Reference Bands for Nonparametrically Estimated Link Functions

Abstract

We explore the shape of the link function in a generalized linear model by estimating the link nonparametrically. We consider the problem of comparing the nonparametrically estimated link with a particular link function. Using reference bands that consist of pointwise confidence intervals for the nonparametrically estimated link centered at the hypothesized parametric link, simple graphical methods are obtained for comparing the two link functions. The resulting diagnostic plots are demonstrated with both artificial and real examples.     

Optimization of Weighted Monte Carlo Methods with Respect to Auxiliary Variables

We consider the problems whose mathematical model is determined by some Markov chain terminating with probability one; moreover, we have to estimate linear functionals of a solution to an integral equation of the second kind with the corresponding substochastic kernel and free term [1]. To construct weighted modifications of numerical statistical models, we supplement the coordinates of the phase space with auxiliary variables whose random values functionally define the transitions in the initial chain. Having implemented each auxiliary random variable, we multiply the weight by the ratio of the corresponding densities of the initial and numerically modeled distributions. We solve the minimization problem for the variances of estimators of linear functionals by choosing the modeled distribution of the first auxiliary random variable.     

Testing for lack of dependence between functional variables

We introduce a test for the lack of dependence between two random variables valued into real Hilbert spaces. Here, we consider lack of dependence in the broader sense, that is, non-correlation. The test statistic is similar to the one proposed by Kokoszka et al. (2008) for testing for no effect in the linear functional model. The asymptotic distribution under the null hypothesis of this statistic is obtained as well as a consistency result for the proposed test. Applications to the case of functional variables are indicated and simulations show, in this context, the performance of the proposed method.     

The nonlinear boundary layer to the Boltzmann equation for cutoff soft potential with physical boundary condition

We consider the nonlinear boundary layer to the Boltzmann equation for cutoff soft potential with physical boundary condition, i.e., the Dirichlet boundary condition with weak diffuse effect. Under the assumption that the distribution function of gas particles tends to a global Maxwellian in the far field, we will show the boundary layer exist if the boundary data satisfy the solvability condition. Moreover, the codimensions of the boundary data which satisfies the solvability condition change with the Mach number of the far field Maxwellian like Chen et al. (2004) [5], Ukai et al. (2003) [6] and Wang et al. (2007) [7].     

Rearrangements and minimization of the principal eigenvalue of a nonlinear Steklov problem

This paper, motivated by Del Pezzo et al. (2006) [1], discusses the minimization of the principal eigenvalue of a nonlinear boundary value problem. In the literature, this type of problem is called Steklov eigenvalue problem. The minimization is implemented with respect to a weight function. The admissible set is a class of rearrangements generated by a bounded function. We merely assume the generator is non-negative in contrast to [1], where the authors consider weights which are positively away from zero, in addition to being two-valued. Under this generality, more physical situations can be modeled. Finally, using rearrangement theory developed by Geoffrey Burton, we are able to prove uniqueness of the optimal solution when the domain of interest is a ball.     

On recent developments in the theory of abstract differential equations with fractional derivatives

This note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives. We show that the existence results in [Agarwal et al. (2009) [1], Belmekki and Benchohra (2010) [2], Darwish et al. (2009) [3], Hu et al. (2009) [4], Mophou and N’Guérékata (2009) [6] and [7], Mophou (2010) [8] and [9], Muslim (2009) [10], Pandey et al. (2009) [11], Rashid and El-Qaderi (2009) [12] and Tai and Wang (2009) [13]] are incorrect since the considered variation of constant formulas is not appropriate. In this note, we also consider a different approach to treat a general class of abstract fractional differential equations.     

Global optimization of multilevel electricity market models including network design and graph partitioning

We consider the combination of a network design and graph partitioning model in a multilevel framework for determining the optimal network expansion and the optimal zonal configuration of zonal pricing electricity markets, which is an extension of the model discussed in Grimm et al. (2019) that does not include a network design problem. The two classical discrete optimization problems of network design and graph partitioning together with nonlinearities due to economic modeling yield extremely challenging mixed-integer nonlinear multilevel models for which we develop two problem-tailored solution techniques. The first approach relies on an equivalent bilevel formulation and a standard KKT transformation thereof including novel primal-dual bound tightening techniques, whereas the second is a tailored generalized Benders decomposition. For the latter, we strengthen the Benders cuts of Grimm et al. (2019) by using the structure of the newly introduced network design subproblem. We prove for both methods that they yield global optimal solutions. Afterward, we compare the approaches in a numerical study and show that the tailored Benders approach clearly outperforms the standard KKT transformation. Finally, we present a case study that illustrates the economic effects that are captured in our model.     

Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of solutions of the generalized equilibrium problems and the set of all common fixed points of a nonexpansive semigroup in the framework of a real Hilbert space. We prove that both approaches converge strongly to a common element of such two sets. Such common element is the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Furthermore, we utilize the main results to obtain two mean ergodic theorems for nonexpansive mappings in a Hilbert space. The results of this paper extend and improve the results of Li et al. (J Nonlinear Anal 70:3065–3071, 2009), Cianciaruso et al. (J Optim Theory Appl 146:491–509, 2010) and many others.     

The role of the dependence between mortality and interest rates when pricing Guaranteed Annuity Options

In this paper we investigate the consequences on the pricing of insurance contingent claims when we relax the typical independence assumption made in the actuarial literature between mortality risk and interest rate risk. Starting from the Gaussian approach of Liu et al. (2014), we consider some multifactor models for the mortality and interest rates based on more general affine models which remain positive and we derive pricing formulas for insurance contracts like Guaranteed Annuity Options (GAOs). In a Wishart affine model, which allows for a non-trivial dependence between the mortality and the interest rates, we go far beyond the results found in the Gaussian case by Liu et al. (2014), where the value of these insurance contracts can be explained only in terms of the initial pairwise linear correlation.     

On improved asymptotic bounds for codes from global function fields

We analyse a binary cyclotomic sequence constructed via generalized cyclotomic classes by Bai et al. (IEEE Trans Inforem Theory 51: 1849–1853, 2005). First we determine the linear complexity of a natural generalization of this binary sequence to arbitrary prime fields. Secondly we consider k-error linear complexity and autocorrelation of these sequences and point out certain drawbacks of this construction. The results show that the parameters for the sequence construction must be carefully chosen in view of the respective application.      

Optimal admission control for two station tandem queues with loss

We consider a two-station tandem queue with a buffer size of one at the first station and a finite buffer size at the second station. Silva et al. (2013) gave a criterion determining the optimal admission control policy for this model. In this paper, we improve the results of Silva et al. (2013) and also solve the problem conjectured by Silva et al. (2013).     

Existence and approximation of solutions to nonlinear hybrid ordinary differential equations

In this note, we consider the initial value problem for first order nonlinear hybrid ordinary differential equations and we discuss the existence and approximation of the solutions. The main results are related to a recent work by Dhage et al. (2014) through the restructuring of some of the hypotheses imposed and the extension of some of the results there. In addition, we provide an example to illustrate the applicability of the abstract results developed.     

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.     

Semiparametric analysis based on weighted estimating equations for transformation models with missing covariates

Missing covariate data are very common in regression analysis. In this paper, the weighted estimating equation method (Qi et al., 2005) [25] is used to extend the so-called unified estimation procedure (Chen et al., 2002) [4] for linear transformation models to the case of missing covariates. The non-missingness probability is estimated nonparametrically by the kernel smoothing technique. Under missing at random, the proposed estimators are shown to be consistent and asymptotically normal, with the asymptotic variance estimated consistently by the usual plug-in method. Moreover, the proposed estimators are more efficient than the weighted estimators with the inverse of true non-missingness probability as weight. Finite sample performance of the estimators is examined via simulation and a real dataset is analyzed to illustrate the proposed methods.     

Duality and optimality conditions for generalized equilibrium problems involving DC functions

