spatially dependent polya tree modeling for survival data with the proliferation of spatially oriented time to event data spatial modeling in the survival context has received increased recent attention a traditional way to capture a spatial pattern is to introduce frailty terms in the linear predictor of a semiparametric model such as proportional hazards or accelerated failure time we propose a new methodology to capture the spatial pattern by assuming a prior based on a mixture of spatially dependent polya trees for the baseline survival in the proportional hazards model thanks to modern markov chain monte carlo mcmc methods this approach remains computationally feasible in a fully hierarchical bayesian framework we compare the spatially dependent mixture of polya trees mpt approach to the traditional spatial frailty approach and illustrate the usefulness of this method with an analysis of iowan breast cancer survival data from the surveillance epidemiology and end results seer program of the national cancer institute our method provides better goodness of fit over the traditional alternatives as measured by log pseudo marginal likelihood lpml the deviance information criterion dic and full sample score fss statistics

inference for mixtures of finite polya tree models mixtures of polya tree models provide a flexible alternative when a parametric model may only hold approximately i provide computational strategies for obtaining full semiparametric inference for mixtures of finite polya tree models given a standard parameterization including models that would be troublesome to fit using dirichlet process mixtures recommendations are put forth on choosing the level of a finite polya tree and model comparison is discussed several examples demonstrate the utility of finite polya tree modeling including data fit to generalized linear mixed models and several survival models

order based dependent dirichlet processes in this article we propose a new framework for bayesian nonparametric modeling with continuous covariates in particular we allow the nonparametric distribution to depend on covariates through ordering the random variables building the weights in the stick breaking representation we focus mostly on the class of random distributions that induces a dirichlet process at each covariate value we derive the correlation between distributions at different covariate values and use a point process to implement a practically useful type of ordering two main constructions with analytically known correlation structures are proposed practical and efficient computational methods are introduced we apply our framework through mixtures of these processes to regression modeling the modeling of stochastic volatility in time series data and spatial geostatistical modeling

on conditional and intrinsic autoregressions summary gaussian conditional autoregressions have been widely used in spatial statistics and bayesian image analysis where they are intended to describe interactions between random variables at fixed sites in euclidean space the main appeal of these distributions is in the markovian interpretation of their full conditionals intrinsic autoregressions are limiting forms that retain the markov property despite being improper they can have advantages over the standard autoregressions both conceptually and in practice for example they often avoid difficulties in parameter estimation without apparent loss or exhibit appealing invariances as in texture analysis however on small arrays and in nonlattice applications both forms of autoregression can lead to undesirable second order characteristics either in the variables themselves or in contrasts among them this paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using dempster s 1972 algorithm or an appropriate modification the approach represents a partial synthesis of standard geostatistical and gaussian markov random field formulations some nonspatial applications are also mentioned

some aspects of polya tree distributions for statistical modelling the definition and elementary properties of polya tree distributions are reviewed two theorems are presented showing that polya trees can be constructed to concentrate arbitrarily closely about any desired pdf and that polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy thereby satisfying a consistency condition such theorems are false for dirichlet processes models are constructed combining partially specified polya trees with other information such as monotonicity or unimodality it is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications a numerical example is given a theorem of diaconis and freedman about dirichlet processes is generalized to polya trees allowing polya trees to be the models for errors in regression problems finally empirical bayes models using dirichlet processes are generalized to polya trees an example from berry and christensen is reanalyzed with a polya tree model

a unified framework for fitting bayesian semiparametric models to arbitrarily censored survival data including spatially referenced data a comprehensive unified approach to modeling arbitrarily censored spatial survival data is presented for the three most commonly used semiparametric models proportional hazards proportional odds

