a semiparametric single index risk score across populations abstractwe consider a problem motivated by issues in nutritional epidemiology across diseases and populations in this area it is becoming increasingly common for diseases to be modeled by a single diet score such as the healthy eating index the mediterranean diet score etc for each disease and for each population a partially linear single index model is fit the partially linear aspect of the problem is allowed to differ in each population and disease however and crucially the single index itself having to do with the diet score is common to all diseases and populations and the nonparametrically estimated functions of the single index are the same up to a scale parameter using b splines with an increasing number of knots we develop a method to solve the problem and display its asymptotic theory an application to the nih aarp study of diet and health is described where we show the advantages of using multiple diseases and populations simultaneously rather than one at a time in understand

generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

a theory of pictures for quasi posets abstract the theory of pictures between posets is known to encode much of the combinatorics of symmetric group representations and related topics such as young diagrams and tableaux many reasons combinatorial e g since semi standard tableaux can be viewed as double quasi posets and topological quasi posets identify with finite topologies lead to extend the theory to quasi posets this is the object of the present article

flows and stochastic taylor series in ito calculus for general stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the ito flow map is given the computation relies on the lift to quasi shuffle algebras of formulas involving products of ito integrals of semimartingales whereas the chen strichartz formula computing the logarithm of the stratonovich flow map is classically expanded as a formal sum indexed by permutations the analogous formula in ito calculus is naturally indexed by surjections this reflects the change of algebraic background involved in the transition between the two integration theories lastly we extend our formula for the quasi shuffle chen strichartz series for the logarithm of the flow map to the non commutative case for linear matrix valued sdes driven by arbitrary semimartingales we obtain a similar formula

flots et series de taylor stochastiques we study the expansion of the solution of a stochastic differential equation as an infinite sum of iterated stochastic stratonovitch integrals this enables us to give a universal and explicit formula for any invariant diffusion on a lie group in terms of lie brackets as well as a universal and explicit formula for the brownian motion on a riemannian manifold in terms of derivatives of the curvature tensor the first of these formulae contains and extends to the non nilpotent case the results of doss 6 sussmann 17 yamato 18 fliess and normand cyrot 7 krener and lobry 19 and kunita 11 on the representation of solutions of stochastic differential equations

pull backs of algebras and crossed products by certain diagonal actions let g be a locally compact group and p q t a principal g bundle if a is a c algebra with primitive ideal space t the pull back p a of a along p is the balanced tensor product co q c t a if 3 g aut a consists of c t module automorphisms and y g aut co q is the natural action then the automorphism group y 0x 3 of co q 0 a respects the balancing and induces the diagonal action p 3 of g on p a we discuss some examples of such actions and study the crossed product p a x p g we suggest a substitute d for the fixed point algebra prove p a x g is strongly morita equivalent to d and investigate the structure of d in various cases in particular we ask when d is strongly morita equivalent to a sometimes but by no means always and investigate the case where a has continuous trace let b be a c algebra and g a locally compact group acting on b as a strongly continuous automorphism group a our goal here is to study the crossed product c algebra b x g for two classes of diagonal actions for which the induced action of g on b is free the first class includes actions of the form y 0x3 on b co q 0 a where p q t is a principal g bundle y is the dual action of g on co q and 3 g aut a is an action of g on another c algebra a we also consider diagonal actions on algebras which are the pull backs of another algebra a along a principal bundle p q t if a is a c algebra with primitive ideal space t then the pull back p a is the balanced tensor product co q cb t a when 3 g aut a consists of c t module automorphisms the product action ty 0 3 preserves the balancing and the diagonal action p 3 is by definition the induced action on p a in general if f x y and q prima y are continuous then cb y acts on co x by composition with f and on a by composition with q and the daunshofmann theorem we can therefore define the pull back f a of a along f as the c algebraic tensor product co x cb y a the reason for the name is that when a is the algebra of sections of some c bundle e over y there is a natural isomorphism of f a onto the algebra of sections of the pull back f e in 1 we discuss this and other basic properties of pull backs and give some evidence to show they are likely to be of interest in particular we show that if g is abelian and a g aut a is locally unitary in the sense of 18 then the crossed product received by the editors october 20 1983 and in revised form february 15 1984 1980 mathematics subject classification primary 46l40 46l55

nonassociative tori and applications to t duality in this paper we initiate the study of c algebras open image in new window endowed with a twisted action of a locally compact abelian lie group open image in new window and we construct a twisted crossed product open image in new window which is in general a nonassociative noncommutative algebra the duality properties of this twisted crossed product algebra are studied in detail and are applied to t duality in type ii string theory to obtain the t dual of a general principal torus bundle with general h flux which we will argue to be a bundle of noncommutative nonassociative tori nonassociativity is interpreted in the context of monoidal categories of modules we also show that this construction of the t dual includes the other special cases already analysed in a series of papers

