On a necessary condition for stability in perturbed linear and convex programming

We show that a necessary condition for stable perturbations in linear and convex programming is valid on an arbitrary region of stability. Using point-to-set mappings, two new regions of stability are identified.Contribution of this author is part of his M.Sc. thesis in Applied Mathematics at McGill University.     

Linearity of the core correspondence

Bloch and de Clippel (J Econ Theory 145:2424–2434, 2010) characterized sets of balanced TU-games on which the core correspondence is linear by means of an equivalence relation. We characterize maximal regions on which the core correspondence is linear in four different ways. First, by finitely many linear equalities and inequalities; thus, the core is piecewise linear. Second, maximal linear regions coincide with closures of equivalence classes (in the sense of Bloch and de Clippel) that are maximal w.r.t. set inclusion. Third, maximal linear regions coincide with closures of equivalence classes of full dimension. Fourth, for every extreme point of the core of a game in the interior of a maximal linear region, the collection of tight core inequalities constitutes a basis.     

缺失数据下线性模型中回归参数的经验似然置信区域

在缺失样本下,构造了线性模型中参数的调整的经验似然置信域,数值模拟表明调整的经验似然置信域有较好的覆盖率和精度.     

Robustness of the efficient DMUs in data envelopment analysis

By means of modified versions of CCR model based on evaluation of a decision making unit (DMU) relative to a reference set grouped by all other DMUs, sensitivity analysis of the CCR model in data envelopment analysis (DEA) is studied in this paper. The methods for sensitivity analysis are linear programming problems whose optimal values yield particular regions of stability. Sufficient and necessary conditions for upward variations of inputs and for downward variations of outputs of an (extremely) efficient DMU which remains efficient are provided. The approach does not require calculation of the basic solutions and of the inverse of the corresponding optimal basis matrix. The approach is illustrated by two numerical examples.     

Nonsensitiveness Regions for Threshold Ellipsoids

The problem is to determine nonsensitiveness regions for threshold ellipsoids within a regular mixed linear model.     

Empirical likelihood confidence regions of the parameters in a partially linear single-index model

In this paper,a partially linear single-index model is investigated,and three empirical log-likelihood ratio statistics for the unknown parameters in the model are sug- gested.It is proved that the proposed statistics are asymptotically standard chi-square un- der some suitable conditions,and hence can be used to construct the confidence regions of the parameters.Our methods can also deal with the confidence region construction for the index in the pure single-index model.A simulation study indicates that,in terms of cov- erage probabilities and average areas of the confidence regions,the proposed methods perform better than the least-squares method.     

EMPIRICAL LIKELIHOOD FOR LINEAR MODELS UNDER m-DEPENDENT ERRORS

In this paper，the empirical likelihood confidence regions for the regression coefficient in a linear model are constructed under m-dependent errors. It is shown that the blockwise empirical likelihood is a good way to deal with dependent samples.     

缺失数据下线性EV模型中参数的经验似然置信域

考虑了在响应变量随机缺失情形下的线性EV模型.通过利用回归借补方法,构造了未知参数的两种经验对数似然比统计量,即估计的经验对数似然比统计量和调整的经验对数似然比统计量.证明了所构造的经验似然比统计量渐近于χ2分布,所得结果可以用来构造未知参数的置信域.     

Linear minimax estimation with ellipsoidal constraints

We consider the simultaneous linear minimax estimation problem in linear models with ellipsoidal constraints imposed on an unknown parameter. Using convex analysis, we derive necessary and sufficient optimality conditions for a matrix to define the linear minimax estimator. For certain regions of the set of characteristics of linear models and constraints, we exploit these optimality conditions and get explicit formulae for linear minimax estimators.     

区间数据情形下线性模型的经验似然推断

§1Introduction Instatisticalapplications,weoftenencounterintervalcensoreddatawhenafailure timeYcannotbeobserved,butcanonlybedeterminedtolieinanintervalobtainedfroma sequenceofexaminationtimes.Forinstance,themaximumdosagewhichpatientscan endureisconcerned.LetYibethemaximumdosagewhichtheithpatientcanendure,Ui,j(j=1,2,...,k)bethedosagewhichthepatienthasbeentested.ItisobviousthatYiis unobservable.SupposetheithpatientisnormalwhenthedosageisUi,j,andhe(orshe)is abnormalwhenthedosageisUi,j+1.Then…     

Optimal two-phase vaccine allocation to geographically different regions under uncertainty

In this article, we consider a decision process in which vaccination is performed in two phases to contain the outbreak of an infectious disease in a set of geographic regions. In the first phase, a limited number of vaccine doses are allocated to each region; in the second phase, additional doses may be allocated to regions in which the epidemic has not been contained. We develop a simulation model to capture the epidemic dynamics in each region for different vaccination levels. We formulate the vaccine allocation problem as a two-stage stochastic linear program (2-SLP) and use the special problem structure to reduce it to a linear program with a similar size to that of the first stage problem. We also present a Newsvendor model formulation of the problem which provides a closed form solution for the optimal allocation. We construct test cases motivated by vaccine planning for seasonal influenza in the state of North Carolina. Using the 2-SLP formulation, we estimate the value of the stochastic solution and the expected value of perfect information. We also propose and test an easy to implement heuristic for vaccine allocation. We show that our proposed two-phase vaccination policy potentially results in a lower attack rate and a considerable saving in vaccine production and administration cost.     

