Dimension Asymptotics for Generalised Bootstrap in Linear Regression

We prove consistency of a class of generalised bootstrap techniques for the distribution of the least squares parameter estimator in linear regression, when the number of parameters tend to infinity with data size and the regressors are random. We show that best results are obtainable with resampling techniques that have not been considered earlier in the literature.     

Covariance estimation under spatial dependence

Correlated multivariate processes have a dependence structure which must be taken into account when estimating the covariance matrix. The natural estimator of the covariance matrix is introduced and is shown that to be biased under the dependence structure. This bias is studied under two different asymptotic models, namely increasing the domain by increasing the number of observations, and increasing the number of observations in the fixed domain. Using the first asymptotic model, we quantify the convergence rate of the bias and of the covariance between the components of the estimated covariance matrix. The second asymptotic model serves to derive a fast and accurate bias correction. As shown, under mild hypotheses, the asymptotic normality of the estimated covariance matrix holds and can be used to test whether the bias is significant, for example, in the sense that the eigenvectors of the estimated and true covariance matrices are significantly different.     

Quantile regression for longitudinal data

The penalized least squares interpretation of the classical random effects estimator suggests a possible way forward for quantile regression models with a large number of “fixed effects”. The introduction of a large number of individual fixed effects can significantly inflate the variability of estimates of other covariate effects. Regularization, or shrinkage of these individual effects toward a common value can help to modify this inflation effect. A general approach to estimating quantile regression models for longitudinal data is proposed employing ?₁ regularization methods. Sparse linear algebra and interior point methods for solving large linear programs are essential computational tools.     

A well-conditioned estimator for large-dimensional covariance matrices

Many applied problems require a covariance matrix estimator that is not only invertible, but also well-conditioned (that is, inverting it does not amplify estimation error). For large-dimensional covariance matrices, the usual estimator—the sample covariance matrix—is typically not well-conditioned and may not even be invertible. This paper introduces an estimator that is both well-conditioned and more accurate than the sample covariance matrix asymptotically. This estimator is distribution-free and has a simple explicit formula that is easy to compute and interpret. It is the asymptotically optimal convex linear combination of the sample covariance matrix with the identity matrix. Optimality is meant with respect to a quadratic loss function, asymptotically as the number of observations and the number of variables go to infinity together. Extensive Monte Carlo confirm that the asymptotic results tend to hold well in finite sample.     

Two approaches to optimal routing and admission control in systems with real-time traffic

This paper addresses the problem of routing and admission control of real-time traffic in a queueing system where customers must begin service within given deadlines (or complete service within given deadlines), otherwise they are considered lost. Performance in such systems is measured by the probability a customer is lost. For a system ofK parallel servers with a probabilistic routing and admission control scheme, the problem of the optimal routing and admission control is considered and two approaches are presented. Assuming the availability of a closed-form expression for the probability of loss at each server, the problem is solved under general conditions and properties of the optimal flow allocation are given. However, such closed-form expressions are often unavailable. This motivates a second approach, which involves a gradient-based stochastic optimization algorithm with on-line gradient estimation. The gradient estimation problem for loss probabilities is solved through a recently-developed smoothed perturbation analysis (SPA) technique. The effectiveness of on-line stochastic optimization using this type of gradient estimator is demonstrated by combining the SPA algorithm with a sampling-controlled stochastic optimization algorithm for the aforementioned routing and admission control problem.This work was supported in part by the Office of Naval Research under Contract N00014-87-K-0304, by the Rome Air Development Center under Contract F30602-88-D-0027, by NASA under Contract NAG 2-595, and by the National Science Foundation under Grant EID-92-12122.The authors are grateful to Don Towsley for several contributions to Section 2 and to an anonymous reviewer for pointing out a redundant assumption in the proof of Lemma 2.1.     