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

estimation for a partial linear single index model in this paper we study the estimation for a partial linear single index model a two stage estimation procedure is proposed to estimate the link function for the single index and the parameters in the single index as well as the parameters in the linear component of the model asymptotic normality is established for both parametric components for the index a constrained estimating equation leads to an asymptotically more efficient estimator than existing estimators in the sense that it is of a smaller limiting variance the estimator of the nonparametric link function achieves optimal convergence rates and the structural error variance is obtained in addition the results facilitate the construction of confidence regions and hypothesis testing for the unknown parameters a simulation study is performed and an application to a real dataset is illustrated the extension to multiple indices is briefly sketched

on deep learning as a remedy for the curse of dimensionality in nonparametric regression assuming that a smoothness condition and a suitable restriction on the structure of the regression function hold it is shown that least squares estimates based on multilayer feedforward neural networks are able to circumvent the curse of dimensionality in nonparametric regression the proof is based on new approximation results concerning multilayer feedforward neural networks with bounded weights and a bounded number of hidden neurons the estimates are compared with various other approaches by using simulated data

stochastic lie group integrators we present lie group integrators for nonlinear stochastic differential equations with noncommutative vector fields whose solution evolves on a smooth finite dimensional manifold given a lie group action that generates transport along the manifold we pull back the stochastic flow on the manifold to the lie group via the action and subsequently pull back the flow to the corresponding lie algebra via the exponential map we construct an approximation to the stochastic flow in the lie algebra via closed operations and then push back to the lie group and then to the manifold thus ensuring that our approximation lies in the manifold we call such schemes stochastic munthe kaas methods after their deterministic counterparts we also present stochastic lie group integration schemes based on castell gaines methods these involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential lie series they become stochastic lie group integrator schemes if we use munthe kaas methods as the underlying ordinary differential integrator further we show that some castell gaines methods are uniformly more accurate than the corresponding stochastic taylor schemes lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes

variable selection for the single index model summary we consider variable selection in the single index model we prove that the popular leave m out crossvalidation method has different behaviour in the single index model from that in linear regression models or nonparametric regression models a new consistent variable selection method called separated crossvalidation is proposed further analysis suggests that the method has better finite sample performance and is computationally easier than leave m out crossvalidation separated crossvalidation applied to the swiss banknotes data and the ozone concentration data leads to single index models with selected variables that have better prediction capability than models based on all the covariates

identifiability of single index models and additive index models summary we provide a proof for the identifiability for both single index models and partially linear single index models assuming only the continuity of the regression function a condition much weaker than the differentiability conditions assumed in the existing literature our discussion is then extended to the identifiability of the additive index models

analysis of panel data partially linear single index models with serially correlated errors abstract in this paper we study a panel data partially linear single index models pdplsim with serially correlated errors a semiparametric minimum average variance estimation method was given to estimate both the unknown parameters and link function the asymptotic properties of the resulting estimators are established then the serially correlated error structure is investigated furthermore a generalized f type test method was proposed to check the index parameters it is shown that under the null hypothesis the proposed test statistic follows asymptotically a 2 distribution a numerical simulation and a real data example show that the proposed methods are effective

homology and k theory methods for classes of branes wrapping nontrivial cycles we apply some methods of homology and k theory to special classes of branes wrapping homologically nontrivial cycles we treat the classification of four geometries in terms of compact stabilizers by analogy with thurston s classification of three geometries and derive the k amenability of lie groups associated with locally symmetric spaces listed in this case more complicated examples of t duality and topology change from fluxes are also considered we analyse d branes and fluxes in type ii string theory on with torsion h flux and demonstrate in detail the conjectured t duality to with no flux in the simple case of t dualizing the circles reduces to the duality between with h flux and with no flux

