Asymptotic Error Research Articles

Asymptotic Standard Errors of IRT Linking Coefficients for the Bifactor Extension of the 3PL Model

The bifactor extension of the three-parameter logistic (B3PL) model has been used in applications of multidimensional item response theory (IRT) such as test equating and vertical scaling. Developing a common multidimensional IRT scale (that is, multidimensional coordinate system) is critical in those applications. Three common-item scale linking methods, the direct least squares (DLS), mean/least squares (MLS), and item response function (IRF) methods, for the B3PL model have been found to be effective in developing a common multidimensional IRT ability scale between two test forms to be linked. In this paper, the asymptotic standard errors (SEs) of IRT linking coefficients estimated by the DLS, MLS, and IRF methods are derived assuming that the B3PL model holds and the asymptotic variance-covariance matrix of item parameter estimates from separate calibrations is available. The delta method is used for the derivations. Computer simulations which investigate the accuracy of the derivations under various conditions are given, showing that the derivations are reasonably accurate when sufficiently large samples are used and that in general the SEs of the IRF method are smaller than those of the DLS and MLS methods. The simulation results also suggest that the SEs of linking coefficient estimates are, approximately, inversely proportional to the square root of the sample size when two test forms are administered to the same number of examinees.

Asymptotic error in Euler’s method with a constant step size

In the gradient descent method, one often focus on the convergence of the sequence generated by the algorithm, but less often on the deviation of these points from the solutions of the original continuous-time differential equation (gradient flow). This also happens when discretizing other ordinary differential equations. In the case of a discretization by explicit Euler’s method with a constant step h, we provide here sufficient conditions, in terms of strong monotonicity and co-coercivity, for the deviation between discrete and continuous solutions to tend asymptotically towards zero. This analysis could shed new light on some applications of the gradient descent algorithm.

Performance Optimization via Sequential Processing for Nonlinear State Estimation of Noisy Systems

We propose a framework for designing observers for noisy nonlinear systems with global convergence properties and performing robustness and noise sensitivity. This framework comes out from the combination of a state norm estimator with a chain of filters, adaptively tuned by the state norm estimator. The state estimate is sequentially processed through the chain of filters. Each filter contributes to improving, by a certain amount, the estimation error performances of the previous filter in terms of noise sensitivity, and this amount is quantitatively evaluated using a comparison criterion, which considers the ratio of the asymptotic error norm bounds of two consecutive filters in the chain. A recursive algorithm is given for implementing the chain of filters and guaranteeing a sequential error performance optimization process. Simulations show the effectiveness of these chains of filters.

Code-to-code SIMMER/FRENETIC comparison for the neutronic simulation of lead-cooled fast reactors

This paper presents a neutronic benchmark between the FRENETIC and the SIMMER codes, performed considering the core of the ALFRED lead-cooled fast reactor. FRENETIC includes a nodal diffusion module developed for fast full-core analyses of safety-relevant transients in liquid-metal fast reactors, while SIMMER is a reference tool for qualifying such kind of systems in accidental conditions. In order to highlight the influence of the different numerical methods available in the two codes on the output responses, macroscopic parameters like the effective multiplication factor and the linear power are compared in an increasingly detailed simulation framework. The 3D full-core FRENETIC results are compared at first to a cylindrical 2D SIMMER core model, to highlight the impact of the absorber rings location. Then, a 3D full-core model is also employed in SIMMER to assess the differences, with respect to FRENETIC, induced by the adoption of a Cartesian mesh and of the discrete ordinates method. The full input data consistency is ensured providing the same set of few-group material constants, generated with the Monte Carlo code Serpent 2. Finally, the W-ε scheme is applied to accelerate the angular and spatial convergence of the keff sequences computed by SIMMER in order to estimate the asymptotic errors with respect to the reference Serpent calculation.

Asymptotically correct boundary conditions for the higher-order theory of plate bending

The construction of refined boundary conditions for the case of edge loading, corresponding to the theory with modified inertia in respect to the order of asymptotic approximation, is considered. The methods of investigation are based on the works of A. L. Goldenveizer and A. V. Kolos. The solution of three-dimensional (3D) problem is constructed as a superposition of the long-wave approximation and boundary layers. The traction on the edge is presented as generalized Fourier expansions, using the Legendre polynomials. This approach allows to obtain explicit expressions for the coefficients of boundary layers via an iteration procedure, in course of which the boundary conditions are satisfied with a pure asymptotic error. As a result, the refined boundary conditions (RBCs) are constructed. Effective stress resultants and couples are introduced, the coefficients of which (the Goldenveizer–Kolos constants) are the simple polynomial functions of Poisson’s ratio. In addition, the contribution of boundary layers to edge displacements is determined. The comparison of the dispersion curve for the edge wave, calculated on the basis of the theory with modified inertia and RBC, with the 3D solution show good agreement in the 10 times wider frequency range than that of the Kirchhoff’s theory. The same result is obtained for the amplitude of a transient edge wave, excited by edge loading. It is also shown that with making use of RBC one can consider the action of a self-equilibrated edge load in the framework of two-dimensional (2D) theory.