Influence of measurement data metrics on the identification of Interval Fields for the representation of spatial variability in finite element models

Due to the exponential growth in computing power, numerical modelling techniques method have gained an increasing amount of interest for engineering and design applications. Nowadays, the deterministic finite element (FE) method, an efficient tool to accurately solve the Partial Differential Equations (PDE) that govern most real-world problems, has become an indispensable tool for an engineer in various design stages. A more recent trend herein is to use the ever increasing computing power incorporate uncertainty and variability, which is omnipresent is all real-live applications, into these FE models. Several advanced techniques for incorporating either variability between nominally identical parts or spatial variability within one part into the FE models, have been introduced in this context. For the representation of spatial variability on the parameters of an FE model in a possibilistic context, the theory of Interval Fields (IF) was proven to show promising results. Following this approach, variability in the input FE model is introduced as the superposition of base vectors, depicting the spatial ‘patterns’, which are scaled by interval factors, which represent the actual variability. Application of this concept, however, requires identification of the governing parameters of these interval fields, i. e. the base vectors and interval scalars. Recent work of the authors therefore was focussed on finding a solution to the inverse problem, where the spatial uncertainty on the output side of the model is known from measurement data, but the spatial variability on the input parameters is unknown. Based on an a priori knowledge on the constituting base vectors of the interval field, the simulated output of the IFFEM computation is compared to measured data, and the input parameters are iteratively adjusted in order to minimize the discrepancy between the variability in simulation and measurement data. This discrepancy is defined based on geometric properties of the convex sets of both measurement and simulation data. However, the robustness of this methodology with respect to the size of the measurement data set that is used for the identification, as yet remains unclear. This paper therefore is focussed on the investigation of this robustness, by performing the identification on different measurement sets, depicting the same variability in the dynamic response of a simple FE model, which contain a decreasing amount of measurement replica. It was found that accurate identification remains feasible, even under a limited amount of measurement replica, which is highly relevant in the context of a non-probabilistic representation of variability in the FE model parameters. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Penalized profile least squares-based statistical inference for varying coefficient partially linear errors-in-variables models

The purpose of this paper is two fold. First, we investigate estimation for varying coefficient partially linear models in which covariates in the nonparametric part are measured with errors. As there would be some spurious covariates in the linear part, a penalized profile least squares estimation is suggested with the assistance from smoothly clipped absolute deviation penalty. However, the estimator is often biased due to the existence of measurement errors, a bias correction is proposed such that the estimation consistency with the oracle property is proved. Second, based on the estimator, a test statistic is constructed to check a linear hypothesis of the parameters and its asymptotic properties are studied. We prove that the existence of measurement errors causes intractability of the limiting null distribution that requires a Monte Carlo approximation and the absence of the errors can lead to a chi-square limit. Furthermore, confidence regions of the parameter of interest can also be constructed. Simulation studies and a real data example are conducted to examine the performance of our estimators and test statistic.     

A Subsampling Method for the Computation of Multivariate Estimators with High Breakdown Point

Abstract

All known robust location and scale estimators with high breakdown point for multivariate samples are very expensive to compute. In practice, this computation has to be carried out using an approximate subsampling procedure. In this article we describe an alternative subsampling scheme, applicable to both the Stahel-Donoho estimator and the minimum volume ellipsoid estimator, with the property that the number of subsamples required can be substantially reduced with respect to the standard subsampling procedures used in both cases. We also discuss some bias and variability properties of the estimator obtained from the proposed subsampling process.     

The combinatorial bandwidth packing problem

In managing a telecommunications network, decisions need to be made concerning the admission of requests submitted by customers to use the network bandwidth. The classical bandwidth packing problem requires that each request submitted by a customer use network resources to establish a one-to-one connection involving one single pair of nodes. We extend the problem to the more practical case where each request submitted by a customer to use the network resources includes a set or combination of calls. This extension suggests that each request requires one-to-many or many-to-many connections to be established between many communicating node pairs. The extension has applications in many important areas such video conferencing and collaborative computing. The combinatorial nature of the requests makes the admission decision more complex because of bandwidth capacity limitations and call routing difficulties. We develop an integer programming formulation of the problem and propose a procedure that can produce verifiably good feasible solutions to the problem. The results of extensive computational experiments over a wide range of problem structures indicate that the procedure provides verifiably good feasible solutions to the problem within reasonable computational times.     