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

the maz ya shaposhnikova limit in the magnetic setting abstract we prove a magnetic version of the maz ya shaposhnikova singular limit of nonlocal norms with vanishing fractional parameter this complements a general convergence result recently obtained by authors when the parameter approaches one

multiplicity results for magnetic fractional problems abstract the paper deals with the existence of multiple solutions for a boundary value problem driven by the magnetic fractional laplacian a s that is a s u f u u in u 0 in r n where is a real parameter f is a continuous function and is an open bounded subset of r n with lipschitz boundary we prove that the problem admits at least two nontrivial weak solutions under two different sets of conditions on the nonlinear term f which are dual in a suitable sense

nonlinear fractional magnetic schrodinger equation existence and multiplicity abstract in this paper we focus our attention on the following nonlinear fractional schrodinger equation with magnetic field e 2 s a e s u v x u f u 2 u in r n where e 0 is a parameter s 0 1 n 3 a s is the fractional magnetic laplacian v r n r and a r n r n are continuous potentials and f r n r is a subcritical nonlinearity by applying variational methods and ljusternick schnirelmann theory we prove existence and multiplicity of solutions for e small

on the fractional schrodinger kirchhoff equations with electromagnetic fields and critical nonlinearity abstract in this paper we consider the fractional schrodinger kirchhoff equations with electromagnetic fields and critical nonlinearity e 2 s m u s a e 2 a e s u v x u u 2 s 2 u h x u 2 u x r n u x 0 as x where a e s is the fractional magnetic operator with 0 s 1 2 s 2 n n 2 s m r 0 r is a continuous nondecreasing function v r n r 0 and a r n r n are the electric and magnetic potentials respectively by using the fractional version of the concentration compactness principle and variational methods we show that the above problem i has at least one solution provided that e e and ii for any m n has m pairs of solutions if e e m where e and e m are sufficiently small positive numbers moreover these solutions u e 0 as e 0

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

characterization of the traces on the boundary of functions in magnetic sobolev spaces abstract we characterize the trace of magnetic sobolev spaces defined in a half space or in a smooth bounded domain in which the magnetic field a is differentiable and its exterior derivative corresponding to the magnetic field da is bounded in particular we prove that for d 1 and p 1 the trace of the magnetic sobolev space w a 1 p r d 1 is exactly w a 1 1 p p r d where a x a 1 a d x 0 for x r d with the convention a a 1 a d 1 when a c 1 r d 1 r d 1 we also characterize fractional magnetic sobolev spaces as interpolation spaces and give extension theorems from a half space to the entire space

modeling spatial survival data using semiparametric frailty models we propose a new class of semiparametric frailty models for spatially correlated survival data specifically we extend the ordinary frailty models by allowing random effects accommodating spatial correlations to enter into the baseline hazard function multiplicatively we prove identifiability of the models and give sufficient regularity conditions we propose drawing inference based on a marginal rank likelihood no parametric forms of the baseline hazard need to be assumed in this semiparametric approach monte carlo simulations and the laplace approach are used to tackle the intractable integral in the likelihood function different spatial covariance structures are explored in simulations and the proposed methods are applied to the east boston asthma study to detect prognostic factors leading to childhood asthma

predictive model selection we consider the problem of selecting one model from a large class of plausible models a predictive bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model using criteria derived from a certain predictive density and a prior specification that emphasizes the observables we implement the proposed methodology for three common problems arising in normal linear models variable subset selection selection of a transformation of predictor variables and estimation of a parametric variance function interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria relationships between the proposed criteria and other well known criteria are examined

traces of weighted sobolev spaces old and new abstract we give short simple proofs of uspenski s results characterizing besov spaces as trace spaces of weighted sobolev spaces we generalize uspenski s results and prove the optimality of these generalizations we next show how classical results on the functional calculus in the besov spaces can be obtained as straightforward consequences of the theory of weighted sobolev spaces

local smoothness of functions and bernstein durrmeyer operators continuous functions satisfying a local lipschitz condition of order 0 1 on any subset of 0 1 are characterized by the local rate of convergence of bernstein durrmeyer operators as an application we give an algorithm for singular detection a new direct estimate for the approximation of continuous functions by bernstein durrmeyer operators and kantorovich operators is also presented

bayesian density regression summary the paper considers bayesian methods for density regression allowing a random probability distribution to change flexibly with multiple predictors the conditional response distribution is expressed as a non parametric mixture of regression models with the mixture distribution changing with predictors a class of weighted mixture of dirichlet process priors is proposed for the uncountable collection of mixture distributions it is shown that this specification results in a generalized polya urn scheme which incorporates weights that are dependent on the distance between subjects predictor values to allow local dependence in the mixture distributions we propose a kernel based weighting scheme a gibbs sampling algorithm is developed for posterior computation the methods are illustrated by using simulated data examples and an epidemiologic application