Rank-Based Greedy Model Averaging for High-Dimensional Survival Data

ABSTRACT Model averaging is an effective way to enhance prediction accuracy. However, most previous works focus on low-dimensional settings with completely observed responses. To attain an accurate prediction for the risk effect of survival data with high-dimensional predictors, we propose a novel method: rank-based greedy (RG) model averaging. Specifically, adopting the transformation model with splitting predictors as working models, we doubly use the smooth concordance index function to derive the candidate predictions and optimal model weights. The final prediction is achieved by weighted averaging all the candidates. Our approach is flexible, computationally efficient, and robust against model misspecification, as it neither requires the correctness of a joint model nor involves the estimation of the transformation function. We further adopt the greedy algorithm for high dimensions. Theoretically, we derive an asymptotic error bound for the optimal weights under some mild conditions. In addition, the summation of weights assigned to the correct candidate submodels is proven to approach one in probability when there are correct models included among the candidate submodels. Extensive numerical studies are carried out using both simulated and real datasets to show the proposed approach’s robust performance compared to the existing regularization approaches. Supplementary materials for this article are available online.

On optimal two-stage testing of multiple mediators.

Mediation analysis in high‐dimensional settings often involves identifying potential mediators among a large number of measured variables. For this purpose, a two‐step familywise error rate procedure called ScreenMin has been recently proposed. In ScreenMin, variables are first screened and only those that pass the screening are tested. The proposed data‐independent threshold for selection has been shown to guarantee asymptotic familywise error rate. In this work, we investigate the impact of the threshold on the finite‐sample familywise error rate. We derive a power maximizing threshold and show that it is well approximated by an adaptive threshold of Wang et al. (2016, arXiv preprint arXiv:1610.03330). We illustrate the investigated procedures on a case‐control study examining the effect of fish intake on the risk of colorectal adenoma. We also apply our procedure in the context of replicability analysis to identify single nucleotide polymorphisms (SNP) associated with crop yield in two distinct environments.

A general approach to the study of the convergence of Picard iteration with an application to Halley's method for multiple zeros of analytic functions

In this paper, we define a new wide class of iteration functions and then we use it to prove a general convergence theorem that provides exact domain of initial approximations to guarantee the high Q-order of convergence of Picard iteration generated by this class of functions. As an application of this theorem, we prove some local convergence theorems about the famous Halley's method for simple and multiple zeros of analytic functions. All obtained results are supplied with a priori and a posteriori error estimates as well as with assessments of the asymptotic error constants.

Stable Simultaneous Inertia and Disturbance Torque Identification for SPMSM Drive Systems Subject to Mismatched Rotor Flux Linkage

In surface-mounted permanent magnet synchronous motor (SPMSM) drives, the distinguished orthogonal-principle-based method is presently popular for handling simultaneous inertia and disturbance torque estimation due to its simplicity and ease of implementation. Nevertheless, it is facing challenges posed by the rotor flux-linkage mismatch, zero-derivative constraint, and operating-condition limitation. To cope with these issues, this article presents an alternative scheme, which explores a stable simultaneous inertia and disturbance torque estimation with the real-time flux-linkage correction. First, an Adaline-based estimator is developed to acquire the disturbance torque. With an input-adaptive learning rate designed by the Lyapunov theory, this estimator can prevent potential convergence failure. Then, an extended sliding-mode observer for inertia estimation is proposed regarding the inertia as a new system state. By using the identified disturbance torque to devise an adaptive feedback gain, this observer guarantees the asymptotic error convergence. Finally, to counteract the flux-linkage mismatch, a flux-linkage observer with a time-varying learning rate is constructed utilizing the Adaline technique. Moreover, it attains robust convergence and only has one to-be-adjusted parameter. The effectiveness of our scheme is validated by abundant simulations and real-time experiments.

