estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

the unregularized gradient flow of the symplectic action the symplectic action can be defined on the space of smooth paths in a symplectic manifold p which join two lagrangian submanifolds of p to pursue a new approach to the variational theory of this function we define on a subset of the path space the flow whose trajectories are given by the solutions of the cauchy riemann equation with respect to a suitable almost complex structure on p in particular we prove compactness and transversality results for the set of bounded trajectories

combining multiple biomarker models in logistic regression in medical research there is great interest in developing methods for combining biomarkers we argue that selection of markers should also be considered in the process traditional model variable selection procedures ignore the underlying uncertainty after model selection in this work we propose a novel model combining algorithm for classification in biomarker studies it works by considering weighted combinations of various logistic regression models five different weighting schemes are considered in the article the weights and algorithm are justified using decision theory and risk bound results simulation studies are performed to assess the finite sample properties of the proposed model combining method it is illustrated with an application to data from an immunohistochemical study in prostate cancer

a note on a general definition of the coefficient of determination summary a generalization of the coefficient of determination r2 to general regression models is discussed a modification of an earlier definition to allow for discrete models is proposed

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

infinitely many homoclinic orbits for the second order hamiltonian systems with general potentials abstract in this paper we study the following nonperiodic second order hamiltonian systems u t l t u t r t u t r where l t may not be uniformly positive definite for all t r under more general conditions on r u t u we prove that the above system has infinitely many homoclinic orbits provided r is even in u

homoclinic solutions of hamiltonian systems with symmetry abstract we prove the existence of infinitely many homoclinic solutions for a first order hamiltonian system symmetric with respect to an action of a compact lie group by means of variational methods we make no convexity assumption on the hamiltonian

connecting orbits for some classes of almost periodic lagrangian systems abstract the existence is proved by means of variational arguments of infinitely many heteroclinic solutions connecting possibly degenerate equilibria for a class of almost periodic lagrangian system an analogous multiplicity result is then established for homoclinic solutions of systems with an almost periodic singular potential

hybrid adaptive splines abstract an adaptive spline method for smoothing is proposed that combines features from both regression spline and smoothing spline approaches one of its advantages is the ability to vary the amount of smoothing in response to the inhomogeneous curvature of true functions at different locations this method can be applied to many multivariate function estimation problems which is illustrated by an application to smoothing temperature data on the globe the method s performance in a simulation study is found to be comparable to the wavelet shrinkage methods proposed by donoho and johnstone the problem of how to count the degrees of freedom for an adaptively chosen set of basis functions is addressed this issue arises also in the mars procedure proposed by friedman and other adaptive regression spline procedures

existence and exponential decay of homoclinics in a nonperiodic superquadratic hamiltonian system abstract this paper deals with existence and exponential decay of homoclinic orbits in the first order hamiltonian system z j h z t z where the hamiltonian function h t z is nonperiodic in t r and superquadratic in z r 2 n with certain mild conditions we obtain the solutions via variational methods for strongly indefinite problems

on the degrees of freedom of the lasso we study the effective degrees of freedom of the lasso in the framework of stein s unbiased risk estimation sure we show that the number of nonzero coefficients is an unbiased estimate forthe degrees of freedom of the lasso a conclusion that requires no special assumption on the predictors in addition the unbiased estimator is shown to be asymptotically consistent with these results on hand various model selection criteria cp aic and bic are available which along with the lars algorithm provide a principled and efficient approach to obtaining the optimal lasso fit with the computational effort of a single ordinary least squares fit

calibration and empirical bayes variable selection for the problem of variable selection for the normal linear model selection criteria such as aic c p bic and ric have fixed dimensionality penalties such criteria are shown to correspond to selection of maximum posterior models under implicit hyperparameter choices for a particular hierarchical bayes formulation based on this calibration we propose empirical bayes selection criteria that use hyperparameter estimates instead of fixed choices for obtaining these estimates both marginal and conditional maximum likelihood methods are considered as opposed to traditional fixed penalty criteria these empirical bayes criteria have dimensionality penalties that depend on the data their performance is seen to approximate adaptively the performance of the best fixed penalty criterion across a variety of orthogonal and nonorthogonal set ups including wavelet regression empirical bayes shrinkage estimators of the selected coefficients are also proposed

