On the numerical solution of the quadratic eigenvalue complementarity problem

The Quadratic Eigenvalue Complementarity Problem (QEiCP) is an extension of the Eigenvalue Complementarity Problem (EiCP) that has been introduced recently. Similar to the EiCP, the QEiCP always has a solution under reasonable hypotheses on the matrices included in its definition. This has been established in a previous paper by reducing a QEiCP of dimension n to a special 2n-order EiCP. In this paper we propose an enumerative algorithm for solving the QEiCP by exploiting this equivalence with an EiCP. The algorithm seeks a global minimum of a special Nonlinear Programming Problem (NLP) with a known global optimal value. The algorithm is shown to perform very well in practice but in some cases terminates with only an approximate optimal solution to NLP. Hence, we propose a hybrid method that combines the enumerative method with a fast and local semi-smooth method to overcome the latter drawback. This algorithm is also shown to be useful for computing a positive eigenvalue for an EiCP under similar assumptions. Computational experience is reported to demonstrate the efficacy and efficiency of the hybrid enumerative method for solving the QEiCP.     

A pseudo-empirical log-likelihood estimator using scrambled responses

In this paper, we propose an empirical log-likelihood estimator for estimating the population mean of a sensitive variable in the presence of an auxiliary variable. A new concept of conditional mean squared error of the empirical likelihood estimator is introduced. The proposed method is valid for simple random and without replacement sampling (SRSWOR) and could easily be extended for complex survey designs. The relative efficiency of the proposed pseudo-empirical log-likelihood estimator with respect to the usual, and to a recent estimator due to Diana and Perri (2009b), has been investigated through a simulation study.     

On the enumerative nature of Gomory’s dual cutting plane method

For 30 years after their invention half a century ago, cutting planes for integer programs have been an object of theoretical investigations that had no apparent practical use. When they finally proved their practical usefulness in the late eighties, that happened in the framework of branch and bound procedures, as an auxiliary tool meant to reduce the number of enumerated nodes. To this day, pure cutting plane methods alone have poor convergence properties and are typically not used in practice. Our reason for studying them is our belief that these negative properties can be understood and thus remedied only based on a thorough investigation of such procedures in their pure form. In this paper, the second in a sequence, we address some important issues arising when designing a computationally sound pure cutting plane method. We analyze the dual cutting plane procedure proposed by Gomory in 1958, which is the first (and most famous) convergent cutting plane method for integer linear programming. We focus on the enumerative nature of this method as evidenced by the relative computational success of its lexicographic version (as documented in our previous paper on the subject), and we propose new versions of Gomory’s cutting plane procedure with an improved performance. In particular, the new versions are based on enumerative schemes that treat the objective function implicitly, and redefine the lexicographic order on the fly to mimic a sound branching strategy. Preliminary computational results are reported.     

Robust estimation in a nonlinear cointegration model

This paper considers the nonparametric M-estimator in a nonlinear cointegration type model. The local time density argument, which was developed by Phillips and Park (1998) [6] and Wang and Phillips (2009) [9], is applied to establish the asymptotic theory for the nonparametric M-estimator. The weak consistency and the asymptotic distribution of the proposed estimator are established under mild conditions. Meanwhile, the asymptotic distribution of the local least squares estimator and the local least absolute distance estimator can be obtained as applications of our main results. Furthermore, an iterated procedure for obtaining the nonparametric M-estimator and a cross-validation bandwidth selection method are discussed, and some numerical examples are provided to show that the proposed methods perform well in the finite sample case.     

NM-QELE for ARMA-GARCH models with non-Gaussian innovations

Although the quasi maximum likelihood estimator based on Gaussian density (Gaussian-QMLE) is widely used to estimate parameters in ARMA models with GARCH innovations (ARMA-GARCH models), it does not perform successfully when error distribution of ARMA-GARCH models is either skewed or leptokurtic. In order to circumvent such defects, Lee and Lee (submitted for publication) proposed the quasi maximum estimated-likelihood estimator using Gaussian mixture-based likelihood (NM-QELE) for GARCH models. In this paper, we adopt the NM-QELE method for estimating parameters in ARMA-GARCH models and demonstrate the validity of NM-QELE by verifying its consistency.     

