a general akaike type criterion for model selection in robust regression akaike s procedure 1970 for selecting a model minimises an estimate of the expected squared error in predicting new independent observations this selection criterion was designed for models fitted by least squares a different model fitting technique such as least absolute deviation regression requires an appropriate model selection procedure this paper presents a general akaike type criterion applicable to a wide variety of loss functions for model fitting it requires only that the function be convex with a unique minimum and twice differentiable in expectation simulations show that the estimators proposed here well approximate their respective prediction errors

minimizing neumann fundamental tones of triangles an optimal poincare inequality abstract the first nonzero eigenvalue of the neumann laplacian is shown to be minimal for the degenerate acute isosceles triangle among all triangles of given diameter hence an optimal poincare inequality for triangles is derived the proof relies on symmetry of the neumann fundamental mode for isosceles triangles with aperture less than 3 antisymmetry is proved for apertures greater than 3

computational survey on a posteriori error estimators for nonconforming finite element methods for the poisson problem this paper compares different a posteriori error estimators for nonconforming first order crouzeix raviart finite element methods for simple second order partial differential equations all suggested error estimators yield a guaranteed upper bound of the discrete energy error up to oscillation terms with explicit constants novel equilibration techniques and an improved interpolation operator for the design of conforming approximations of the discrete nonconforming finite element solution perform very well in an error estimator competition with six benchmark examples

a numerical study of the spectral gap we present a numerical study of the spectral gap of the dirichlet laplacian k 2 k 1 k of a planar convex region k besides providing supporting numerical evidence for the long standing gap conjecture that k 3 2 d2 k where d k denotes the diameter of k our study suggests new types of bounds and several conjectures regarding the dependence of the gap not only on the diameter but also on the perimeter and the area one of these conjectures is a stronger version of the gap conjecture mentioned above a similar study is carried out for the quotient of the first two dirichlet eigenvalues

refined fully explicit a posteriori residual based error control the explicit residual based a posteriori error estimator for elliptic partial differential equations is reliable up to the multiplication of some generic constant which needs to be involved for full error control the present mathematical literature takes this constant from the stability and approximation properties of clement type quasi interpolation operators and so results in an overestimation of the error which is bigger than for implicit and more expensive a posterori error estimators this paper propagates a paradigm shift to start with an equilibration error estimator technique followed by its efficiency analysis the outcome is a refined residual based a posteriori error estimate with explicit constants which leads to slightly sharper error control than the work of veeser and verfurth in 2009 a first application to guaranteed explicit error estimation for two dimensional nonconforming and a generalization to higher order finite element methods conclude the paper

on the range of the first two dirichlet and neumann eigenvalues of the laplacian in this paper we study the set of points in the plane defined by x y lambda 1 omega lambda 2 omega omega 1 where lambda 1 omega lambda 2 omega are either the two first eigenvalues of the dirichlet laplacian or the two first non trivial eigenvalues of the neumann laplacian we consider the case of general open sets together with the case of convex open domains we give some qualitative properties of these sets show some pictures obtained through numerical computations and state several open problems

linear model selection by cross validation abstract we consider the problem of selecting a model having the best predictive ability among a class of linear models the popular leave one out cross validation method which is asymptotically equivalent to many other model selection methods such as the akaike information criterion aic the c p and the bootstrap is asymptotically inconsistent in the sense that the probability of selecting the model with the best predictive ability does not converge to 1 as the total number of observations n we show that the inconsistency of the leave one out cross validation can be rectified by using a leave n v out cross validation with n v the number of observations reserved for validation satisfying n v n 1 as n this is a somewhat shocking discovery because nv n 1 is totally opposite to the popular leave one out recipe in cross validation motivations justifications and discussions of some practical aspects of the use of the leave n v out cross validation method are provided and results

superconvergence of both the crouzeix raviart and morley elements in this paper a new method is proposed to prove the superconvergence of both the crouzeix raviart and morley elements the main idea is to fully employ equivalences with the first order raviart thomas element and the first order hellan herrmann johnson element respectively in this way some special conformity of discrete stresses is explored and superconvergence of mixed elements can be used to analyze superconvergence of nonconforming elements finally the superconvergence of one and a half order by postprocessing is proved for both nonconforming elements

