity score in observational studies for causal effects. Biometrika 70, 41-55] proposed an adjustment method for pre-treatment variables using propensity scores. Imbens [2000. The role of the propensity score in estimating dose-response functions. Biometrika 87, 706-710] extended the propensity score methodology for estimation of average treatment effects with multivalued treatments.However, these studies focused only on estimating the marginal mean structure. In many substantive sciences such as the biological and social sciences, a general estimation method is required to deal with more complex analyses other than regression, such as testing group differences on latent variables. For latent variable models, the EM algorithm or the traditional Monte Carlo methods are necessary. However, in propensity score adjustment, these methods cannot be used because the full distribution is not specified.In this paper, we propose a quasi-Bayesian estimation method for general parametric models that integrate out the distributions of covariates using propensity scores. Although the proposed Bayes estimates are shown to be consistent, they can be calculated by existing Markov chain Monte Carlo methods such as Gibbs sampler. The proposed method is useful to estimate parameters in latent variable models, while the previous methods were unable to provide valid estimates for complex models such as latent variable models.We also illustrated the procedure using the data obtained from the US National Longitudinal Survey of Children and Youth (NLSY1979-2002) for estimating the effect of maternal smoking during pregnancy on the development of the child's cognitive functioning.     

A polynomial approximation-based approach for chance-constrained optimization

We proposed a polynomial approximation-based approach to solve a specific type of chance-constrained optimization problem that can be equivalently transformed into a convex programme. This type of chance-constrained optimization is in great needs of many applications and most solution techniques are problem-specific. Our key contribution is to provide an all-purpose solution approach through Monte Carlo and establish the linkage between our obtained optimal solution with the true optimal solution. Our approach performs well because: First, our method controls approximation errors for both the function value and its gradient (or subgradient) at the same time. This is the primary advantage of our method in comparison to the commonly used finite difference method. Second, the approximation error is well bounded in our method and, with a properly chosen algorithm, the total computational complexity will be polynomial. We also address issues associated with Monte Carlo, such as discontinuity and nondifferentiability of the function. Thanks to fast-advancing computer hardware, our method would be increasingly appealing to businesses, including small businesses. We present the numerical results to show that our method with Monte Carlo will yield high-quality, timely, and stable solutions.     

Autonomous control systems: monitoring, diagnosis and tuning

A systematic and unified approach which accomplishes performance monitoring, performance improvement and fault prediction in control systems is proposed. The feature vector which is a vector formed of the coefficients of the estimate of the sensitivity function and the influence matrix which is the Jacobian of the feature vector with respect to the physical parameter are shown to contain the relevant information to realize an autonomous control system. The feature vector is estimated using a robust, accurate and reliable linear predictive coding algorithm. The influence matrix is computed by perturbing the physical parameters one at a time and estimating the feature vectors for each case. The proposed scheme is evaluated both on simulated as well as on actual control systems     

Monte Carlo analysis of ridged waveguides with transformation media

A computational model is presented for Monte Carlo simulation of waveguides with ridges, by combining the principles of transformation electromagnetics and the finite methods (such as finite element or finite difference methods). The principle idea is to place a transformation medium around the ridge structure, so that a single and easy‐to‐generate mesh can be used for each realization of the Monte Carlo simulation. Hence, this approach leads to less computational resources. The technique is validated by means of various finite element simulations in the context of 3D waveguides of uniform cross‐section. © 2013 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2013.