Hypothesis tests for high-dimensional covariance structures

We consider hypothesis testing for high-dimensional covariance structures in which the covariance matrix is a (i) scaled identity matrix, (ii) diagonal matrix, or (iii) intraclass covariance matrix. Our purpose is to systematically establish a nonparametric approach for testing the high-dimensional covariance structures (i)–(iii). We produce a new common test statistic for each covariance structure and show that the test statistic is an unbiased estimator of its corresponding test parameter. We prove that the test statistic establishes the asymptotic normality. We propose a new test procedure for (i)–(iii) and evaluate its asymptotic size and power theoretically when both the dimension and sample size increase. We investigate the performance of the proposed test procedure in simulations. As an application of testing the covariance structures, we give a test procedure to identify an eigenvector. Finally, we demonstrate the proposed test procedure by using a microarray data set.

     

A unified approach for solving nonlinear regular perturbation problems

This aritcle describes a simple alternative unified method of solving nonlinear regular perturbation problems. The procedure is based upon the manipulation of Taylor's approximation for the expansion of the nonlinear term in the perturbed equation.

An essential feature of this technique is the relative simplicity used and the associated unified computational procedure that is employed. As such it should be of interest to teachers of applied mathematics courses, particularly those courses which include perturbation methods.

One of the merits of this approach is that it leads on naturally to a scheme based on Taylor's expansions and, consequently, allows the regular perturbation method to be introduced into a course much earlier than is currently common.

The method is illustrated by implementing it to four perturbation problems, including two algebraic equations and two initial-value problems. The new approach is compared and contrasted with the traditional perturbation scheme in order to demonstrate its relative merit.     

Approximating the two-level facility location problem via a quasi-greedy approach

We propose a quasi-greedy algorithm for approximating the classical uncapacitated 2-level facility location problem (2-LFLP). Our algorithm, unlike the standard greedy algorithm, selects a sub-optimal candidate at each step. It also relates the minimization 2-LFLP problem, in an interesting way, to the maximization version of the single level facility location problem. Another feature of our algorithm is that it combines the technique of randomized rounding with that of dual fitting. This new approach enables us to approximate the metric 2-LFLP in polynomial time with a ratio of 1.77, a significant improvement on the previously known approximation ratios. Moreover, our approach results in a local improvement procedure for the 2-LFLP, which is useful in improving the approximation guarantees for several other multi-level facility location problems. An additional result of our approach is an O(ln (n))-approximation algorithm for the non-metric 2-LFLP, where n is the number of clients. This is the first non-trivial approximation for a non-metric multi-level facility location problem. An extended abstract of this paper appeared in the Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA), January 2004.     

Importance Sampling for p-value Computations in Multivariate Tests

Abstract

An importance sampling procedure is developed to approximate the distribution of an arbitrary function of the eigenvalues for a matrix beta random matrix or a Wishart random matrix. The procedure is easily implemented and provides confidence intervals for the p-values of many of the commonly used test statistics in multivariate analysis. An adaptive procedure allows for the control of either absolute error or relative error in this p-value estimation through the choice of importance sample size.     

A fractional version of the peano-sard theorem

Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials.

In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual.

The standard version of the Peano-Sard theory is contained in our results as a special case.     

Boundary conditions in acoustic scattering with respect to the reciprocity relation

A new concept associated with the reciprocity relation in acoustic scattering is introduced. Motivated by this well-known relation, which holds in all the classical cases, more general boundary value problems for the scalar Helmholtz equation are studied. These generalized boundary conditions are characterized by the validity of the reciprocity relation and seem to be an appropriate unification of a large class of boundary value problems in acoustic scattering. The well-known results for the classical problems can be extended to this general case and also new connections with the question about existence and uniqueness of solutions to the boundary value problems can be derived. A constructive procedure shows that this approach is applicable for a large class of boundary conditions.     