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

estimation and testing for partially linear single index models in partially linear single index models we obtain the semiparametrically efficient profile least squares estimators of regression coefficients we also employ the smoothly clipped absolute deviation penalty scad approach to simultaneously select variables and estimate regression coefficients we show that the resulting scad estimators are consistent and possess the oracle property subsequently we demonstrate that a proposed tuning parameter selector bic identifies the true model consistently finally we develop a linear hypothesis test for the parametric coefficients and a goodness of fit test for the nonparametric component respectively monte carlo studies are also presented

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

direct estimation of the index coefficient in a single index model single index modeling is widely applied in for example econometric studies as a compromise between too restrictive parametric models and flexible but hardly estimable purely nonparametric models by such modeling the statistical analysis usually focuses on estimating the index coefficients the average derivative estimator ade of the index vector is based on the fact that the average gradient of a single index function f x t is proportional to the index vector unfortunately a straightforward application of this idea meets the so called curse of dimensionality problem if the dimensionality d of the model is larger than 2 however prior information about the vector can be used for improving the quality of gradient estimation by extending the weighting kernel in a direction of small directional derivative the method proposed in this paper consists of such iterative improvements of the original ade the whole procedure requires at most 2 log n iterations and the resulting estimator is n consistent under relatively mild assumptions on the model independently of the dimensionality d

t duality for torus bundles with h fluxes via noncommutative topology it is known that the t dual of a circle bundle with h flux given by a neveu schwarz 3 form is the t dual circle bundle with dual h flux however it is also known that torus bundles with h flux do not necessarily have a t dual which is a torus bundle a big puzzle has been to explain these mysterious missing t duals here we show that this problem is resolved using noncommutative topology it turns out that every principal t2 bundle with h flux does indeed have a t dual but in the missing cases which we characterize the t dual is non classical and is a bundle of noncommutative tori the duality comes with an isomorphism of twisted k theories just as in the classical case the isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology

aberrant crypt foci and semiparametric modeling of correlated binary data motivated by the spatial modeling of aberrant crypt foci acf in colon carcinogenesis we consider binary data with probabilities modeled as the sum of a nonparametric mean plus a latent gaussian spatial process that accounts for short range dependencies the mean is modeled in a general way using regression splines the mean function can be viewed as a fixed effect and is estimated with a penalty for regularization with the latent process viewed as another random effect the model becomes a generalized linear mixed model in our motivating data set and other applications the sample size is too large to easily accommodate maximum likelihood or restricted maximum likelihood estimation reml so pairwise likelihood a special case of composite likelihood is used instead we develop an asymptotic theory for models that are sufficiently general to be used in a wide variety of applications including but not limited to the problem that motivated this work the splines have penalty parameters that must converge to zero asymptotically we derive theory for this along with a data driven method for selecting the penalty parameter a method that is shown in simulations to improve greatly upon standard devices such as likelihood crossvalidation finally we apply the methods to the data from our experiment acf we discover an unexpected location for peak formation of acf

quasi shuffle products revisited abstract quasi shuffle products introduced by the first author have been useful in studying multiple zeta values and some of their analogues and generalizations the second author together with kajikawa ohno and okuda significantly extended the definition of quasi shuffle algebras so it could be applied to multiple q zeta values this article extends some of the algebraic machinery of the first author s original paper to the more general definition and demonstrates how various algebraic formulas in the quasi shuffle algebra can be obtained in a transparent way some applications to multiple zeta values interpolated multiple zeta values multiple q zeta values and multiple polylogarithms are given

polynomial splines and their tensor products in extended linear modeling 1994 wald memorial lecture analysis of variance type models are considered for a regression function or for the logarithm of a probability function conditional probability function density function conditional density function hazard function conditional hazard function or spectral density function polynomial splines are used to model the main effects and their tensor products are used to model any interaction components that are included in the special context of survival analysis the baseline hazard function is modeled and nonproportionality is allowed in general the theory involves the l 2 rate of convergence for the fitted model and its components the methodology involves least squares and maximum likelihood estimation stepwise addition of basis functions using rao statistics stepwise deletion using wald statistics and model selection using the bayesian information criterion cross validation or an independent test set publicly available software written in c and interfaced to s s plus is used to apply this methodology to real data