A simple smoothing spline,III

Summary  Computational methods for spline smoothing are studied in the context of the linear smoothing spline. Comparisons are made between two efficient methods for computing the estimator using band-limited basis functions and the Kalman filter. In particular, the Kalman filter approach is shown to be an efficient method for computing under the Kimeldorf-Wahba representation for the estimator. Run time comparisons are made between band-limited B-spline and Kalman filter based algorithms.     

On the comparison of the pre-test and shrinkage phi-divergence test estimators for the symmetry model of categorical data

The estimation problem of the parameters in a symmetry model for categorical data has been considered for many authors in the statistical literature (for example, Bowker (1948) [1], Ireland et al. (1969) [2], Quade and Salama (1975) [3], Cressie and Read (1988) [4], Menéndez et al. (2005) [5]) without using uncertain prior information. It is well known that many new and interesting estimators, using uncertain prior information, have been studied by a host of researchers in different statistical models, and many papers have been published on this topic (see Saleh (2006) [9] and references therein). In this paper, we consider the symmetry model of categorical data and we study, for the first time, some new estimators when non-sample information about the symmetry of the probabilities is considered. The decision to use a “restricted” estimator or an “unrestricted” estimator is based on the outcome of a preliminary test, and then a shrinkage technique is used. It is interesting to note that we present a unified study in the sense that we consider not only the maximum likelihood estimator and likelihood ratio test or chi-square test statistic but we consider minimum phi-divergence estimators and phi-divergence test statistics. Families of minimum phi-divergence estimators and phi-divergence test statistics are wide classes of estimators and test statistics that contain as a particular case the maximum likelihood estimator, likelihood ratio test and chi-square test statistic. In an asymptotic set-up, the biases and the risk under the squared loss function for the proposed estimators are derived and compared. A numerical example clarifies the content of the paper.     

On generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables

ABSTRACT

We define and discuss different enumerative methods to compute solutions of generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. We propose both branch-and-bound methods based on merit functions for the mixed-integer game, and branch-and-prune methods that exploit the concept of dominance to make effective cuts. We show that under mild assumptions the equilibrium set of the game is finite and we define an enumerative method to compute the whole of it. We show that our branch-and-prune method can be suitably modified in order to make a general equilibrium selection over the solution set of the mixed-integer game. We define an application in economics that can be modelled as a Nash game with linear coupling constraints and mixed-integer variables, and we adapt the branch-and-prune method to efficiently solve it.     

On an enumerative algorithm for solving eigenvalue complementarity problems

In this paper, we discuss the solution of linear and quadratic eigenvalue complementarity problems (EiCPs) using an enumerative algorithm of the type introduced by Júdice et al. (Optim. Methods Softw. 24:549–586, 2009). Procedures for computing the interval that contains all the eigenvalues of the linear EiCP are first presented. A nonlinear programming (NLP) model for the quadratic EiCP is formulated next, and a necessary and sufficient condition for a stationary point of the NLP to be a solution of the quadratic EiCP is established. An extension of the enumerative algorithm for the quadratic EiCP is also developed, which solves this problem by computing a global minimum for the NLP formulation. Some computational experience is presented to highlight the efficiency and efficacy of the proposed enumerative algorithm for solving linear and quadratic EiCPs.     

Partial γ-positivity for quasi-Stirling permutations of multisets

We prove that the enumerative polynomials of quasi-Stirling permutations of multisets with respect to the statistics of plateaux, descents and ascents are partial γ-positive, thereby confirming a recent conjecture posed by Lin, Ma and Zhang. This is accomplished by proving the partial γ-positivity of the enumerative polynomials of certain ordered labeled trees, which are in bijection with quasi-Stirling permutations of multisets. As an application, we provide an alternative proof of the partial γ-positivity of the enumerative polynomials on Stirling permutations of multisets.     