Efficient computation of generalized median estimators

The generalized median (GM) estimator is a family of robust estimators that balances the competing demands of statistical efficiency and robustness. By choosing a kernel that is efficient for the parameter, the GM estimator gains robustness by computing the median of the kernel evaluated at all possible subsets from the sample. The GM estimator is often computationally infeasible because the number of subsets can be large for even modest sample sizes. Writing the estimator in terms of the quantile function facilitates an approximation using a sample of all possible subsets. While both sampling with and without replacement are feasible, sampling without replacement is preferred because of the reduction in variance from the sampling fraction. The proposed algorithm uses sequential sampling to compute an approximation within a user-chosen margin of error.     

Second order asymptotics in nonlinear regression

It is a well known part of statistical knowledge that first order asymptotically efficient procedures can be misleading for moderate sample sizes. Usually this is demonstrated for some popular special cases including numerical comparisons. Typically the situation is worse if nuisance parameters are present. In this paper we give second order asymptotically efficient tests, confidence regions, and estimators for the nonlinear regression model which are based on the least-squares estimator and the residual sum of squares.     

Confidence sets centered at C p -estimators

Suppose X _n is an observation, or average of observations, on a discretized signal _n that is measured at n time points. The random vector X _n has a N( _n , ² _n I) distribution, the mean and variance being unknown. Under squared error loss, the unbiased estimator X _n of _n can be improved by variable-selection. Consider the candidate estimator _n (A) whose i-th component equals the i-th component of X _n whenever i/(n+1) lies in A and vanishes otherwise. Allow the set A to range over a large collection of possibilities. A C _p -estimator is a candidate estimator that minimizes estimated quadratic loss over A. This paper constructs confidence sets that are centered at a C _p -estimator, have correct asymptotic coverage probabiligy for _n , and are geometrically smaller than or equal to the competing confidence balls centered at X _n . The asymptotics are locally uniform in the parameters ( _n , ² _n ). The results illustrate an approach to inference after variable-selection.     

A class of weighted estimators for additive hazards model in case-cohort studies

Case-cohort sampling is a commonly used and efficient method for studying large cohorts. In many situations, some covariates are easily measured on all cohort subjects, and surrogate measurements of the expensive covariates also may be observed. In this paper, to make full use of the covariate data collected outside the case-cohort sample, we propose＇a class of weighted estimators with general time-varying weights for the additive hazards model, and the estimators are shown to be consistent and asymptotically normal. We also identify the estimator within this class that maximizes efficiency, and simulation studies show that the efficiency gains of the proposed estimator over the existing ones can be substantial in practical situations. A real example is provided.     

Looking in the wrong place for healthcare improvements: A system dynamics study of an accident and emergency department

Accident and Emergency (A&E) units provide a route for patients requiring urgent admission to acute hospitals. Public concern over long waiting times for admissions motivated this study, whose aim is to explore the factors which contribute to such delays. The paper discusses the formulation and calibration of a system dynamics model of the interaction of demand pattern, A&E resource deployment, other hospital processes and bed numbers; and the outputs of policy analysis runs of the model which vary a number of the key parameters. Two significant findings have policy implications. One is that while some delays to patients are unavoidable, reductions can be achieved by selective augmentation of resources within, and relating to, the A&E unit. The second is that reductions in bed numbers do not increase waiting times for emergency admissions, their effect instead being to increase sharply the number of cancellations of admissions for elective surgery. This suggests that basing A&E policy solely on any single criterion will merely succeed in transferring the effects of a resource deficit to a different patient group.     

On the asymptotic analysis of statistical multiplexers with hyper-bursty arrivals

We consider multiplexers in discrete time fed by the superposition of Ternary Markov Sources. Such sources are the natural extension of the Binary Markov Sources (BMS) recently used to model bursty arrivals in a high speed environment. Unlike BMS, we allow sources to have arbitrary (large) variance in the duration of their OFF (silence) or ON (burst) periods.This paper focuses mainly on the impact of large variability either in the ON or OFF period on the performance. Following some asymptotic analysis, simple results on the tail behavior of the number of cells queued in the multiplexer are given.Our results indicate that ignoring the variability in the ON period may grossly underestimate the cell buildup in the multiplexer queue for all levels of the utilization. Furthermore, the impact of large variability of the OFF period depends very much on the utilization of the system. For a lightly-loaded multiplexer (utilization below a given threshold), the impact of large variability of the OFF period is minimal. However, for a heavy-loaded multiplexer (utilization above the threshold) the impact of the large variability in the OFF period is similar to that of the ON period.     