stratified fiber bundles quinn homology and brane stability of hyperbolic orbifolds we revisit the problem of stability of string vacua involving hyperbolic orbifolds using methods from homotopy theory and k homology we propose a definition of type ii string theory on such backgrounds that further carry stratified systems of fiber bundles which generalize the more conventional orbifold and symmetric string backgrounds together with a classification of wrapped branes by a suitable generalized homology theory for spaces stratified fibered over hyperbolic orbifolds we use the algebraic k theory of their fundamental groups and quinn homology to derive criteria for brane stability in terms of an atiyah hirzebruch type spectral sequence with its lift to k homology stable d branes in this setting carry stratified charges which induce new additive structures on the corresponding k homology groups we extend these considerations to backgrounds which support h flux where we use k groups of twisted group algebras of the fundamental groups to analyze stability of locally symmetric spaces with k amenable isometry groups and derive stability conditions for branes wrapping the fibers of an eilenberg maclane spectrum functor

profile kernel versus backfitting in the partially linear models for longitudinal clustered data summary we study the profile kernel and backfitting methods in partially linear models for clustered longitudinal data for independent data despite the potential root n inconsist ency of the backfitting estimator noted by rice 1986 the two estimators have the same asymptotic variance matrix as shown by opsomer ruppert 1999 in this paper theoretical comparisons of the two estimators for multivariate responses are investigated we show that for correlated data backfitting often produces a larger asymptotic variance than the profile kernel method that is for clustered data in addition to its bias problem the backfitting estimator does not have the same asymptotic efficiency as the profile kernel estimator consequently the common practice of using the backfitting method to compute profile kernel estimates is no longer advised we illustrate this in detail by following zeger diggle 1994 and lin carroll 2001 with a working independence covariance structure for nonparametric estimation and a correlated covariance structure for para metric estimation numerical performance of the two estimators is investigated through a simulation study their application to an ophthalmology dataset is also described

optimal smoothing in single index models single index models generalize linear regression they have applications to a variety of fields such as discrete choice analysis in econometrics and dose response models in biometrics where high dimensional regression models are often employed single index models are similar to the first step of projection pursuit regression a dimension reduction method in both cases the orientation vector can be estimated root n consistently even if the unknown univariate function or nonparametric link function is assumed to come from a large smoothness class however as we show in the present paper the similarities end there in particular the amount of smoothing necessary for root n consistent orientation estimation is very different in the two cases we suggest a simple empirical rule for selecting the bandwidth appropriate to single index models this rule is studied in a small simulation study and an application in binary response models

penalized spline estimation for partially linear single index models single index models are potentially important tools for multivariate nonparametric regression they generalize linear regression by replacing the linear combination 0tx with a nonparametric component 0 0tx where 0 is an unknown univariate link function by reducing the dimensionality from that of a general covariate vector x to a univariate index 0tx single index models avoid the so called curse of dimensionality we propose penalized spline p spline estimation of 0 in partially linear single index models where the mean function has the form 0 0tx 0tz the p spline approach offers a number of advantages over other fitting methods for single index models all parameters in the p spline single index model can be estimated simultaneously by penalized nonlinear least squares as a direct least squares fitting method our approach is rapid and computationally stable standard nonlinear least squares software can be used moreover joint inference for 0 0 and 0 is possible by

variable selection and function estimation in additive nonparametric regression using a data based prior abstract a hierarchical bayesian approach is proposed for variable selection and function estimation in additive nonparametric gaussian regression models and additive nonparametric binary regression models the prior for each component function is an integrated wiener process resulting in a posterior mean estimate that is a cubic smoothing spline each of the explanatory variables is allowed to be in or out of the model and the regression functions are estimated by model averaging to allow variable selection and model averaging data based priors are used for the smoothing parameter and the slope at 0 of each component function a two step markov chain monte carlo method is used to efficiently obtain the data based prior and to carry out variable selection and function estimation it is shown by simulation that significant improvements in the function estimators can be obtained over an approach that estimates all the unknown functions simultaneously the methodology is illustrated for a binary regressio

estimation of a regression function by maxima of minima of linear functions in this paper estimation of a regression function from independent and identically distributed random variables is considered estimates are defined by minimization of the empirical l2 risk over a class of functions which are defined as maxima of minima of linear functions results concerning the rate of convergence of the estimates are derived in particular it is shown that for smooth regression functions satisfying the assumption of single index models the estimate is able to achieve up to some logarithmic factor the corresponding optimal one dimensional rate of convergence hence under these assumptions the estimate is able to circumvent the so called curse of dimensionality the small sample behavior of the estimates is illustrated by applying them to simulated data