Nonparametric estimation of an extreme-value copula in arbitrary dimensions

Inference on an extreme-value copula usually proceeds via its Pickands dependence function, which is a convex function on the unit simplex satisfying certain inequality constraints. In the setting of an i.i.d. random sample from a multivariate distribution with known margins and an unknown extreme-value copula, an extension of the Capéraà-Fougères-Genest estimator was introduced by D. Zhang, M. T. Wells and L. Peng [Nonparametric estimation of the dependence function for a multivariate extreme-value distribution, Journal of Multivariate Analysis 99 (4) (2008) 577-588]. The joint asymptotic distribution of the estimator as a random function on the simplex was not provided. Moreover, implementation of the estimator requires the choice of a number of weight functions on the simplex, the issue of their optimal selection being left unresolved.A new, simplified representation of the CFG-estimator combined with standard empirical process theory provides the means to uncover its asymptotic distribution in the space of continuous, real-valued functions on the simplex. Moreover, the ordinary least-squares estimator of the intercept in a certain linear regression model provides an adaptive version of the CFG-estimator whose asymptotic behavior is the same as if the variance-minimizing weight functions were used. As illustrated in a simulation study, the gain in efficiency can be quite sizable.     

Statistical inference for panel data semiparametric partially linear regression models with heteroscedastic errors

We consider a panel data semiparametric partially linear regression model with an unknown parameter vector for the linear parametric component, an unknown nonparametric function for the nonlinear component, and a one-way error component structure which allows unequal error variances (referred to as heteroscedasticity). We develop procedures to detect heteroscedasticity and one-way error component structure, and propose a weighted semiparametric least squares estimator (WSLSE) of the parametric component in the presence of heteroscedasticity and/or one-way error component structure. This WSLSE is asymptotically more efficient than the usual semiparametric least squares estimator considered in the literature. The asymptotic properties of the WSLSE are derived. The nonparametric component of the model is estimated by the local polynomial method. Some simulations are conducted to demonstrate the finite sample performances of the proposed testing and estimation procedures. An example of application on a set of panel data of medical expenditures in Australia is also illustrated.     

Presmoothing the transition probabilities in the illness-death model

One major goal in clinical applications of multi-state models is the estimation of transition probabilities. In a recent paper, Meira-Machado et al. (2006) introduce a substitute for the Aalen-Johansen estimator in the case of a non-Markov illness-death model. The idea behind their estimator is to weight the data by the Kaplan-Meier weights pertaining to the distribution of the total survival time of the process. In this paper we propose a modification of Meira-Machado et al. (2006) estimator based on presmoothing. Consistency is established. We investigate the finite sample performance of the new estimator through simulations. Data from a study on colon cancer are used for illustration purposes.     

A new approach to solving three combinatorial enumeration problems on planar graphs

The purpose of this paper is to show how the technique of delta-wye graph reduction provides an alternative method for solving three enumerative function evaluation problems on planar graphs. In particular, it is shown how to compute the number of spanning trees and perfect matchings, and how to evaluate energy in the Ising “spin glass” model of statistical mechanics. These alternative algorithms require O(n²) arithmetic operations on an n-vertex planar grapha, and are relatively easy to implement.     

Combining conditional and unconditional moment restrictions with missing responses

Many statistical models, e.g. regression models, can be viewed as conditional moment restrictions when distributional assumptions on the error term are not assumed. For such models, several estimators that achieve the semiparametric efficiency bound have been proposed. However, in many studies, auxiliary information is available as unconditional moment restrictions. Meanwhile, we also consider the presence of missing responses. We propose the combined empirical likelihood (CEL) estimator to incorporate such auxiliary information to improve the estimation efficiency of the conditional moment restriction models. We show that, when assuming responses are strongly ignorable missing at random, the CEL estimator achieves better efficiency than the previous estimators due to utilization of the auxiliary information. Based on the asymptotic property of the CEL estimator, we also develop Wilks’ type tests and corresponding confidence regions for the model parameter and the mean response. Since kernel smoothing is used, the CEL method may have difficulty for problems with high dimensional covariates. In such situations, we propose an instrumental variable-based empirical likelihood (IVEL) method to handle this problem. The merit of the CEL and IVEL are further illustrated through simulation studies.     