Intermediate Approach to Comparison of Some Goodness-of-Fit Tests

In this paper we present the intermediate approach to investigating asymptotic power and measuring the efficiency of nonparametric goodness-of-fit tests for testing uniformity. Contrary to the classical Pitman approach, the intermediate approach allows the explicit quantitative comparison of powers and calculation of efficiencies. For standard tests, like the Cramér-von Mises test, an intermediate approach gives conclusions consistent with qualitative results obtained using the Pitman approach. For other more complicated cases the Pitman approach does not give the right picture of power behaviour. An example is the data driven Neyman test we present in this paper. In this case the intermediate approach gives results consistent with finite sample results. Moreover, using this setting, we prove that the data driven Neyman test is asymptotically the most powerful and efficient under any smooth departures from uniformity. This result shows that, contrary to classical tests being efficient and the most powerful under one particular type of departure from uniformity, the new test is an adaptive one.     

Classical solutions to phase transition problems are smooth

We prove that classical C¹–solutions to phase transition problems, which include the two–phase Stefan problem, are smooth. The problem is reduced to a fixed domain using von Mises variables. The estimates are obtained by frozen coefficients and new L_p estimates for linear parabolic equations with dynamic boundary condition. Crucial ingredients are the observation that a certain function is a Fourier multiplier, an approximation procedure of norms in Besov spaces and Meyer' approach to Nemytakij operators.     

Numerical analysis of incompressible wormhole propagation with mass-preserving characteristic mixed finite element procedure

In this article, we construct a new combined characteristic mixed finite element procedure to simulate the incompressible wormhole propagation. In this procedure, we use the classical mixed finite element method to solve the pressure equation and a modified mass-preserving characteristic finite element method for the solute transport equation, and solve the porosity function straightly by the given concentration. This combined method not only keeps mass balance globally but also preserves maximum principle for the porosity. We considered the corresponding convergence and derive the optimal L²-norm error estimate. Finally, we present some numerical examples to confirm theoretical analysis.

     

Machine learning for global optimization

In this paper we introduce the LeGO (Learning for Global Optimization) approach for global optimization in which machine learning is used to predict the outcome of a computationally expensive global optimization run, based upon a suitable training performed by standard runs of the same global optimization method. We propose to use a Support Vector Machine (although different machine learning tools might be employed) to learn the relationship between the starting point of an algorithm and the final outcome (which is usually related to the function value at the point returned by the procedure). Numerical experiments performed both on classical test functions and on difficult space trajectory planning problems show that the proposed approach can be very effective in identifying good starting points for global optimization.     

Newsvendor problem under complete uncertainty: a case of innovative products

The paper presents a new scenario-based decision rule for the classical version of the newsvendor problem (NP) under complete uncertainty (i.e. uncertainty with unknown probabilities). So far, NP has been analyzed under uncertainty with known probabilities or under uncertainty with partial information (probabilities known incompletely). The novel approach is designed for the sale of new, innovative products, where it is quite complicated to define probabilities or even probability-like quantities, because there are no data available for forecasting the upcoming demand via statistical analysis. The new procedure described in the contribution is based on a hybrid of Hurwicz and Bayes decision rules. It takes into account the decision maker’s attitude towards risk (measured by coefficients of optimism and pessimism) and the dispersion (asymmetry, range, frequency of extremes values) of payoffs connected with particular order quantities. It does not require any information about the probability distribution.

     

A two-stage rank test using density estimation

For the one-sample problem, a two-stage rank test is derived which realizes a required power against a given local alternative, for all sufficiently smooth underlying distributions. This is achieved using asymptotic expansions resulting in a precision of orderm ^–1, wherem is the size of the first sample. The size of the second sample is derived through a number of estimators of e. g. integrated squared densities and density derivatives, all based on the first sample. The resulting procedure can be viewed as a nonparametric analogue of the classical Stein's two-stage procedure, which uses at-test and assumes normality for the underlying distribution. The present approach also generalizes earlier work which extended the classical method to parametric families of distributions.     

Nonparametric tests for conditional independence in two-way contingency tables

Testing for the independence between two categorical variables R and S forming a contingency table is a well-known problem: the classical chi-square and likelihood ratio tests are used. Suppose now that for each individual a set of p characteristics is also observed. Those explanatory variables, likely to be associated with R and S, can play a major role in their possible association, and it can therefore be interesting to test the independence between R and S conditionally on them. In this paper, we propose two nonparametric tests which generalise the chi-square and the likelihood ratio ideas to this case. The procedure is based on a kernel estimator of the conditional probabilities. The asymptotic law of the proposed test statistics under the conditional independence hypothesis is derived; the finite sample behaviour of the procedure is analysed through some Monte Carlo experiments and the approach is illustrated with a real data example.     