smoothing parameter selection in nonparametric regression using an improved akaike information criterion summary many different methods have been proposed to construct nonparametric estimates of a smooth regression function including local polynomial convolution kernel and smoothing spline estimators each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set in this paper an improved version of a criterion based on the akaike information criterion aic termed aicc is derived and examined as a way to choose the smoothing parameter unlike plug in methods aicc can be used to choose smoothing parameters for any linear smoother including local quadratic and smoothing spline estimators the use of aicc avoids the large variability and tendency to undersmooth compared with the actual minimizer of average squared error seen when other classical approaches such as generalized cross validation or the aic are used to choose the smoothing parameter monte carlo simulations demonstrate that the aicc based smoothing parameter is competitive with a plug in method assuming that one exists when the plug in method works well but also performs well when the plug in approach fails or is unavailable

semiparametric regression using variational approximations semiparametric regression offers a flexible framework for modeling nonlinear relationships between a response and covariates a prime example are generalized additive models gams where splines s

empirical likelihood confidence regions in a partially linear single index model summary empirical likelihood based inference for the parameters in a partially linear singleindex model is investigated unlike existing empirical likelihood procedures for other simpler models if there is no bias correction the limit distribution of the empirical likelihood ratio cannot be asymptotically tractable to attack this difficulty we propose a bias correction to achieve the standard x2 limit the bias corrected empirical likelihood ratio shares some of the desired features of the existing least squares method the estimation of the parameters is not needed when estimating nonparametric functions in the model undersmoothing for ensuring in consistency of the estimator of the parameters is avoided the bias corrected empirical likelihood is self scale invariant and no plug in estimator for the limiting variance is needed furthermore since the index is of norm 1 we use this constraint as information to increase the accuracy of the confidence regions smaller regions at the same nominal level as a by product our approach of using bias correction may also shed light on nonparametric estimation in model checking for other semiparametric regression models a simulation study is carried out to assess the performance of the bias corrected empirical likelihood an application to a real data set is illustrated

nonparametric estimation via empirical risk minimization a general notion of universal consistency of nonparametric estimators is introduced that applies to regression estimation conditional median estimation curve fitting pattern recognition and learning concepts general methods for proving consistency of estimators based on minimizing the empirical error are shown in particular distribution free almost sure consistency of neural network estimates and generalized linear estimators is established

flexible estimation of covariance function by penalized spline with application to longitudinal family data longitudinal data are routinely collected in biomedical research studies a natural model describing longitudinal data decomposes an individual s outcome as the sum of a population mean function and random subject specific deviations when parametric assumptions are too restrictive methods modeling the population mean function and the random subject specific functions nonparametrically are in demand in some applications it is desirable to estimate a covariance function of random subject specific deviations in this work flexible yet computationally efficient methods are developed for a general class of semiparametric mixed effects models where the functional forms of the population mean and the subject specific curves are unspecified we estimate nonparametric components of the model by penalized spline p spline biometrics 2001 57 253 259 and reparameterize the random curve covariance function by a modified cholesky decomposition biometrics 2002 58 121 128 which allows for unconstrained estimation of a positive semidefinite matrix to provide smooth estimates we penalize roughness of fitted curves and derive closed form solutions in the maximization step of an em algorithm in addition we present models and methods for longitudinal family data where subjects in a family are correlated and we decompose the covariance function into a subject level source and observation level source we apply these methods to the multi level framingham heart study data to estimate age specific heritability of systolic blood pressure nonparametrically copyright 2011 john wiley sons ltd

generalized partially linear single index model for zero inflated count data count data often arise in biomedical studies while there could be a special feature with excessive zeros in the observed counts the zero inflated poisson model provides a natural approach to accounting for the excessive zero counts in the semiparametric framework we propose a generalized partially linear single index model for the mean of the poisson component the probability of zero or both we develop the estimation and inference procedure via a profile maximum likelihood method under some mild conditions we establish the asymptotic properties of the profile likelihood estimators the finite sample performance of the proposed method is demonstrated by simulation studies and the new model is illustrated with a medical care dataset

on the asymptotics of penalized splines summavry we study the asymptotic behaviour of penalized spline estimators in the univariate case we use b splines and a penalty is placed on mth order differences of the coefficients the number of knots is assumed to converge to infinity as the sample size increases we show that penalized splines behave similarly to nadaraya watson kernel estimators with equivalent kernels depending upon m the equivalent kernels we obtain for penalized splines are the same as those found by silverman for smoothing splines the asymptotic distribution of the penalized spline estimator is gaussian and we give simple expressions for the asymptotic mean and variance provided that it is fast enough the rate at which the number of knots converges to infinity does not affect the asymptotic distribution the optimal rate of convergence of the penalty parameter is given penalized splines are not design adaptive