An adaptive estimator of the memory parameter and the goodness-of-fit test using a multidimensional increment ratio statistic

The increment ratio (IR) statistic was first defined and studied in Surgailis et al. (2007) [19] for estimating the memory parameter either of a stationary or an increment stationary Gaussian process. Here three extensions are proposed in the case of stationary processes. First, a multidimensional central limit theorem is established for a vector composed by several IR statistics. Second, a goodness-of-fit χ²-type test can be deduced from this theorem. Finally, this theorem allows to construct adaptive versions of the estimator and the test which are studied in a general semiparametric frame. The adaptive estimator of the long-memory parameter is proved to follow an oracle property. Simulations attest to the interesting accuracies and robustness of the estimator and the test, even in the non Gaussian case.     

Inference for intermediate Haezendonck–Goovaerts risk measure

Recently Haezendonck–Goovaerts (H–G) risk measure has received much attention in actuarial science. Nonparametric inference has been studied by Ahn and Shyamalkumar (2014) and Peng et al. (2015) when the risk measure is defined at a fixed level. In risk management, the level is usually set to be quite near one by regulators. Therefore, especially when the sample size is not large enough, it is useful to treat the level as a function of the sample size, which diverges to one as the sample size goes to infinity. In this paper, we extend the results in Peng et al. (2015) from a fixed level to an intermediate level. Although the proposed maximum empirical likelihood estimator for the H–G risk measure has a different limit for a fixed level and an intermediate level, the proposed empirical likelihood method indeed gives a unified interval estimation for both cases. A simulation study is conducted to examine the finite sample performance of the proposed method.     

Preprocessing techniques and column generation algorithms for stochastically efficient demand

We construct a discrete-time, multi-period replenishment plan that integrates the inventory, production and distribution functions and that satisfies the conditions of a very demanding cycle service level. The corresponding optimization problem takes the form of a very complex mixed-integer stochastic program. We develop a new enumerative algorithm that identifies the stochastically efficient demand trajectories at an authorized level of stockout, and derive three algorithmic preprocessing techniques used to discriminate the above trajectories. The application of the enumerative and preprocessing algorithmic approaches transforms the stochastic program into a disjunctive integer program solved through a column generation that reduces the risk of a bottleneck in the distribution resources of the supply chain. Computational results evaluate the efficiency of the algorithmic developments proposed in this paper, and attest the quality and robustness of the solution method. The solution methodology is validated on a real-life problem.     

Adaptive Fourier tester for statistical estimation

Based on Takenaka–Malmquist (TM) system, a new nonparametric estimator for probability density function is proposed. The TM estimation method is completely different from the existent density estimation methods in that the estimator depends on an approximate system with poles in a complex plane. Compared with the classic Fourier estimator, the TM estimator will offer more flexibility and adaptivity for real data due to the poles and nonlinearity of the phase of TM system. We compare the TM estimator with kernel, wavelet, and spline estimators by simulations. It shows that the introduced TM estimator is a more promising method than the existing and commonly used methods. Copyright © 2016 John Wiley & Sons, Ltd.     

An enumerative method for the solution of linear complementarity problems

In this paper an enumerative method for the solution of the Linear Complementarity Problem (LCP) is presented. This algorithm either finds all the solutions of any general LCP (that is, no assumption is made concerning the class of the matrix), or it establishes that no such solution exists. The method is extended to the Second Linear Complementarity Problem (SLCP), A new problem which has been introduced for the solution of a general quadratic program.