General results on preferential attachment and clustering coefficient

This is a review paper that covers some recent results on the behavior of the clustering coefficient in preferential attachment networks and scale-free networks in general. The paper focuses on general approaches to network science. In other words, instead of discussing different fully specified random graph models, we describe some generic results which hold for classes of models. Namely, we first discuss a generalized class of preferential attachment models which includes many classical models. It turns out that some properties can be analyzed for the whole class without specifying the model. Such properties are the degree distribution and the global and average local clustering coefficients. Finally, we discuss some surprising results on the behavior of the global clustering coefficient in scale-free networks. Here we do not assume any underlying model.     

Steepest Descent Method with Random Step Lengths

The paper studies the steepest descent method applied to the minimization of a twice continuously differentiable function. Under certain conditions, the random choice of the step length parameter, independent of the actual iteration, generates a process that is almost surely R-convergent for quadratic functions. The convergence properties of this random procedure are characterized based on the mean value function related to the distribution of the step length parameter. The distribution of the random step length, which guarantees the maximum asymptotic convergence rate independent of the detailed properties of the Hessian matrix of the minimized function, is found, and its uniqueness is proved. The asymptotic convergence rate of this optimally created random procedure is equal to the convergence rate of the Chebyshev polynomials method. Under practical conditions, the efficiency of the suggested random steepest descent method is degraded by numeric noise, particularly for ill-conditioned problems; furthermore, the asymptotic convergence rate is not achieved due to the finiteness of the realized calculations. The suggested random procedure is also applied to the minimization of a general non-quadratic function. An algorithm needed to estimate relevant bounds for the Hessian matrix spectrum is created. In certain cases, the random procedure may surpass the conjugate gradient method. Interesting results are achieved when minimizing functions having a large number of local minima. Preliminary results of numerical experiments show that some modifications of the presented basic method may significantly improve its properties.     

Subspace-stabilized sequential quadratic programming

The stabilized sequential quadratic programming (SQP) method has nice local convergence properties: it possesses local superlinear convergence under very mild assumptions not including any constraint qualifications. However, any attempts to globalize convergence of this method indispensably face some principal difficulties concerned with intrinsic deficiencies of the steps produced by it when relatively far from solutions; specifically, it has a tendency to produce long sequences of short steps before entering the region where its superlinear convergence shows up. In this paper, we propose a modification of the stabilized SQP method, possessing better “semi-local” behavior, and hence, more suitable for the development of practical realizations. The key features of the new method are identification of the so-called degeneracy subspace and dual stabilization along this subspace only; thus the name “subspace-stabilized SQP”. We consider two versions of this method, their local convergence properties, as well as a practical procedure for approximation of the degeneracy subspace. Even though we do not consider here any specific algorithms with theoretically justified global convergence properties, subspace-stabilized SQP can be a relevant substitute for the stabilized SQP in such algorithms using the latter at the “local phase”. Some numerical results demonstrate that stabilization along the degeneracy subspace is indeed crucially important for success of dual stabilization methods.     

Classical Backfitting for Smooth-Backfitting Additive Models

Smooth backfitting has been shown to have better theoretical properties than classical backfitting for fitting additive models based on local linear regression. In this article, we show that the smooth backfitting procedure in the local linear case can be alternatively performed as a classical backfitting procedure with a different type of smoother matrices. These smoother matrices are symmetric and shrinking and some established results in the literature are readily applicable. The connections allow the smooth backfitting algorithm to be implemented in a much simplified way, give new insights on the differences between the two approaches in the literature, and provide an extension to local polynomial regression. The connections also give rise to a new estimator at data points. Asymptotic properties of general local polynomial smooth backfitting estimates are investigated, allowing for different orders of local polynomials and different bandwidths. Cases of oracle efficiency are discussed. Computer-generated simulations are conducted to demonstrate finite sample behaviors of the methodology and a real data example is given for illustration. Supplementary materials for this article are available online.     

Kadec–Klee properties of some quasi-Banach function spaces

We study Kadec–Klee properties with respect to global (local) convergence in measure in quasi-Banach function spaces. First, we prove some general results which can be of independent interest. Next, we investigate these properties in symmetrizations \(E^{\left( *\right) }\). Finally, we apply general results to study these properties in Marcinkiewicz and Lorentz spaces.     