New optimality conditions and a scalarization approach for a nonconvex semi-vectorial bilevel optimization problem

In this paper, we are concerned with the optimistic formulation of a semivectorial bilevel optimization problem. Introducing a new scalarization technique for multiobjective programs, we transform our problem into a scalar-objective optimization problem by means of the optimal value reformulation and establish its theoretical properties. Detailed necessary conditions, to characterize local optimal solutions of the problem, were then provided, while using the weak basic CQ together with the generalized differentiation calculus of Mordukhovich. Our approach is applicable to nonconvex problems and is different from the classical scalarization techniques previously used in the literature and the conditions obtained are new.

     

Robust estimation in single-index models when the errors have a unimodal density with unknown nuisance parameter

This paper develops a robust profile estimation method for the parametric and nonparametric components of a single-index model when the errors have a strongly unimodal density with unknown nuisance parameter. We derive consistency results for the link function estimators as well as consistency and asymptotic distribution results for the single-index parameter estimators. Under a log-Gamma model, the sensitivity to anomalous observations is studied using the empirical influence curve. We also discuss a robust K-fold cross-validation procedure to select the smoothing parameters. A numerical study carried on with errors following a log-Gamma model and for contaminated schemes shows the good robustness properties of the proposed estimators and the advantages of considering a robust approach instead of the classical one. A real data set illustrates the use of our proposal.

     

A new Bayesian procedure for testing point null hypotheses

Testing point null hypotheses is a very common activity in various applied situations. However, the existing Bayesian testing procedure may give evidence which does not agree with the classical frequentist p-value in many point null testing situations. A typical example for this is the well known Lindley’s paradox (Lindley in Biometrika 44:187–192, 1957). In this paper we propose an alternative testing procedure in the Bayesian framework. It is shown that for many classical testing examples, the Bayesian evidence derived by our new testing procedure is not contradictory to its frequentist counterpart any more. In fact, the new Bayesian evidence under the noninformative prior is usually coincident with the frequentist observed significance level.     

IDEMPOTENTS AND MORITA-TAKEUCHI THEORY

ABSTRACT

We offer an approach to basic coalgebras with inspiration in the classical theory of idempotents for finite dimensional algebras. Our theory is based upon the fact that the co-hom functors associated to direct summands of the coalgebra can be easily described in terms of idempotents of the convolution algebra. Our approach is shown to be equivalent to that given by W. Chin and S. Montgomery by using co-endomorphism coalgebras of minimal injective cogenerators.     

ON THE METRIZATION LEMMA FOR UNIFORM SPACES

Abstract

We show a numeric generalization of the well-known metrization lemma for uniform spaces, thereby answering a question left open in [2] to the affirmative. An application of the new lemma in the setting of approach theory yields the conclusion that every approach uniformity has a basis of pseudo-metrics.     

Increasing the total net revenue for single machine order acceptance and scheduling problems using an artificial bee colony algorithm

The order acceptance and scheduling (OAS) problem is an important topic for make-to-order production systems with limited production capacity and tight delivery requirements. This paper proposes a new algorithm based on Artificial Bee Colony (ABC) for solving the single machine OAS problem with release dates and sequence-dependent setup times. The performance of the proposed ABC-based algorithm was validated by a benchmark problem set of test instances with up to 100 orders. Experimental results showed that the proposed ABC-based algorithm outperformed three state-of-art metaheuristic-based algorithms from the literature. It is believed that this study successfully demonstrates a high-performance algorithm that can serve as a new benchmark approach for future research on the OAS problem addressed in this study.

     

The arithmetic‐geometric mean inequality: a short proof

This article gives a simple inductive proof of the following classical inequality connecting the arithmetic and geometric means of the n positive quantities a ₁, a ₂,......a_n :

by using the fact that the geometric mean of the n — 1 numbers

is the same as that of a ₁, a ₂,......a_n : An inductive procedure is then applicable to proving the inequality from the case n — 1 to the case n by repeated applications of a simple inequality involving two pairs of positive numbers with the same product.