robustified maximum likelihood estimation in generalized partial linear mixed model for longitudinal data summary in this article we study the robust estimation of both mean and variance components in generalized partial linear mixed models based on the construction of robustified likelihood function under some regularity conditions the asymptotic properties of the proposed robust estimators are shown some simulations are carried out to investigate the performance of the proposed robust estimators just as expected the proposed robust estimators perform better than those resulting from robust estimating equations involving conditional expectation like sinha 2004 journal of the american statistical association99 451 460 and qin and zhu 2007 journal of multivariate analysis98 1658 1683 in the end the proposed robust method is illustrated by the analysis of a real data set

a dynamic structure for high dimensional covariance matrices and its application in portfolio allocation abstractestimation of high dimensional covariance matrices is an interesting and important research topic in this article we propose a dynamic structure and develop an estimation procedure for high dimensional covariance matrices asymptotic properties are derived to justify the estimation procedure and simulation studies are conducted to demonstrate its performance when the sample size is finite by exploring a financial application an empirical study shows that portfolio allocation based on dynamic high dimensional covariance matrices can significantly outperform the market from 1995 to 2014 our proposed method also outperforms portfolio allocation based on the sample covariance matrix the covariance matrix based on factor models and the shrinkage estimator of covariance matrix supplementary materials for this article are available online

semiparametric gee analysis in partially linear single index models for longitudinal data in this article we study a partially linear single index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases a semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations gee is introduced to estimate the two parameter vectors as well as the unknown link function under some mild conditions we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data we also discuss the estimation of the covariance or weight matrices involved in the semiparametric gee method furthermore we provide some numerical studies including monte carlo simulation and an empirical application to illustrate our methodology and theory

variance function partially linear single index models we consider heteroscedastic regression models where the mean function is a partially linear single index model and the variance function depends upon a generalized partially linear single index model we do not insist that the variance function depend only upon the mean function as happens in the classical generalized partially linear single index model we develop efficient and practical estimation methods for the variance function and for the mean function asymptotic theory for the parametric and nonparametric parts of the model is developed simulations illustrate the results an empirical example involving ozone levels is used to further illustrate the results and is shown to be a case where the variance function does not depend upon the mean function

estimating the interest rate term structure of corporate debt with a semiparametric penalized spline model this article provides a new methodology for estimating the term structure of corporate debt using a semiparametric penalized spline model the method is applied to a case study of at t bonds typically very few data are available on individual corporate bond prices too little to find a nonparametric estimate of term structure from these bonds alone this problem is solved by borrowing strength from treasury bond data more specifically we combine a nonparametric model for the term structure of treasury bonds with a parametric component for the credit spread our methodology generalizes the work of fisher nychka and zervos in several ways first their model was developed for treasury bonds only and cannot be applied directly to corporate bonds second we more fully investigate the problem of choosing the smoothing parameter a problem that is complicated because the forward rate is the derivative log d t where the discount function d is the function fit to the data in our case study estimatio

the geometry of the space of branched rough paths we construct an explicit transitive free action of a banach space of holder functions on the space of branched rough paths which yields in particular a bijection between theses two spaces this endows the space of branched rough paths the structure of a principal homogeneous space over a banach space and allows to characterize its automorphisms the construction is based on the baker campbell hausdorff formula and on the hairer kelly map which allows to describe branched rough paths in terms of anisotropic geometric rough paths

semiparametric least squares sls and weighted sls estimation of single index models abstract for the class of single index models i construct a semiparametric estimator of coefficients up to a multiplicative constant that exhibits 1 n consistency and asymptotic normality this class of models includes censored and truncated tobit models binary choice models and duration models with unobserved individual heterogeneity and random censoring i also investigate a weighting scheme that achieves the semiparametric efficiency bound

quantum hall effect on the hyperbolic plane in this paper we study both the continuous model and the discrete model of the quantum hall effect qhe on the hyperbolic plane the hall conductivity is identified as a geometric invariant associated to an imprimitivity algebra of observables we define a twisted analogue of the kasparov map which enables us to use the pairing between k theory and cyclic cohomology theory to identify this geometric invariant with a topological index thereby proving the integrality of the hall conductivity in this case