Fully Optimal Bases and the Active Bijection in Graphs,Hyperplane Arrangements,and Oriented Matroids

In this note, we present the main results of a series of forthcoming papers, dealing with bi-jective generalizations of some counting formulas. New intrinsic constructions in oriented matroids on a linearly ordered set of elements establish notably structural links between counting regions and linear programming. We introduce fully optimal bases, which have a simple combinatorial characterization, and strengthen the well-known optimal bases of linear programming. Our main result is that every bounded region of an ordered hyperplane arrangement, or ordered oriented matroid, has a unique fully optimal basis, providing the active bijection between bounded regions and uniactive internal bases. The active bijec-tion is extended to an activity preserving mapping between all reorientations and all bases of an ordered oriented matroid. It gives a bijective interpretation of the equality of two expressions for the Tutte polynomial, as well as a new expression of this polynomial in terms of beta invariants of minors. There are several refinements, such as an activity preserving bijection between regions (acyclic reorientations) and no-broken-circuit subsets, and others in terms of hyperplane arrangements, graphs, and permutations.     

An empirical likelihood method in a partially linear single-index model with right censored data

Empirical-likelihood-based inference for the parameters in a partially linear single-index model with randomly censored data is investigated. We introduce an estimated empirical likelihood for the parameters using a synthetic data approach and show that its limiting distribution is a mixture of central chi-squared distribution. To attack this difficulty we propose an adjusted empirical likelihood to achieve the standard χ2-limit. Furthermore, since the index is of norm 1, we use this constraint to reduce the dimension of parameters, which increases the accuracy of the confidence regions. A simulation study is carried out to compare its finite-sample properties with the existing method. An application to a real data set is illustrated.     

Modelling structural recovery in amorphous polymers

Constitutive equations are derived for linear viscoelastic response and enthalpy relaxation in amorphous polymers in the vicinity of the glass transition temperature. According to the concept of cooperative relaxation, a glassy polymer is treated as an ensemble of weakly-connected relaxing regions. Mechanical relaxation in a region occurs when the thermally activated flow unit reaches some liquid-like state. Structural recovery is modeled as a sequence of hops in which rearranging regions change their traps. A constitutive model for the linear viscoelastic behavior and enthalpy relaxation in a glassy polymer is validated using experimental data in mechanical and calorimetric tests on polycarbonate and poly (methyl methacrylate). Fair agreement is demonstrated between observations and results of numerical simulation.     

Empirical likelihood-based subset selection for partially linear autoregressive models

Based on the empirical likelihood method, the subset selection and hypothesis test for parameters in a partially linear autoregressive model are investigated. We show that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. We then present the definitions of the empirical likelihood-based Bayes information criteria （EBIC） and Akaike information criteria （EAIC）. The results show that EBIC is consistent at selecting subset variables while EAIC is not. Simulation studies demonstrate that the proposed empirical likelihood confidence regions have better coverage probabilities than the least square method, while EBIC has a higher chance to select the true model than EAIC.     

Empirical likelihood for partial linear models

In this paper the empirical likelihood method due to Owen (1988,Biometrika,75, 237–249) is applied to partial linear random models. A nonparametric version of Wilks' theorem is derived. The theorem is then used to construct confidence regions of the parameter vector in the partial linear models, which has correct asymptotic coverage. A simulation study is conducted to compare the empirical likelihood and normal approximation based method. Research supported by NNSF of China and a grant to the first author for his excellent Ph.D. dissertation work in China. Research supported by Hong Kong RGC CERG No. HKUST6162/97P.     

Self‐organization of genic and intergenic sequence lengths in genomes: Statistical properties and linguistic coherence

In a genome, genes (coding constituents) are interrupted by intergenic regions (noncoding constituents). This study provides a general picture of the large‐scale self‐organization of coding, noncoding, and total constituent lengths in genomes. Ten model genomes were examined and strong correlations between the number of genomic constituents and the constituent lengths were observed. The analysis was carried out by adopting a linguistic distribution model and a structural analogy between linguistic and genomic constructs. The proposed linguistic‐based statistical analysis may provide a fundamental basis for both understanding the linear structural formation of genomic constituents and developing insightful strategies to figure out the function of genic and intergenic regions in genomic sequences. © 2014 Wiley Periodicals, Inc. Complexity 21: 268–282, 2015     

Subdifferentiability and the Duality Gap

We point out a connection between sensitivity analysis and the fundamental theorem of linear programming by characterizing when a linear programming problem has no duality gap. The main result is that the value function is subdifferentiable at the primal constraint if and only if there exists an optimal dual solution and there is no duality gap. To illustrate the subtlety of the condition, we extend Kretschmer's gap example to construct (as the value function of a linear programming problem) a convex function which is subdifferentiable at a point but is not continuous there. We also apply the theorem to the continuum version of the assignment model.     

The geometry and number of the root invariant regions for linear systems

The stability domain is a feasible set for numerous optimization problems. D-decomposition technique is targeted to describe the stability domain in the parameter space for linear parameter-dependent systems. This technique is very simple and efficient for robust stability analysis and design of low-order controllers. However, the geometry of the arising parameter space decomposition into root invariant regions has not been studied in detail; it is an objective of the present paper. We estimate the number of root invariant regions and provide examples, where this number is attained.     

An application of DEA for comparative analysis and ranking of regions in Serbia with regards to social-economic development

In this paper, we use data envelopment analysis (DEA) to estimate how well regions in Serbia utilize their resources. Based on data for four inputs and four outputs we applied an output-oriented CCR DEA model and it appears that 17 out of 30 regions are efficient. For each inefficient unit, DEA identifies the sources and level of inefficiency for each input and output. An output-oriented set of targets is determined for 13 inefficient regions. In addition, the possibilities of combining DEA and linear discriminant analysis (LDA) in evaluating performance are explored. The efficient regions are ranked using a cross efficiency matrix and an output-oriented version of Andersen–Petersen’s DEA model and the results are analyzed and compared.