We consider a generalized equilibrium problem involving DC functions which is called (GEP). For this problem we establish two new dual formulations based on Toland-Fenchel-Lagrange duality for DC programming problems. The first one allows us to obtain a unified dual analysis for many interesting problems. So, this dual coincides with the dual problem proposed by Martinez-Legaz and Sosa (J Glob Optim 25:311–319, 2006) for equilibrium problems in the sense of Blum and Oettli. Furthermore it is equivalent to Mosco’s dual problem (Mosco in J Math Anal Appl 40:202–206, 1972) when applied to a variational inequality problem. The second dual problem generalizes to our problem another dual scheme that has been recently introduced by Jacinto and Scheimberg (Optimization 57:795–805, 2008) for convex equilibrium problems. Through these schemes, as by products, we obtain new optimality conditions for (GEP) and also, gap functions for (GEP), which cover the ones in Antangerel et al. (J Oper Res 24:353–371, 2007, Pac J Optim 2:667–678, 2006) for variational inequalities and standard convex equilibrium problems. These results, in turn, when applied to DC and convex optimization problems with convex constraints (considered as special cases of (GEP)) lead to Toland-Fenchel-Lagrange duality for DC problems in Dinh et al. (Optimization 1–20, 2008, J Convex Anal 15:235–262, 2008), Fenchel-Lagrange and Lagrange dualities for convex problems as in Antangerel et al. (Pac J Optim 2:667–678, 2006), Bot and Wanka (Nonlinear Anal to appear), Jeyakumar et al. (Applied Mathematics research report AMR04/8, 2004). Besides, as consequences of the main results, we obtain some new optimality conditions for DC and convex problems.     

Worst VaR scenarios: A remark

Theorem 15 of Embrechts et al. [Embrechts, Paul, Höing, Andrea, Puccetti, Giovanni, 2005. Worst VaR scenarios. Insurance: Math. Econom. 37, 115-134] proves that comonotonicity gives rise to the on-average-most-adverse Value-at-Risk scenario for a function of dependent risks, when the marginal distributions are known but the dependence structure between the risks is unknown. This note extends this result to the case where, rather than no information, partial information is available on the dependence structure between the risks. A result of Kaas et al. [Kaas, Rob, Dhaene, Jan, Goovaerts, Marc J., 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econom. 23, 151-168] is also generalized for this purpose.     

Truncation,amplitude, and jitter errors on R for sampling series derivatives

In this paper, we investigate the error analysis of the derivative of the classical sampling theorem of bandlimited functions. We consider truncation, amplitude, and time-jitter errors. Both pointwise and uniform estimates are given. We derive analogues of the results of Piper (1975), Brown (1969), Jagerman (1966) and Li (1998) in a generalized manner. The amplitude and time-jitter errors are studied in the view of the works of Butzer (1983) and Butzer et al. (1988), provided that the bandlimited function satisfies some decay properties.     

A new discrete dynamical system of signed integer partitions

In Brylawski (1973) Brylawski described the covering property for the domination order on non-negative integer partitions by means of two rules. Recently, in Bisi et al. (in press), Cattaneo et al. (2014), Cattaneo et al. (2015) the two classical Brylawski covering rules have been generalized in order to obtain a new lattice structure in the more general signed integer partition context. Moreover, in Cattaneo et al. (2014), Cattaneo et al. (2015), the covering rules of the above signed partition lattice have been interpreted as evolution rules of a discrete dynamical model of a two-dimensional p–n semiconductor junction in which each positive number represents a distribution of holes (positive charges) located in a suitable strip at the left semiconductor of the junction and each negative number a distribution of electrons (negative charges) in a corresponding strip at the right semiconductor of the junction. In this paper we introduce and study a new sub-model of the above dynamical model, which is constructed by using a single vertical evolution rule. This evolution rule describes the natural annihilation of a hole–electron pair at the boundary region of the two semiconductors. We prove several mathematical properties of such new discrete dynamical model and we provide a discussion of its physical properties.     

A note on the paper “The trapezoidal fuzzy soft set and its application in MCDM”

We propose a new concept which is a generalization of fuzzy soft subset and fuzzy soft equal. Using such notions, we will be able to consider the distributive law of fuzzy soft sets. Using the distributive law of fuzzy soft sets, we point out that the distributive law of trapezoidal fuzzy soft sets as proposed by Xiao et al. (2012) is not true. The correction will further improve further extensions of the results of Xiao et al. (2012). We will also establish the generalized distributive law of trapezoidal fuzzy soft sets along with illustrative examples.     

A blow-up criterion for 3D compressible viscoelastic flow with large initial data

In this paper, we consider a Cauchy problem for the three-dimensional compressible viscoelastic flow with large initial data. We establish a blow-up criterion for the strong solutions in terms of the gradient of velocity only, which is similar to the Beale-Kato-Majda criterion for ideal incompressible flow (cf. Beale et al. (1984) [20]) and the blow-up criterion for the compressible Navier-Stokes equations (cf. Huang et al. (2011) [21]).