Optimal subsampling for large‐sample quantile regression with massive data

AbstractTo balance the explosive growth of data volume and limited budgets for computational resources, one of the popular methods is downscaling the data volume by subsampling a subdataset that inherits the relevant property of the full data. As an alternative to the mean regression model, the quantile regression model has been studied extensively when the data are independent and the data scale is medium. This article focuses on quantile regression with massive data where the sample size n (greater than in general) is extraordinarily large but the dimension d (smaller than 20 in general) is small. We first formulate the general subsampling procedure and establish the asymptotic property of the resultant estimator. Then, with the help of optimality criteria in experimental design, we derive two subsampling probabilities that are optimal in the sense of smallest asymptotic mean square error. Since the optimal subsampling probabilities depend on the full data estimator, we develop a two‐step optimal subsampling algorithm and study the consistency and asymptotic normality of the resultant estimator. The empirical performance of the optimal subsampling algorithm is evaluated with synthetic and real datasets.

Derivation of a Numerical Method with Free Second-order Derivatives

We have proposed the second-derivative-free numerical method and determined the control parameters to converge cubically. In addition, we have developed the order of convergence and the asymptotic error constant. Applying this iterative scheme to a variety of examples, numerical results have shown a successful asymptotic error constants with cubic convergence.

Near-Optimal Coding for Many-User Multiple Access Channels

This paper considers the Gaussian multiple-access channel (MAC) in the asymptotic regime where the number of users grows linearly with the code length. We propose efficient coding schemes based on random linear models with approximate message passing (AMP) decoding and derive the asymptotic error rate achieved for a given user density, user payload (in bits), and user energy. The tradeoff between energy-per-bit and achievable user density (for a fixed user payload and target error rate) is studied, and it is demonstrated that in the large system limit, a spatially coupled coding scheme with AMP decoding achieves near-optimal tradeoffs for a wide range of user densities. Furthermore, in the regime where the user payload is large, we also study the tradeoff between energy-per-bit and spectral efficiency and discuss methods to reduce decoding complexity.

Bayesian and Non-Bayesian Reliability Estimation of Stress-Strength Model for Power-Modified Lindley Distribution.

A two-parameter continuous distribution, namely, power-modified Lindley (PML), is proposed. Various structural properties of the new distribution, including moments, moment-generating function, conditional moments, mean deviations, mean residual lifetime, and mean past lifetime, are provided. The reliability of a system is discussed when the strength of the system and the stress imposed on it are independent. Maximum-likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Bayesian estimation methods of the parameters with independent gamma prior are discussed based on symmetric and asymmetric loss functions. We proposed using the MCMC technique with the Metropolis–Hastings algorithm to approximate the posteriors of the stress-strength parameters for Bayesian calculations. The confidence interval for likelihood and the Bayesian estimation method is obtained for the parameter of the model and stress-strength reliability. We prove empirically the importance and flexibility of the new distribution in modeling with real data applications.

A computationally efficient adaptive online state-of-charge observer for Lithium-ion battery for electric vehicle

Due to the large number of cells installed in electric vehicle (EV), its battery management system (BMS) requires efficient state of charge (SOC) estimation algorithm. Since there is an impetus to reduce the computational burden (while retaining an acceptable accuracy), this paper proposes a simple and fast online adaptive observer for SOC estimation of Lithium-ion battery. The observer has several attractive features: first, its stability is proven by Lyapunov approach where asymptotic error convergence is guaranteed. Second, the computational requirements are low since it contains a few simple recursive equations without matrix inversion. Third, it is adaptive and achieves simultaneous online estimation of SOC and most of the battery parameters. The practical implementation using a 3 Ah battery proves the effectiveness of the proposed observer under dynamic stress test (DST). The testing with real EV profiles (supplemental federal test procedure which is known as US06 and federal urban driving schedule FUDS) is also performed to show the reliability. It is confirmed that the computation time of the proposed algorithm is reduced by approximately 2.5 times in compared to the extended Kalman filter-recursive least square (EKF-RLS) method. Despite the reduction in computation time, the errors are comparable to the latter. The low computational cost is significant when considering the need to accurately estimate the SOC of a large number of cells in a battery pack of an EV.

The Sceptical Bayes Factor for the Assessment of Replication Success

AbstractReplication studies are increasingly conducted but there is no established statistical criterion for replication success. We propose a novel approach combining reverse-Bayes analysis with Bayesian hypothesis testing: a sceptical prior is determined for the effect size such that the original finding is no longer convincing in terms of a Bayes factor. This prior is then contrasted to an advocacy prior (the reference posterior of the effect size based on the original study), and replication success is declared if the replication data favour the advocacy over the sceptical prior at a higher level than the original data favoured the sceptical prior over the null hypothesis. The sceptical Bayes factor is the highest level where replication success can be declared. A comparison to existing methods reveals that the sceptical Bayes factor combines several notions of replicability: it ensures that both studies show sufficient evidence against the null and penalises incompatibility of their effect estimates. Analysis of asymptotic properties and error rates, as well as case studies from the Social Sciences Replication Project show the advantages of the method for the assessment of replicability.