Random walks with non-convolution equivalent increments and their applications

This paper mainly presents some global and local asymptotic estimates for the tail probabilities of the supremum and overshoot of a random walk in “the intermediate case”, where the related distributions of the increments of the random walk may not belong to the convolution equivalent distribution class. Some of the obtained results can include the classical results. For this, the paper first introduces some new distribution classes using the γ-transform of distributions, and investigates their properties and relations with some other existing distribution classes. Based on the above results, some equivalent conditions for the global and local asymptotics of the γ-transform of the distribution of the supremum of the above random walk are given. Applying these results to risk theory and infinitely divisible laws, the paper obtains some asymptotic estimates for the ruin probability and the local ruin probability of the renewal risk model with non-convolution equivalent claims, and the global and local asymptotics of an infinitely divisible law with a non-convolution equivalent Lévy measure.     

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.     

Interweaving Markov Chain Monte Carlo Strategies for Efficient Estimation of Dynamic Linear Models

In dynamic linear models (DLMs) with unknown fixed parameters, a standard Markov chain Monte Carlo (MCMC) sampling strategy is to alternate sampling of latent states conditional on fixed parameters and sampling of fixed parameters conditional on latent states. In some regions of the parameter space, this standard data augmentation (DA) algorithm can be inefficient. To improve efficiency, we apply the interweaving strategies of Yu and Meng to DLMs. For this, we introduce three novel alternative DAs for DLMs: the scaled errors, wrongly scaled errors, and wrongly scaled disturbances. With the latent states and the less well known scaled disturbances, this yields five unique DAs to employ in MCMC algorithms. Each DA implies a unique MCMC sampling strategy and they can be combined into interweaving and alternating strategies that improve MCMC efficiency. We assess these strategies using the local level model and demonstrate that several strategies improve efficiency relative to the standard approach and the most efficient strategy interweaves the scaled errors and scaled disturbances. Supplementary materials are available online for this article.     

Adaptive local linear quantile regression

In this paper we propose a new method of local linear adaptive smoothing for nonparametric conditional quantile regression. Some theoretical properties of the procedure are investigated. Then we demonstrate the performance of the method on a simulated example and compare it with other methods. The simulation results demonstrate a reasonable performance of our method proposed especially in situations when the underlying image is piecewise linear or can be approximated by such images. Generally speaking, our method outperforms most other existing methods in the sense of the mean square estimation (MSE) and mean absolute estimation (MAE) criteria. The procedure is very stable with respect to increasing noise level and the algorithm can be easily applied to higher dimensional situations.     

Two-Stage Likelihood Ratio and Union–Intersection Tests for One-Sided Alternatives Multivariate Mean with Nuisance Dispersion Matrix

For a multinormal distribution with an unknown dispersion matrix, union-intersection (UI) tests for the mean against one-sided alternatives are considered. The null distribution of the UI test statistic is derived and its power monotonicity properties are studied. A Stain-type two-stage procedure is proposed to eliminate some of the inherent drawbacks of such tests. Some comparisons are also made with some recently proposed alternative conditional likelihood ratio tests.     

Nonparametric regression estimation with general parametric error covariance

The asymptotic distribution for the local linear estimator in nonparametric regression models is established under a general parametric error covariance with dependent and heterogeneously distributed regressors. A two-step estimation procedure that incorporates the parametric information in the error covariance matrix is proposed. Sufficient conditions for its asymptotic normality are given and its efficiency relative to the local linear estimator is established. We give examples of how our results are useful in some recently studied regression models. A Monte Carlo study confirms the asymptotic theory predictions and compares our estimator with some recently proposed alternative estimation procedures.     

Propagation-Separation Approach for Local Likelihood Estimation

The paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models, including e.g. the regression, density, Poisson and binary response model. The method extends the adaptive weights smoothing (AWS) procedure introduced in Polzehl and Spokoiny (2000) in context of image denoising. The main idea of the method is to describe a greatest possible local neighborhood of every design point X_i in which the local parametric assumption is justified by the data. The method is especially powerful for model functions having large homogeneous regions and sharp discontinuities. The performance of the proposed procedure is illustrated by numerical examples for density estimation and classification. We also establish some remarkable theoretical nonasymptotic results on properties of the new algorithm. This includes the ``propagation' property which particularly yields the root-n consistency of the resulting estimate in the homogeneous case. We also state an ``oracle' result which implies rate optimality of the estimate under usual smoothness conditions and a ``separation' result which explains the sensitivity of the method to structural changes.     