variable selection for 1d regression models variable selection the search for j relevant predictor variables from a group of p candidates is a standard problem in regression analysis the class of 1d regression models is a broad class that includes generalized linear models we show that existing variable selection algorithms originally meant for multiple linear regression and based on ordinary least squares and mallows s cp can also be used for 1d models graphical aids for variable selection are also provided

two applications of analytic functors we apply the theory of analytic functors to two topics related to theoretical computer science one is a mathematical foundation of certain syntactic well quasi orders and well orders appearing in graph theory the theory of term rewriting systems and proof theory the other is a new verification of the lagrange good inversion formula using several ideas appearing in semantics of lambda calculi especially the relation between categorical traces and fixpoint operators copyright 2002 elsevier science b v

single index model selections we derive a new model selection criterion for single index models aicc by minimising the expected kullback leibler distance between the true and candidate models the proposed criterion selects not only relevant variables but also the smoothing parameter for an unknown link function thus it is a general selection criterion that provides a unified approach to model selection across both parametric and nonparametric functions monte carlo studies demonstrate that aicc performs satisfactorily in most situations we illustrate the practical use of aicc with an empirical example for modelling the hedonic price function for cars in addition we extend the applicability of aicc to partially linear and additive single index models language en

a robust version of mallows s c p abstract we present a robust version of mallows s c p for regression models it is defined by rc p w p 2 u p v p where w p i 2 i r 2 i is a weighted residual sum of squares computed from a robust fit of model p 2 is a robust and consistent estimator of 2 in the full model and u p and v p are constants depending on the weight function and the number of parameters in model p good subset models are those with rc p close to v p or smaller than v p when the weights are identically 1 w p becomes the residual sum of squares of a least squares fit and rc p reduces to mallows s c p the robust model selection procedure based on rc p allows us to choose the models that fit the majority of the data by taking into account the presence of outliers and possible departures from the normality assumption on the error distribution together with the classical c p the robust version suggests several models from which we can choose

variable selection for cox s proportional hazards model and frailty model a class of variable selection procedures for parametric models via nonconcave penalized likelihood was proposed in fan and li 2001a it has been shown there that the resulting procedures perform as well as if the subset of significant variables were known in advance such a property is called an oracle property the proposed procedures were illustrated in the context of linear regression robust linear regression and generalized linear models in this paper the nonconcave penalized likelihood approach is extended further to the cox proportional hazards model and the cox proportional hazards frailty model two commonly used semi parametric models in survival analysis as a result new variable selection procedures for these two commonly used models are proposed it is demonstrated how the rates of convergence depend on the regularization parameter in the penalty function further with a proper choice of the regularization parameter and the penalty function the proposed estimators possess an oracle property standard error formulae are derived and their accuracies are empirically tested simulation studies show that the proposed procedures are more stable in prediction and more effective in computation than the best subset variable selection and they reduce model complexity as effectively as the best subset variable selection compared with the lasso which is the penalized likelihood method with the l 1 penalty proposed by tibshirani the newly proposed approaches have better theoretic properties and finite sample performance

graphics for assessing the adequacy of regression models abstract graphical paradigms for assessing the adequacy of nearly any regression model are discussed the fundamental idea is to examine the fit of a model using a sequence of marginal model plots on each plot nonparametric estimates of fit derived from the model are compared to nonparametric fits based on the observed data several examples are given

on the hot spots conjecture of j rauch rodrigo banuelos research partially supported by nsf grant dms 9400854 krzysztof burdzy s research partially supported by nsf grant dms 9322689

maximizing neumann fundamental tones of triangles we prove sharp isoperimetric inequalities for neumann eigenvalues of the laplacian on triangular domains the first nonzero neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter and hence among all triangles of given area similar results are proven for the harmonic and arithmetic means of the first two nonzero eigenvalues

guaranteed lower eigenvalue bounds for the biharmonic equation the computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate s vibrations this paper shows that the eigenvalue provided by the nonconforming morley finite element analysis which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense is not always a lower bound a fully explicit error analysis of the morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound this paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower order terms