Pointwise comparison of two multivariate density functions

AbstractTesting the equality of two density functions based on independent samples is a classical problem in statistics. While the focus is often on global equality, it is also of interest to conduct local comparisons of density functions. Typically a type of Wald statistic is employed, where the local difference in densities is standardized by an estimate of the asymptotic standard error of that difference. We study the null distribution of this test statistic. The literature has suggested that this will be asymptotically standard normal, but we show that this is by no means always the case. In particular, when using bandwidth matrices of optimal order (for estimation), we prove that the asymptotic mean of this null distribution is nonzero when either the sample sizes differ, or when the Hessian matrices of the densities differ at the point where the densities are equal. In numerical studies we find the erroneous use of the standard normal null distribution in such cases can severely corrupt the test size. We show that these problems can be managed effectively by using common bandwidths when the Hessian matrices are equal, and applying adjusted undersmoothing bandwidth matrices when they are not.

Information geometry approach to parameter estimation in hidden Markov model

We consider the estimation of the transition matrix of a hidden Markovian process by using information geometry with respect to transition matrices. In this paper, only the histogram of k-memory data is used for the estimation. To establish our method, we focus on a partial observation model with the Markovian process and we propose an efficient estimator whose asymptotic estimation error is given as the inverse of projective Fisher information of transition matrices. This estimator is applied to the estimation of the transition matrix of the hidden Markovian process. In this application, we carefully discuss the equivalence problem for hidden Markovian process on the tangent space.

Unknown input observer-based appointed-time funnel control for quadrotors

In this paper, an unknown input observer-based appointed-time funnel control policy is proposed for quadrotors suffering from environmental disturbances and parametric uncertainties. To improve the robustness of quadrotors, unknown input observers (UIOs) are respectively devised in trajectory and attitude levels to online identify unknown system dynamics via utilizing filtering manipulations, possessing a concise structure, low calculation consumption and asymptotic error convergence. Then, an appointed-time funnel control consisting of a modified funnel variable is proposed to regulate trajectory tracking error profiles within preassigned boundary without incurring singularity problem, where a continuous piecewise function is embedded to specify the funnel envelop, such that the stringent temporal constraints can be arbitrarily prescribed by operators, independent of initial states and system arguments. All error signals involved in quadrotor dynamics are demonstrated to be ultimately uniformly bounded (UUB) and trajectory tracking errors are constrained to designed funnel. Eventually, simulations and comparisons validate the effectiveness of proposed control approach.

Exponential Stability With RISE Controllers

A class of continuous robust controllers termed Robust Integral of the Sign of the Error (RISE) have been published over the past two decades as a means to yield asymptotic tracking error convergence and asymptotic identification of time-varying uncertainties, for classes of nonlinear systems that are subject to sufficiently smooth bounded exogenous disturbances and/or modeling uncertainties. Despite the wide application of RISE-based techniques, an open question that has eluded researchers during this time-span is whether the asymptotic tracking error convergence is also uniform or exponential. This question has remained open due to certain limitations in the traditional construction of a Lyapunov function for RISE-based error systems. In this letter, new insights on the construction of a Lyapunov function are used that result in an exponential stability result for RISE-based controllers. As an outcome of this breakthrough, the inherent learning capability of RISE-based controllers is shown to yield exponential identification of state-dependent disturbances/uncertainty.

Lyapunov-Derived Control and Adaptive Update Laws for Inner and Outer Layer Weights of a Deep Neural Network

Lyapunov-based real-time update laws are well-known for neural network (NN)-based adaptive controllers that control nonlinear dynamical systems using single-hidden-layer NNs. However, developing real-time weight update laws for deep NNs (DNNs) remains an open question. This letter presents the first result with Lyapunov-based real-time weight adaptation laws for each layer of a feedforward DNN-based control architecture, with stability guarantees. Additionally, the developed method allows nonsmooth activation functions to be used in the DNN to facilitate improved transient performance. A nonsmooth Lyapunov-based stability analysis proves global asymptotic tracking error convergence. Simulation results are provided for a nonlinear system using DNNs with leaky rectified linear unit (LReLU) and hyperbolic tangent activation functions to demonstrate the efficacy of the developed method.