parameter estimation of partial differential equation models partial differential equation pde models are commonly used to model complex dynamic systems in applied sciences such as biology and finance the forms of these pde models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system parameters in pde models often have interesting scientific interpretations but their values are often unknown and need to be estimated from the measurements of the dynamic system in the presence of measurement errors most pdes used in practice have no analytic solutions and can only be solved with numerical methods currently methods for estimating pde parameters require repeatedly solving pdes numerically under thousands of candidate parameter values and thus the computational load is high in this article we propose two methods to estimate parameters in pde models a parameter cascading method and a bayesian approach in both methods the underlying dynamic process modeled with the pde model is represented via basis function

rate optimal estimation for a general class of nonparametric regression models with unknown link functions this paper discusses a nonparametric regression model that naturally generalizes neural network models the model is based on a finite number of one dimensional transformations and can be estimated with a one dimensional rate of convergence the model contains the generalized additive model with unknown link function as a special case for this case it is shown that the additive components and link function can be estimated with the optimal rate by a smoothing spline that is the solution of a penalized least squares criterion

stochastic expansions and hopf algebras we study solutions to nonlinear stochastic differential systems driven by a multi dimensional wiener process a useful algorithm for strongly simulating such stochastic systems is the castell gaines method which is based on the exponential lie series when the diffusion vector fields commute it has been proved that at low orders this method is more accurate in the mean square error than corresponding stochastic taylor methods however it has also been shown that when the diffusion vector fields do not commute this is not true for strong order one methods here we prove that when there is no drift and the diffusion vector fields do not commute the exponential lie series is usurped by the sinh log series in other words the mean square error associated with a numerical method based on the sinh log series is always smaller than the corresponding stochastic taylor error in fact to all orders our proof uses the underlying hopf algebra structure of these series and a two alphabet associative algebra of shuffle and concatenation operations we illustrate the benefits of the proposed series in numerical studies

a multiple index model and dimension reduction dimension reduction can be used as an initial step in statistical modeling further specification of model structure is imminent and important when the reduced dimension is still greater than 1 in this article we investigate one method of specification that involves separating the linear component from the nonlinear components leading to further dimension reduction in the unknown link function and thus better estimation and easier interpretation of the model the specified model includes the popular econometric multiple index model and the partially linear single index model as its special cases a criterion is developed to validate the model specification an algorithm is proposed to estimate the model directly asymptotic distributions for the estimators of the parameters and the nonparametric link function are derived air pollution data in chicago are used to illustrate the modeling procedure and to demonstrate its advantages over the existing dimension reduction approaches

bayesian smoothing and regression splines for measurement error problems in the presence of covariate measurement error estimating a regression function nonparametrically is extremely difficult the problem being related to deconvolution various frequentist approaches exist for this problem but to date there has been no bayesian treatment in this article we describe bayesian approaches to modeling a flexible regression function when the predictor variable is measured with error the regression function is modeled with smoothing splines and regression p splines two methods are described for exploration of the posterior the first called the iterative conditional modes icm is only partially bayesian icm uses a componentwise maximization routine to find the mode of the posterior it also serves to create starting values for the second method which is fully bayesian and uses markov chain monte carlo mcmc techniques to generate observations from the joint posterior distribution use of the mcmc approach has the advantage that interval estimates that directly model and

sparse modal additive model sparse additive models have been successfully applied to high dimensional data analysis due to the flexibility and interpretability of their representation however the existing methods are often formulated using the least squares loss with learning the conditional mean which is sensitive to data with the non gaussian noises e g skewed noise heavy tailed noise and outliers to tackle this problem we propose a new robust regression method called as sparse modal additive model spmam by integrating the modal regression metric the data dependent hypothesis space and the weighted lq 1 norm regularizer q 1 into the additive models specifically the modal regression metric assures the model robustness to complex noises via learning the conditional mode the data dependent hypothesis space offers the model adaptivity via sample based presentation and the lq 1 norm regularizer addresses the algorithmic interpretability via sparse variable selection in theory the proposed spmam enjoys statistical guarantees on asymptotic consistency for regression estimation and variable selection simultaneously experimental results on both synthetic and real world benchmark data sets validate the effectiveness and robustness of the proposed model

algebraic structures and stochastic differential equations driven by levy processes we construct an efficient integrator for stochastic differential systems driven by lvy processes an efficient integrator is a strong approximation that is more accurate than the corresponding stoc