maximum likelihood estimation for the proportional odds model with random effects in this article we study the semiparametric proportional odds model with random effects for correlated right censored failure time data we establish that the maximum likelihood estimators for the parameters of this model are consistent and asymptotically gaussian furthermore the limiting variances achieve the semiparametric efficiency bounds and can be consistently estimated simulation studies show that the asymptotic approximations are accurate for practical sample sizes and that the efficiency gains of the proposed estimators over those of cai cheng and wei can be substantial a real example is provided to illustrate the proposed methods

a bayesian semiparametric accelerated failure time model a bayesian semiparametric approach is described for an accelerated failure time model the error distribution is assigned a polya tree prior and the regression parameters a noninformative hierarchical prior two cases are considered the first assumes error terms are exchangeable the second assumes that error terms are partially exchangeable a markov chain monte carlo algorithm is described to obtain a predictive distribution for a future observation given both uncensored and censored data

on the structure of spectra of periodic elliptic operators the paper discusses the problem of absolute continuity of spectra of periodic elliptic operators a new result on absolute continuity for a matrix operator of schrodinger type is obtained it is shown that all types of operators for which the absolute continuity has previously been established can be reduced to this one it is also discovered that some natural generalizations stumble upon an obstacle in the form of non triviality of a certain analytic bundle on the two dimensional torus

semiclassical limit for nonlinear schrodinger equations with electromagnetic fields abstract we study the existence of standing waves for a class of nonlinear schrodinger equations in r n with both an electric and a magnetic field under suitable non degeneracy assumptions on the critical points of an auxiliary function related to the electric field we prove the existence and the multiplicity of complex valued solutions in the semiclassical limit we show that in the semiclassical limit the presence of a magnetic field produces a phase in the complex wave but it does not influence the location of peaks of the modulus of these waves

approximation theory and functional analysis part 1 research and survey articles on bernstein durrineyer polynomials with jacob weights h berens and y xu an overview of wavelets c chui a note on weak inequalities in orlicz and lorentz spaces g a edgar and l sucheston bivariate birkhoff interpolation a survey r a lorentz linear approximations of functions with several restricted derivatives y makovoz some characterizations theorems for measures associated with orthogonal polynomials on the unit circle k pan and e b saff box splines cardinal series and wavelets s d reimenschneider and z shen some aspects of the subspace structure of infinite dimensional banach space h rosenthal fairness and monotone curvature j roulier et al projections on 2 dimensional spaces n tomczak jacgermann real versus complex best rational approximation r s varga and a ruttan

hierarchical generalized linear models and frailty models with bayesian nonparametric mixing summary this paper proposes bayesian nonparametric mixing for some well known and popular models the distribution of the observations is assumed to contain an unknown mixed effects term which includes a fixed effects term a function of the observed covariates and an additive or multiplicative random effects term typically these random effects are assumed to be independent of the observed covariates and independent and identically distributed from a distribution from some known parametric family this assumption may be suspect if either there is interaction between observed covariates and unobserved covariates or the fixed effects predictor of observed covariates is misspecified another cause for concern might be simply that the covariates affect more than just the location of the mixed effects distribution as a consequence the distribution of the random effects could be highly irregular in modality and skewness leaving parametric families unable to model the distribution adequately this paper therefore proposes a bayesian nonparametric prior for the random effects to capture possible deviances in modality and skewness and to explore the observed covariates effect on the distribution of the mixed effects

a minimax theorem a new general criterion is given for ensuring that a closed saddle function has a nonempty compact set of saddlepoints under this criterion it is shown also that every minimaximizing sequence clusters around some saddlepoint a comparable theorem is given for semicontinuous quasi saddle functions the new criterion is applied to constrained saddlepoint problems and to the fenchel rockafellar duality model for constrained convex minimization finally the relationship to existing saddlepoint results is explored in detail

strong converse inequalities for meyer konig and zeller operators abstract in this paper we give the strong converse inequality of type b for meyer konig and zeller operators

weighted sobolev spaces and pseudodifferential operators with smooth symbols absmacr let u i be the fefferman stein sharp function of u and for 1 0 such that the pointwise estimate au x cm u x holds for all x e ri and all schwartz functions u this estimate implies the boundedness of 0 order pseudodifferential operators on weighted lp spaces whenever the weight function belongs to muckenhoupt s class ap having established this we construct weighted sobolev spaces of fractional order in r and on a compact manifold prove a version of sobolev s theorem and exhibit coercive weighted estimates for elliptic pseudodifferential operators