Local linear regression for data with AR errors

In many statistical applications, data are collected over time, and they are likely correlated. In this paper, we investigate how to incorporate the correlation information into the local linear regression. Under the assumption that the error process is an auto-regressive process, a new estimation procedure is proposed for the nonparametric regression by using local linear regression method and the profile least squares techniques. We further propose the SCAD penalized profile least squares method to determine the order of auto-regressive process. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed procedure, and to compare the performance of the proposed procedures with the existing one. From our empirical studies, the newly proposed procedures can dramatically improve the accuracy of naive local linear regression with working-independent error structure. We illustrate the proposed methodology by an analysis of real data set.     

A Branch-and-Bound procedure to minimize total tardiness on one machine with arbitrary release dates

In this paper, we present a Branch-and-Bound procedure to minimize total tardiness on one machine with arbitrary release dates. We introduce new lower bounds and we generalize some well-known dominance properties. Our procedure handles instances as large as 500 jobs although some 60 jobs instances remain open. Computational results show that the proposed approach outperforms the best known procedures.     

Abelian and Tauberian results for the one-dimensional modified Stockwell transforms

The aim of this paper is to study the asymptotic behavior of one- dimensional modified Stockwell transform of a tempered distribution signal through the quasiasymptotic behavior at origin or infinity of the signal itself. More precisely, we give some Abelian results which mean that we derive the asymptotic properties of the S-transform of a tempered signal from the quasiasymptotic properties of the signal itself and we do also the opposite. So, we also give some Tauberian results which describe some quasiasymptotic properties of the tempered signal by means of the asymptotic properties of its Stockwell transform.     

On the (N+1)-dimensional local fractional reduced differential transform method and its applications

In this paper, we generalize the (N+1)-dimensional local fractional reduced differential transform method (LFRDTM) within the local fractional derivative sense. First, we show some new properties, lemmas, theorems and corollariesfor the (N+1)-dimensional LFRDTM. Second, these new properties, lemmas and theorems can be proved immediately after. Thirdly, we used two examples to state that this approach is efficient and simple to find numerical solutions to higher dimensional local fractional partial differential equations. Finally, we can be seen that this work can be looked as an extension of the prior work.     

Basic Homogenization Results for a Biconnected ε-Periodic Structure

We present, in a bounded domain, a model of an l -periodic structure composed of two phases, both being connected but only one reaching the boundary of the domain, avoiding in this way the local type convergences of the homogenization process. In this framework we revise some basic tools of the homogenization theory in porous media: the extension and the restriction operators, the Ne ) as inequality. Moreover, we obtain some compacity properties which reduce the proof of the pressure type convergences from the homogenization of fluid flows through porous media to the expected procedure of a priori estimations and two-scale convergences. As all the properties can be proved without much technical difficulties, avoiding annoying hypotheses and the use of Kolmogorov's criterion of compacity, the present structure seems one of the most convenient realistic models of porous media that can be studied with the methods of homogenization.     

Direct reconstruction of complex refractive index distribution from boundary measurements

We obtain the reconstruction of the refractive index distribution of body based on the intensity and normal derivative of the intensity measurements. The Helmholtz equation is inverted either directly or indirectly through repeated implementation of the forward operator and its adjoint, for recovering the complex refractive index distribution. We do not adopt the procedure of recovery of phase (normally required for complete knowledge distribution). We derive certain sensitivity relations which is used for the easy computation of the Jacobian. Our procedure successfully reconstructs the real and imaginary parts of the complex refractive index from the measurement of the two data types derived from the complex amplitude at the boundary. Our other interest is the reconstruction of the spectroscopic variations of optical absorption coefficients and visco-elastic properties of a tissue which is extremely useful in diagnostic medicines. The research is on progress and some results are available. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)