Effects of system parameters on the optimal policy structure in a class of queueing control problems

This paper studies a class of queueing control problems involving commonly used control mechanisms such as admission control and pricing. It is well established that in a number of these problems, there is an optimal policy that can be described by a few parameters. From a design point of view, it is useful to understand how such an optimal policy varies with changes in system parameters. We present a general framework to investigate the policy implications of the changes in system parameters by using event-based dynamic programming. In this framework, the control model is represented by a number of common operators, and the effect of system parameters on the structured optimal policy is analyzed for each individual operator. Whenever a queueing control problem can be modeled by these operators, the effects of system parameters on the optimal policy follow from this analysis.      

Cluster synchronization in community network with hybrid coupling

A general model of community network with hybrid coupling is proposed in this paper. In the community network model with hybrid coupling, the inner connections are in the same type of coupling within the same community and in different types of coupling in different communities. The connections between different pair of communities are also nonidentical. Cluster synchronization of community network with hybrid coupling is investigated via adaptive couplings control scheme. Effective controllers are designed for constructing an effective control scheme and adjusting automatically the adaptive external coupling strength by taking external coupling strength as adaptive variables on a small fraction of network edges. Moreover, the impact of the topology on the synchronizability of community network is investigated. The numerical results reveal that the number of links between communities and the degree of the connector nodes have significant effects on the synchronization performance.     

Robust Parameter Estimation for Stochastic Differential Equations

We consider the estimation of parameters in stochastic differential equations (SDEs). The problem is treated in the setting of nonlinear filtering theory with a degenerate diffusion matrix. A robust stochastic Feynman–Kac representation for solutions of SDEs of Zakai-type is derived. It is verified that these solutions are conditional densities for the conditional measures defined by degenerate filtering problems. We show that the corresponding estimator for the parameters is robust in the following sense: It depends continuously on both the measurement path and on the intensity of the measurement noise. An algorithm based on a Monte-Carlo approach is given for the practical application of the estimator, and numerical results are reported. Mathematics Subject Classifications (2000) Primary: 62M05, 62M20; secondary: 62F15.     

Resurgent aspects of applied exponential asymptotics

In many physical problems, it is important to capture exponentially small effects that lie beyond-all-orders of an algebraic asymptotic expansion; when collected, the full asymptotic expansion is known as a trans-series. Applied exponential asymptotics has been enormously successful in developing practical tools for studying the leading exponentials of a trans-series expansion, typically for singularly perturbed nonlinear differential or integral equations. Separately to applied exponential asymptotics, there exists a related line of research known as Écalle's theory of resurgence, which, via Borel resummation, describes the connection between trans-series and a certain class of holomorphic functions known as resurgent functions. Most applications and examples of Écalle's resurgence theory focus mainly on nonparametric asymptotic expansions (i.e., differential equations without a parameter). The relationships between these latter areas with applied exponential asymptotics have not been thoroughly examined—largely due to differences in language and emphasis. In this work, we establish these connections as an alternative framework to the factorial-over-power ansatz procedure in applied exponential asymptotics and clarify a number of aspects of applied exponential asymptotic methodology, including Van Dyke's rule and the universality of factorial-over-power ansatzes. We provide a number of useful tools for probing more pathological problems in exponential asymptotics and establish a framework for future applications to nonlinear and multidimensional problems in the physical sciences.     

The precise asymptotic behavior of parameter estimators in Ornstein-Uhlenbeck process

In the present paper, we study the asymptotic behavior for estimator of the drift parameter in an Ornstein-Uhlenbeck process. The Lr-convergence rate and the precise asymptotics in the law of iterated logarithm and in the law of logarithm for the estimator are obtained. Moreover, we also get the complete moment convergence of this estimator. The main method of this paper is the deviation inequality for the quadratic functional.