adaptive model selection and assessment for exponential family distributions in many scientific and engineering problems selecting the optimal model from a large pool of candidate models is important particularly in data mining in the literature model assessment in the context of nonnormal distributions has not yet received a lot of attention indeed many existing model selection criteria such as the bayes information criterion and cp may not be suitable for a situation in which the conditional mean and variance of the response are dependent such as in generalized linear model regression in this article we propose a new adaptive model selection criterion and construct an approximately unbiased kullback leibler loss estimator for model assessment in the context of exponential family distributions this permits comparing any arbitrary complex modeling procedures our proposal uses a concept called generalized degrees of freedom that generalizes the concept originally proposed for the normal distribution the proposed procedure is implemented for the binomial and poisson distr

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

smoothing spline anova models for large data sets with bernoulli observations and the randomized gacv we propose the randomized generalized approximate cross validation rangacv method for choosing multiple smoothing parameters in penalized likelihood estimates for bernoulli data the method is intended for application with penalized likelihood smoothing spline anova models in addition we propose a class of approximate numerical methods for solving the penalized likelihood variational problem which in conjunction with the rangacv method allows the application of smoothing spline anova models with bernoulli data to much larger data sets than previously possible these methods are based on choosing an approximating subset of the natural representer basis functions for the variational problem simulation studies with synthetic data including synthetic data mimicking demographic risk factor data sets is used to examine the properties of the method and to compare the approach with the grkpack code of wang 1997c bayesian confidence intervals are obtained for the fits and are shown in the simulation studies to have the across the function property usually claimed for these confidence intervals finally the method is applied to an observational data set from the beaver dam eye study with scientifically interesting results

on measuring and correcting the effects of data mining and model selection abstract in the theory of linear models the concept of degrees of freedom plays an important role this concept is often used for measurement of model complexity for obtaining an unbiased estimate of the error variance and for comparison of different models i have developed a concept of generalized degrees of freedom gdf that is applicable to complex modeling procedures the definition is based on the sum of the sensitivity of each fitted value to perturbation in the corresponding observed value the concept is nonasymptotic in nature and does not require analytic knowledge of the modeling procedures the concept of gdf offers a unified framework under which complex and highly irregular modeling procedures can be analyzed in the same way as classical linear models by using this framework many difficult problems can be solved easily for example one can now measure the number of observations used in a variable selection process different modeling procedures such as a tree based regression and a

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

first order elliptic systems and the existence of homoclinic orbits in hamiltonian systems and h r x r 2 r is smooth by a homoclinic or a doubly asymptotic orbit to y where y is a periodic solution of 1 we mean a solution x y of 1 such that i x 0 y t l 0 as i tl oo finding homoclinic orbits in systems like 1 can be quite difficult in the case when y is a hyperbolic orbit this is equivalent to showing that the stable and unstable manifold of y intersect in some situation this can be done by melnikov s method recently some progress has been made by applying variational methods coti zelati and ekeland showed in 3 using dual variational methods that 1 has at least one homoclinic orbit if h satisfies some convexity and growth assumptions in their proof the convexity assumption enters in a crucial way already in the set up of the variational problem variational methods have been used by benci and giannoni 2 who studied homoclinic orbits on compact riemannian manifolds and by p rabinowitz in 9 who proved existence of heteroclinic orbits on the n dimensional torus as well as their multiplicity in 2 and 9 only special hamiltonian of lagrangian type are studied the associated variational problem is then at least semi definite the

some results on connecting orbits for a class of hamiltonian systems abstract the existence of various kinds of connecting orbits is established for a certain hamiltonian system as well as its time dependent analogue for the autonomous case our main assumption is that v has a global maximum e g at x o and we find a various kinds of orbits terminating at o for the time dependent case v has a local but not global maximum at x o and we find a homoclinic orbit emanating from and terminating at o keywords periodic solution

differential geometric least angle regression a differential geometric approach to sparse generalized linear models sparsity is an essential feature of many contemporary data problems remote sensing various forms of automated screening and other high throughput measurement devices collect a large amount of information typically about few independent statistical subjects or units in certain cases it is reasonable to assume that the underlying process generating the data is itself sparse in the sense that only a few of the measured variables are involved in the process we propose an explicit method of monotonically decreasing sparsity for outcomes that can be modelled by an exponential family in our approach we generalize the equiangular condition in a generalized linear model although the geometry involves the fisher information in a way that is not obvious in the simple regression setting the equiangular condition turns out to be equivalent with an intuitive condition imposed on the rao score test statistics in certain special cases the method can be tweaked to obtain l1 penalized generalized linear model solution paths but the method itself defines sparsity more directly although the computation of the solution paths is not trivial the method compares favourably with other path following algorithms

integer linear programming formulation for a vehicle routing problem abstract a mixed integer linear programming formulation is given for the vehicle routing problem constrained by vehicle capacity and distance travelled this corrects an error in the formulation recently proposed by kulkarni and bhave

nonstationary homoclinic orbits for an infinite dimensional hamiltonian system we study the following system of diffusion equations on r rn tu xu v x u g t x v tv xv v x v f t x u where z u v r rn r2 v c rn r g t x v f t x u are periodic in t x and superlinear at infinity by a variant linking theorem and concentration compactness arguments we establish the existence of nonstationary homoclinic type solutions of least energy for this systems

homoclinic orbits on compact manifolds abstract under a suitable assumption on the potential energy v we prove the existence of a homoclinic orbit of the equation dt x t grad v x t 0 x m where m is a compact riemannian manifold our assumption is satisfied for instance if the function v has a unique nondegenerate maximum point

tensor product space anova models to deal with the curse of dimensionality in high dimensional nonparametric problems we consider using tensor product space anova models which extend the popular additive models and are able to capture interactions of any order the multivariate function is given an anova decomposition that is it is expressed as a constant plus the sum of functions of one variable main effects plus the sum of functions of two variables two factor interactions and so on we assume the interactions to be in tensor product spaces we show in both regression and white noise settings the optimal rate of convergence for the tps anova model is within a log factor of the one dimensional optimal rate and that the penalized likelihood estimator in tps anova achieves this rate of convergence the quick optimal rate of the tps anova model makes it very preferable in high dimensional function estimation many properties of the tensor product space of sobolev hilbert spaces are also given

adaptive model selection most model selection procedures use a fixed penalty penalizing an increase in the size of a model these nonadaptive selection procedures perform well only in one type of situation for instance bayesian information criterion bic with a large penalty performs well for small models and poorly for large models and akaike s information criterion aic does just the opposite this article proposes an adaptive model selection procedure that uses a data adaptive complexity penalty based on a concept of generalized degrees of freedom the proposed procedure combining the benefit of a class of nonadaptive procedures approximates the best performance of this class of procedures across a variety of different situations this class includes many well known procedures such as aic bic mallows s cp and risk inflation criterion ric the proposed procedure is applied to wavelet thresholding in nonparametric regression and variable selection in least squares regression simulation results and an asymptotic

homoclinic solutions for some second order non autonomous hamiltonian systems without the globally superquadratic condition abstract this paper deals with the existence of homoclinic solutions for the following second order non autonomous hamiltonian system hs q v q t q f t where v c 1 r r n r v t q k t q w t q is t periodic in t f is aperiodic and belongs to l 2 r r n under the assumptions that k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w t q is not globally superquadratic on q and some additionally reasonable assumptions we give a new existence result to guarantee that hs has a homoclinic solution q t emanating from 0 the homoclinic solution q t is obtained as a limit of 2 k t periodic solutions of a sequence of the second order differential equations and these periodic solutions are obtained by the use of a standard version of the mountain pass theorem recent results in the literature are generalized and significantly improved