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

optimal lower bounds for eigenvalues of linear and nonlinear neumann problems in this paper we prove a sharp lower bound for the first non trivial neumann eigenvalue 1 for the p laplace operator p 1 in a lipschitz bounded domain in n our estimate does not require any convexity assumption on and it involves the best isoperimetric constant relative to in a suitable class of convex planar domains our bound turns out to be better than the one provided by the payne weinberger inequality

robust residual based a posteriori arnold winther mixed finite element analysis in elasticity this paper presents a residual based a posteriori error estimator for the arnold winther mixed finite element that utilises a post processing for the skew symmetric part of the strain numerical experiments verify the proven reliability and efficiency for suitable approximation of the skew symmetric deformation gradient numerical evidence supports that the l2l2 stress error estimator is robust in the poisson ratio and allows stable error control even in the incompressible limit

applications of robust distances for regression the dd plot introduced by rousseeuw and van driessen 1999 is a plot of classical vs robust mahalanobis distances mdi vs rdi the dd plot can be used as a diagnostic for multivariate normality and elliptical symmetry and to assess the success of numerical transformations towards elliptical symmetry in the regression context many procedures can be adversely aected if strong nonlinearities are present in the predictors even if strong nonlinearities are present the robust distances can be used to help visualize important regression models such as generalized linear models

precise bounds and asymptotics for the first dirichlet eigenvalue of triangles and rhombi abstract we study the asymptotic expansion of the first dirichlet eigenvalue of certain families of triangles and of rhombi as a singular limit is approached in certain cases which include isosceles and right triangles we obtain the exact value of all the coefficients of the unbounded terms in the asymptotic expansion as the angle opening approaches zero plus the constant term and estimates on the remainder for rhombi and other triangle families such as isosceles triangles where now the angle opening approaches we have the first two terms plus bounds on the remainder these results are based on new upper and lower bounds for these domains whose asymptotic expansions coincide up to the orders mentioned apart from being accurate near the singular limits considered our lower bounds for the rhombus improve upon the bound by hooker and protter for angles up to approximately 22 and in the range 31 54 these results also show that the asymptotic expansion around the degenerate case of the isosceles triangle with vanishing angle opening depends on the path used to approach it

on the buckling eigenvalue problem we prove a density result which allows us to justify the application of the method of fundamental solutions to solve the buckling eigenvalue problem of a plate we address an example of an analytic convex domain for which the first eigenfunction does change the sign and present a large scale numerical study with polygons providing numerical evidence to some new conjectures

sums of laplace eigenvalues rotationally symmetric maximizers in the plane abstract the sum of the first n 1 eigenvalues of the laplacian is shown to be maximal among triangles for the equilateral triangle maximal among parallelograms for the square and maximal among ellipses for the disk provided the ratio area 3 moment of inertia for the domain is fixed this result holds for both dirichlet and neumann eigenvalues and similar conclusions are derived for robin boundary conditions and schrodinger eigenvalues of potentials that grow at infinity a key ingredient in the method is the tight frame property of the roots of unity for general convex plane domains the disk is conjectured to maximize sums of neumann eigenvalues

nonlinear confounding in high dimensional regression it is not uncommon to find nonlinear patterns in the scatterplots of regressor variables but how such findings affect standard regression analysis remains largely unexplored this article offers a theory on nonlinear confounding a term for describing the situation where a certain nonlinear relationship in regressors leads to difficulties in modeling and related analysis of the data the theory begins with a measure of nonlinearity between two regressor variables it is then used to assess nonlinearity between any two projections from the high dimensional regressor and a method of finding most nonlinear projections is given nonlinear confounding is addressed by taking a fresh new look at fundamental issues such as the validity of prediction and inference diagnostics regression surface approximation model uncertainty and fisher information loss

more comments on c p we discuss the interpretation of c p plots and show how they can be calibrated in several ways we comment on the practice of using the display as a basis for formal selection of a subset regression model and extend the range of application of the device to encompass arbitrary linear estimates of the regression coefficients for example ridge estimates