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

adapting for the missing link we consider the fitting of generalized linear models in wich the link function is assumed to be unknown and propose the following computational method first estimate regression coefficients using the canonical link then estimate the link via a kernel smoother treating the direction in the predictor space determined by the regression coefficients as known then reestimate the direction using the estimated link and alternate between these two steps we show that under fairly general conditions n 1 2 consistent estimates of the direction are obtained a small monte carlo study is presented

spline based tests in survival analysis this paper examines a method for testing hypotheses on covariate effects in a proportional hazards model and also on how effects change over time in regression analysis of survival data the technique used is very general and can be applied to testing many other aspects of parametric and semiparametric models the basic idea is to formulate a flexible parametric alternative using fixed knot splines together with a penalty function that penalizes noisy alternatives more than smooth ones to focus the power of the tests toward smooth alternatives the test statistics are the analogs of ordinary likelihood based statistics only computed from a penalized likelihood formed by subtracting the penalty function from the ordinary log likelihood large sample approximations to the distributions are found when the number of knots is held fixed as the sample size increases numerical results suggest these approximations may be adequate with moderate sized samples

miscellanea on the optimal amount of smoothing in penalised spline regression summary the optimal amount of smoothing in penalised spline regression is investigated in particular a simple closed form approximation to the optimal smoothing parameter is derived comparisons with its exact counterpart show it to be a useful starting point for measuring the optimal amount of smoothing in penalised spline regression it also lends itself to the development of quick and simple rules for automatic smoothing parameter selection

on partial sufficient dimension reduction with applications to partially linear multi index models partial dimension reduction is a general method to seek informative convex combinations of predictors of primary interest which includes dimension reduction as its special case when the predictors in the remaining part are constants in this article we propose a novel method to conduct partial dimension reduction estimation for predictors of primary interest without assuming that the remaining predictors are categorical to this end we first take the dichotomization step such that any existing approach for partial dimension reduction estimation can be employed then we take the expectation step to integrate over all the dichotomic predictors to identify the partial central subspace as an example we use the partially linear multi index model to illustrate its applications for semiparametric modeling simulations and real data examples are given to illustrate our methodology

nonparametric regression based on hierarchical interaction models in this paper we introduce the so called hierarchical interaction models where we assume that the computation of the value of a function m mathbb r d rightarrow mathbb r is done in several layers where in each layer a function of at most d inputs computed by the previous layer is evaluated we investigate two different regression estimates based on polynomial splines and on neural networks and show that if the regression function satisfies a hierarchical interaction model and all occurring functions in the model are smooth the rate of convergence of these estimates depends on d and not on d hence in this case the estimates can achieve good rate of convergence even for large d and are in this sense able to circumvent the so called curse of dimensionality

the exponential lie series for continuous semimartingales we consider stochastic differential systems driven by continuous semimartingales and governed by non commuting vector fields we prove that the logarithm of the flowmap is an exponential lie series this relies on a natural change of basis to vector fields for the associated quadratic covariation processes analogous to stratonovich corrections the flowmap can then be expanded as a series in compositional powers of vector fields and the logaritm of the flowmap can thus be expanded in the lie algebra of vector fields further we give a direct self contained proof of the corresponding chen strichartz formula which provides an explicit formula for the lie series coefficients such exponential lie series are important in the development of strong lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves

variable selection in semiparametric regression modeling in this paper we are concerned with how to select significant variables in semiparametric modeling variable selection for semiparametric regression models consists of two components model selection for nonparametric components and select significant variables for parametric portion thus it is much more challenging than that for parametric models such as linear models and generalized linear models because traditional variable selection procedures including stepwise regression and the best subset selection require model selection to nonparametric components for each submodel this leads to very heavy computational burden in this paper we propose a class of variable selection procedures for semiparametric regression models using nonconcave penalized likelihood the newly proposed procedures are distinguished from the traditional ones in that they delete insignificant variables and estimate the coefficients of significant variables simultaneously this allows us to establish the sampling properties of the resulting estimate we first establish the rate of convergence of the resulting estimate with proper choices of penalty functions and regularization parameters we then establish the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures perform as well as an oracle procedure semiparametric generalized likelihood ratio test is proposed to select significant variables in the nonparametric component we investigate the asymptotic behavior of the proposed test and demonstrate its limiting null distribution follows a chi squared distribution which is independent of the nuisance parameters extensive monte carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedures