Class of tight bounds on the Q‐function with closed‐form upper bound on relative error

In this paper, we propose a novel class of parametric bounds on the Q‐function, which are lower bounds for 1 ≤ a < 3 and x > x_t = (a (a‐1) / (3‐a))^1/2, and upper bound for a = 3. We prove that the lower and upper bounds on the Q‐function can have the same analytical form that is asymptotically equal, which is a unique feature of our class of tight bounds. For the novel class of bounds and for each particular bound from this class, we derive the beneficial closed‐form expression for the upper bound on the relative error. By comparing the bound tightness for moderate and large argument values not only numerically, but also analytically, we demonstrate that our bounds are tighter compared with the previously reported bounds of similar analytical form complexity.     

Spectral approximation for drainage of an Elastico‐viscous liquid and error analysis

A Fourier‐Galerkin spectral method is proposed and used to analyze a system of quasilinear partial differential equations governing the drainage of liquids of the Oldroyd four‐constant type. It is shown that, Fourier‐Galerkin approximations are convergent with spectral accuracy. An efficient and accurate algorithm based on the Fourier‐Galerkin approximations to the system of quasilinear partial differential equations are developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 492–505, 2012     

Selection of smoothing parameters in<Emphasis Type="Italic">B</Emphasis>-spline nonparametric regression models using information criteria

We consider the use ofB-spline nonparametric regression models estimated by the maximum penalized likelihood method for extracting information from data with complex nonlinear structure. Crucial points inB-spline smoothing are the choices of a smoothing parameter and the number of basis functions, for which several selectors have been proposed based on cross-validation and Akaike information criterion known as AIC. It might be however noticed that AIC is a criterion for evaluating models estimated by the maximum likelihood method, and it was derived under the assumption that the ture distribution belongs to the specified parametric model. In this paper we derive information criteria for evaluatingB-spline nonparametric regression models estimated by the maximum penalized likelihood method in the context of generalized linear models under model misspecification. We use Monte Carlo experiments and real data examples to examine the properties of our criteria including various selectors proposed previously.     

Automatic detection and identification of shocks in Gaussian state‐space models: a Bayesian approach

An automatic monitoring and intervention algorithm that permits the supervision of very general aspects in an univariate linear Gaussian state–space model is proposed. The algorithm makes use of a model comparison and selection approach within a Bayesian framework. In addition, this algorithm incorporates the possibility of eliminating earlier interventions when subsequent evidence against them comes to light. Finally, the procedure is illustrated with two empirical examples taken from the literature. Copyright © 2005 John Wiley & Sons, Ltd.     

A Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient: regularization and error estimates

In this paper, we consider a Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient We show that this problem is ill‐posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under a priori and a posteriori parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd.     

A FE‐BE coupling for a fluid‐structure interaction problem: Hierarchical a posteriori error estimates

In this article, we establish a hierarchical a posteriori error estimate for a coupling of finite elements and boundary elements for a fluid‐structure interaction problem posed in two and three dimensions. These methods combine boundary elements for the exterior fluid and finite elements for the elastic structure. We consider two weak formulations, a nonsymmetric one and a symmetric one, which are both uniquely solvable. We present the reliability and efficiency of the error estimates. For the two dimensional case, we compute local error indicators which allow us to develop an adaptive mesh refinement strategy on triangles. For the three dimensional case, we use hexahedrons as elements. Numerical experiments underline our theoretical results. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

‘Statistical methods for automatic crack detection based on vibrothermography sequence‐of‐image data’ by M. Li,S. D. Holland and W. Q. Meeker: Discussion 2

The paper written by M. Li, S. D. Holland and W. Q. Meeker [Applied Stochastic Models in Business and Industry] presents statistical methods for automatic Crack detection based on vibrothermography sequence‐of‐image data. In particular, a matched filter used to increase the signal‐to‐noise ratio is developed. The review gives suggestions about physical approach and detection criteria. Copyright © 2010 John Wiley & Sons, Ltd.     

A recovery‐based error estimator for finite volume methods of interface problems: Conforming linear elements

In this article, we present a recovery‐based a posteriori error estimator for finite volume methods which use the conforming linear trial functions to approximate elliptic interface problems. The reliability and efficiency bounds for the error estimator are established by recovering the flux from a weighted L² projection to H(div) conforming finite element spaces. Numerical experiments are given to support the conclusions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Selection of variables in two-group discriminant analysis by error rate and Akaike's information criteria

This paper deals with two criteria for selection of variables for the discriminant analysis in the case of two multivariate normal populations with different means and a common covariance matrix. One is based on the estimated error rate of misclassification. The other uses Akaike's information criterion. The asymptotic distributions and error rate risks of the criteria are obtained. The result will prove that the two criteria are asymptotically equivalent in the sense of their asymptotic distributions and error rate risks being identical.     

Log‐symmetric regression models: information criteria and application to movie business and industry data with economic implications

This work deals with log‐symmetric regression models, which are particularly useful when the response variable is continuous, strictly positive, and following an asymmetric distribution, with the possibility of modeling atypical observations by means of robust estimation. In these regression models, the distribution of the random errors is a member of the log‐symmetric family, which is composed by the log‐contaminated‐normal, log‐hyperbolic, log‐normal, log‐power‐exponential, log‐slash and log‐Student‐t distributions, among others. One way to select the best family member in log‐symmetric regression models is using information criteria. In this paper, we formulate log‐symmetric regression models and conduct a Monte Carlo simulation study to investigate the accuracy of popular information criteria, as Akaike, Bayesian, and Hannan‐Quinn, and their respective corrected versions to choose adequate log‐symmetric regressions models. As a business application, a movie data set assembled by authors is analyzed to compare and obtain the best possible log‐symmetric regression model for box offices. The results provide relevant information for model selection criteria in log‐symmetric regressions and for the movie industry. Economic implications of our study are discussed after the numerical illustrations.     

A class of pseudo‐parabolic equations: existence,uniqueness of weak solutions,and error estimates for the Euler‐implicit discretization

In this paper, we investigate a class of pseudo‐parabolic equations. Such equations model two‐phase flow in porous media where dynamic effects are included in the capillary pressure. The existence and uniqueness of a weak solution are proved, and error estimates for an Euler implicit time discretization are obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

A posteriori error estimators for optimal distributed control governed by the first‐order linear hyperbolic equation: DG method

In this article, We analyze the ‐version of the discontinuous Galerkin finite element method (DGFEM) for the distributed first‐order linear hyperbolic optimal control problems. We derive a posteriori error estimators on general finite element meshes which are sharp in the mesh‐width . These error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems. For the DGFEM we admit very general irregular meshes. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Falling and explosive,dormant, and rising markets via multiple‐regime financial time series models

A multiple‐regime threshold nonlinear financial time series model, with a fat‐tailed error distribution, is discussed and Bayesian estimation and inference are considered. Furthermore, approximate Bayesian posterior model comparison among competing models with different numbers of regimes is considered which is effectively a test for the number of required regimes. An adaptive Markov chain Monte Carlo (MCMC) sampling scheme is designed, while importance sampling is employed to estimate Bayesian residuals for model diagnostic testing. Our modeling framework provides a parsimonious representation of well‐known stylized features of financial time series and facilitates statistical inference in the presence of high or explosive persistence and dynamic conditional volatility. We focus on the three‐regime case where the main feature of the model is to capturing of mean and volatility asymmetries in financial markets, while allowing an explosive volatility regime. A simulation study highlights the properties of our MCMC estimators and the accuracy and favourable performance as a model selection tool, compared with a deviance criterion, of the posterior model probability approximation method. An empirical study of eight international oil and gas markets provides strong support for the three‐regime model over its competitors, in most markets, in terms of model posterior probability and in showing three distinct regime behaviours: falling/explosive, dormant and rising markets. Copyright © 2009 John Wiley & Sons, Ltd.     

Finite‐volume schemes for Friedrichs systems in multiple space dimensions: A priori and a posteriori error estimates

We consider a class of finite‐volume schemes on unstructured meshes for symmetric hyperbolic linear systems of balance laws in two and three space dimensions. This class of schemes has been introduced and analyzed by Vila and Villedieu ( 5 ). They have proven an a priori error estimate for approximations of smooth solutions. We extend the results to weak solutions. This is the base to derive an a posteriori error estimate for finite‐volume approximations of weak solutions. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Assessing the Error Probability of the Model Selection Test

The asymptotic error probability of Linhart's model selection test isevaluated, and compared with the nominal significance level. We examine thecase where the expected discrepancies of the candidate models from the truemodel are asymptotically equal. The local alternatives method is employed inthe limiting operation of the asymptotic evaluation. Although the errorprobability under the null hypothesis is actually shown to be equal to orless than the level for most situations, intolerable violations of the errorcontrol are observed for nested models: It is often erroneously concludedthat the smaller model is significantly better than the larger model. Toprevent this violation, a modification of Linhart's test statistic isproposed. The effectiveness of the proposed test is confirmed throughtheoretical analysis and numerical simulations.     

Optimal error estimates of the local discontinuous Galerkin method for nonlinear second‐order elliptic problems on Cartesian grids

In this paper, we study the local discontinuous Galerkin (LDG) methods for two‐dimensional nonlinear second‐order elliptic problems of the type u_xx + u_yy = f(x, y, u, u_x, u_y) , in a rectangular region Ω with classical boundary conditions on the boundary of Ω . Convergence properties for the solution and for the auxiliary variable that approximates its gradient are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L² norm achieve optimal (p + 1)th order convergence, when tensor product polynomials of degree at most p are used. Moreover, we prove that the gradient of the LDG solution is superclose with order p + 1 toward the gradient of Gauss–Radau projection of the exact solution. The results are valid in two space dimensions on Cartesian meshes using tensor product polynomials of degree p ≥ 1 , and for both mixed Dirichlet–Neumann and periodic boundary conditions. Preliminary numerical experiments indicate that our theoretical findings are optimal.     

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM

Averaging techniques are popular tools in adaptive finite element methods since they provide efficient a posteriori error estimates by a simple postprocessing. In the second paper of our analysis of their reliability, we consider conforming -FEM of higher (i.e., not of lowest) order in two or three space dimensions. In this paper, reliablility is shown for conforming higher order finite element methods in a model situation, the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of local averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.

     

An optimal burn‐in policy based on a degradation model

Burn‐in tests help manufacturers detect defective items and remove them before being sold to customers. In a competitive marketplace, cost is a major consideration and not employing a burn‐in test may result in higher and needless expenses. With this in mind, we consider degradation‐based burn‐in tests in which the degradation path follows a Wiener process and weak items are identified when the process crosses a piecewise linear function. We also study linear functions as a special case of such a piecewise linear barrier. Within this setup, we apply a cost model to determine the optimal burn‐in test. Finally, we discuss an illustrative example using GaAs laser degradation data and present an optimal burn‐in test for it.     

The effects of mental model formation on group decision making: An agent‐based simulation

We investigated dynamics of group decision making on complex problems when agents can form mental models of others from discussion history. Results indicated that as the agents' memory capacity increases, the group reaches superficial consensus more easily. Surprisingly, however, the shared mental model of the problem develops only within a limited area of the problem space, because incorporating knowledge from others into one's own knowledge quickly creates local agreement on where relevant solutions are, leaving other potentially useful solutions beyond the scope of discussion. The mechanisms stifling group‐level exploration and their implications for decision making research are discussed. © 2010 Wiley Periodicals, Inc. Complexity 16: 49–57, 2011     

Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM

Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conforming, nonconforming, and mixed low order finite element methods in a model situation: the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.