a semilinear schrodinger equation in the presence of a magnetic field we consider the semilinear stationary schrodinger equation in a magnetic field i a 2u v x u g x u u in n where v is the scalar or electric potential and a is the vector or magnetic potential we study the existence of nontrivial solutions both in the critical and in the subcritical case respectively g x u u 2 2 and g x u c 1 u p 2 where 2 p 2 the results are obtained by variational methods for g critical we use constrained minimization and for subcritical g we employ a minimax type argument in the latter case we also study the existence of infinitely many geometrically distinct solutions

a bayesian hierarchical model for related densities by using polya trees bayesian hierarchical models are used to share information between related samples and to obtain more accurate estimates of sample level parameters common structure and variation between samples when the parameter of interest is the distribution or density of a continuous variable a hierarchical model for continuous distributions is required various such models have been described in the literature using extensions of the dirichlet process and related processes typically as a distribution on the parameters of a mixing kernel we propose a new hierarchical model based on the polya tree which enables direct modelling of densities and enjoys some computational advantages over the dirichlet process the polya tree also enables more flexible modelling of the variation between samples providing more informed shrinkage and permitting posterior inference on the dispersion function which quantifies the variation between sample densities we also show how the model can be extended to cluster samples in situations where the observed samples are believed to have been drawn from several latent populations

the strong converse inequality for bernstein kantorovich operators abstract we give a solution to a problem posed by totik at the 1992 texas conference concerning the strong converse inequality for approximation by bernstein kantorovich operators the approximation behaviour of these operators is characterized for 1 p by using an appropriate k functional which for 1 p is equivalent to a second order modulus and an extra term crucial in our approach are estimates for the derivatives of iterated kantorovich operators

data augmentation priors for bayesian and semi bayes analyses of conditional logistic and proportional hazards regression data augmentation priors have a long history in bayesian data analysis formulae for such priors have been derived for generalized linear models but their accuracy depends on two approximation steps this note presents a method for using offsets as well as scaling factors to improve the accuracy of the approximations in logistic regression this method produces an exceptionally simple form of data augmentation that allows it to be used with any standard package for conditional logistic or proportional hazards regression to perform bayesian and semi bayes analyses of matched and survival data the method is illustrated with an analysis of a matched case control study of diet and breast cancer

kernel estimation of hazard functions when observations have dependent and common covariates abstract we propose a hazard model where dependence between events is achieved by assuming dependence between covariates this model allows for correlated variables specific to observations as well as macro variables which all observations share this setup better fits many economic and financial applications where events are not independent nonparametric estimation of the hazard function is then studied kernel estimators proposed in nielsen and linton 1995 and linton et al 2003 are shown to have similar asymptotic properties compared with the i i d case mixing conditions ensure the asymptotic results follow these results depend on adjustments to bandwidth conditions simulations are conducted which verify the impact of dependence on estimators bandwidth selection accounting for dependence is shown to improve performance in an empirical application trade intensity in high frequency financial data is estimated

mixtures of polya trees for flexible spatial frailty survival modelling mixtures of polya trees offer a very flexible nonparametric approach for modelling time to event data many such settings also feature spatial association that requires further sophistication either at the point level or at the lattice level in this paper we combine these two aspects within three competing survival models obtaining a data analytic approach that remains computationally feasible in a fully hierarchical bayesian framework using markov chain monte carlo methods we illustrate our proposed methods with an analysis of spatially oriented breast cancer survival data from the surveillance epidemiology and end results program of the national cancer institute our results indicate appreciable advantages for our approach over competing methods that impose unrealistic parametric assumptions ignore spatial association or both

modeling regression error with a mixture of polya trees we model the error distribution in the standard linear model as a mixture of absolutely continuous polya trees constrained to have median 0 by considering a mixture we smooth out the partitioning effects of a simple polya tree and the predictive error density has a derivative everywhere except 0 the error distribution is centered around a standard parametric family of distributions and thus may be viewed as a generalization of standard models in which important data driven features such as skewness and multimodality are allowed by marginalizing the polya tree exact inference is possible up to markov chain monte carlo error