eliminating leading corrections to scaling in the three dimensional o n symmetric φ 4 model n 3 and 4 we study corrections to scaling in the o 3 and o 4 symmetric 4 model on the three dimensional simple cubic lattice with nearest neighbour interactions for this purpose we use monte carlo simulations in connection with a finite size scaling method we find that a finite value of the coupling exists for both values of n where leading corrections to scaling vanish as a first application we compute the critical exponents 0 710 2 and 0 0380 10 for n 3 and 0 749 2 and 0 0365 10 for n 4

contour regression a general approach to dimension reduction we propose a novel approach to sufficient dimension reduction in regression based on estimating contour directions of small variation in the response these directions span the orthogonal complement of the minimal space relevant for the regression and can be extracted according to two measures of variation in the response leading to simple and general contour regression scr and gcr methodology in comparison with existing sufficient dimension reduction techniques this contour based methodology guarantees exhaustive estimation of the central subspace under ellipticity of the predictor distribution and mild additional assumptions while maintaining sqrt n consistency and computational ease moreover it proves robust to departures from ellipticity we establish population properties for both scr and gcr and asymptotic properties for scr simulations to compare performance with that of standard techniques such as ordinary least squares sliced inverse regression principal hessian directions and sliced average variance estimation confirm the advantages anticipated by the theoretical analyses we demonstrate the use of contour based methods on a data set concerning soil evaporation

testing single index restrictions with a focus on average derivatives abstract this paper considers a situation where the violation of a single index restriction is a concern only to the extent that it causes bias to the estimates of the average derivatives we propose a method to construct tests that concentrate their asymptotic powers upon only such interesting alternatives the test has a 1 2 limiting distribution under the null hypothesis and even accommodates the case where the parameter estimates have a convergence rate slower than n as in the case of maximum score estimation the testing procedure can be easily modified when the main interest lies in average increment effects of binary covariates multivariate average derivatives or linear restrictions other than those of average derivatives results from monte carlo experiments show that the asymptotic theory is a reasonable approximation of the finite sample distributions and an application of our methods to female labor market participation illustrates the importance of this non omnibus approach

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

constrained inverse regression for incorporating prior information inverse regression methods facilitate dimension reduction analyses of high dimensional data by extracting a small number of factors that are linear combinations of the original predictor variables but the estimated factors may not lend themselves readily to interpretation consistent with prior information our approach to solving this problem is to first incorporate prior information via theory or data driven constraints on model parameters and then apply the proposed method constrained inverse regression cir to extract factors that satisfy the constraints we provide chi squared and t tests to assess the significance of each factor and its estimated coefficients and we also generalize cir to other inverse regression methods in situations where both dimension reduction and factor interpretation are important finally we investigate cir s small sample performance test data driven constraints and present a marketing example to illustrate its use in discovering meaningful factors that influence th

how sensitive are average derivatives average derivatives are the mean slopes of regression functions in practice they are estimated via a nonparametric smoothing technique every smoothing method needs a calibration parameter that determines the finite sample performance in this paper we use the kernel estimation method and develop a formula for the bandwidth that describes the sensitivity of the average derivative estimator one can determine an optimal smoothing parameter from this formula which tries out to undersmooth the density of the regression variable

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

a monte carlo study of leading order scaling corrections of phi 4 theory on a three dimensional lattice we present a monte carlo study of the one component 4 model on a cubic lattice in three dimensions leading order scaling corrections are studied using the finite size scaling method we compute the corrections to scaling exponent with high precision we determine the value of the coupling at which leading order corrections to scaling vanish using this result we obtain estimates for critical exponents that are more precise than those obtained with field theoretic methods

a theoretical framework for proving accuracy results for deferred corrections general techniques are described for proving accuracy results for deferred correction solutions to differential equations these techniques apply also to computational estimates of the local discretization error the proofs avoid the necessity of demonstrating the existence of asymptotic expansions of the global error in powers of some meshsize parameter

sure independence screening in generalized linear models with np dimensionality ultrahigh dimensional variable selection plays an increasingly important role in contemporary scientific discoveries and statistical research among others fan and lv 2008 propose an independent screening framework by ranking the marginal correlations they showed that the correlation ranking procedure possesses a sure independence screening property within the context of the linear model with gaussian covariates and responses in this paper we propose a more general version of the independent learning with ranking the maximum marginal likelihood estimates or the maximum marginal likelihood itself in generalized linear models we show that the proposed methods with fan and lv 2008 as a very special case also possess the sure screening property with vanishing false selection rate the conditions under which that the independence learning possesses a sure screening is surprisingly simple this justifies the applicability of such a simple method in a wide spectrum we quantify explicitly the extent to which the dimensionality can be reduced by independence screening which depends on the interactions of the covariance matrix of covariates and true parameters simulation studies are used to illustrate the utility of the proposed approaches in addition we supported in part by grant nsf grants dms 0714554 and dms 0704337 the bulk of the work was conducted when rui song was a postdoctoral research fellow at princeton university the authors would like to thank the associate editor and two referees for their constructive comments that improve the presentation and the results of the paper ams 2000 subject classifications primary 68q32 62j12 secondary 62e99 60f10

critical exponents of the n vector model recently the series for two renormalization group functions corresponding to the anomalous dimensions of the fields and of the three dimensional field theory have been extended to next order seven loops by murray and nickel we examine the influence of these additional terms on the estimates of critical exponents of the n vector model using some new ideas in the context of the borel summation techniques the estimates have slightly changed but remain within the errors of the previous evaluation exponents such as related to the field anomalous dimension which were poorly determined in the previous evaluation of le guillou zinn justin have seen their apparent errors significantly decrease more importantly perhaps summation errors are better determined the change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values finally because an error has been discovered in the last order of the published expansions order we have also re analysed the determination of exponents from the expansion the conclusion is that the general agreement between expansion and three dimensional series has improved with respect to le guillou zinn justin

nonparametric checks for single index models in this paper we study goodness of fit testing of single index models the large sample behavior of certain score type test statistics is investigated as a by product we obtain asymptotically distribution free maximin tests for a large class of local alternatives furthermore characteristic function based goodness of fit tests are proposed which are omnibus and able to detect peak alternatives simulation results indicate that the approximation through the limit distribution is acceptable already for moderate sample sizes applications to two real data sets are illustrated

identification and estimation of nonseparable single index models in panel data with correlated random effects the identification in a nonseparable single index models with correlated random effects is considered in panel data with a fixed number of time periods the identification assumption is based on the correlated random effects structure under this assumption the parameters of interest are identified up to a multiplicative constant and could be estimated by an average difference of derivatives estimator based on the local polynomial smoothing we suggest to linearly combine the estimators obtained for different orders of differences and derive the variance minimizing weighting scheme the asymptotic distribution of the proposed estimators is derived both for stationary and non stationary explanatory variables along with a test of the stationarity finally monte carlo experiments reveal finite sample properties of the proposed estimator

sliced inverse regression with regularizations in high dimensional data analysis sliced inverse regression sir has proven to be an effective dimension reduction tool and has enjoyed wide applications the usual sir however cannot work with problems where the number of predictors p exceeds the sample size n and can suffer when there is high collinearity among the predictors in addition the reduced dimensional space consists of linear combinations of all the original predictors and no variable selection is achieved in this article we propose a regularized sir approach based on the least squares formulation of sir the l2 regularization is introduced and an alternating least squares algorithm is developed to enable sir to work with n p and highly correlated predictors the l1 regularization is further introduced to achieve simultaneous reduction estimation and predictor selection both simulations and the analysis of a microarray expression data set demonstrate the usefulness of the proposed method

a model free test for reduced rank in multivariate regression we propose a test of dimension in multivariate regression this test is in the spirit of tests on the rank of the coefficient matrix in a multivariate linear model but it does not require a prespecified model the test may be particularly useful at the outset of an analysis before a multivariate model is posited because it can lead to low dimensional summary plots that are inferred to contain all of the sample information on the multivariate mean function

graphics for regressions with a binary response abstract central dimension reduction subspaces which characterize the dependence of a response variable on one or more predictors are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension reduction subspace

the xy model and the three state antiferromagnetic potts model in three dimensions critical properties from fluctuating boundary conditions we present the results of a monte carlo study of the three dimensionalxy model and the three dimensional antiferromagnetic three state potts model in both cases we compute the difference of the free energies of a system with periodic and a system with antiperiodic boundary conditions in a neighborhood of the critical coupling from the finite size scaling behaviour of this quantity we extract values for the critical temperature and the critical exponentv that are compatible with recent high statistics monte carlo studies of the models the results for the free energy difference at the critical temperature and for the exponentv confirm that both models belong to the same universality class

extending sliced inverse regression sliced inverse regression sir and an associated chi squared test for dimension have been introduced as a method for reducing the dimension of regression problems whose predictor variables are normal in this article the assumptions on the predictor distribution under which the chi squared test was proved to apply are relaxed and the result is extended a general weighted chi squared test that does not require normal regressors for the dimension of a regression is given simulations show that the weighted chi squared test is more reliable than the chi squared test when the regressor distribution digresses from normality significantly and that it compares well with the chi squared test when the regressors are normal

structure adaptive approach for dimension reduction we propose a new method of effective dimension reduction for a multiindex model which is based on iterative improvement of the family of average derivative estimates the procedure is computationally straightforward and does not require any prior information about the structure of the underlying model we show that in the case when the effective dimension m of the index space does not exceed 3 this space can be estimated with the rate n 1 2 under rather mild assumptions on the model

numerical integration of non linear two point boundary value problems using iterated deferred corrections i abstract in several recent papers the idea has been proposed of using implicit runge kutta formulae for the approximate integration of non linear two point boundary value problems these formulae often take a very special form so that computational efficiencies are obtained in certain well defined circumstances in this paper we examine the relative merits of these different classes of formulae and discuss their efficient implementation in a deferred correction framework a theoretical comparison of these formulae is important firstly because they call for different computational tasks to be performed and secondly because they are not always applicable to exactly the same classes of problems the most efficient of these formulae are then compared with methods based on finite differences and with methods based on spline collocation for some simple smooth test functions on the basis of operation counts and numerical experimentation it is concluded that runge kutta methods can be generally competitive with the other methods considered

aggregate inverse mean estimation for sufficient dimension reduction many well known sufficient dimension reduction methods investigate the inverse conditional moments of the predictors given the response the required linearity condition the number and ar

testing predictor contributions in sufficient dimension reduction we develop tests of the hypothesis of no effect for selected predictors in regression without assuming a model for the conditional distribution of the response given the predictors predictor effects need not be limited to the mean function and smoothing is not required the general approach is based on sufficient dimension reduction the idea being to replace the predictor vector with a lower dimensional version without loss of information on the regression methodology using sliced inverse regression is developed in detail

a note on shrinkage sliced inverse regression summary we employ lasso shrinkage within the context of sufficient dimension reduction to obtain a shrinkage sliced inverse regression estimator which provides easier interpretations and better pre diction accuracy without assuming a parametric model the shrinkage sliced inverse regression approach can be employed for both single index and multiple index models simulation studies suggest that the new estimator performs well when its tuning parameter is selected by either the bayesian information criterion or the residual information criterion

on a projective resampling method for dimension reduction with multivariate responses consider the dimension reduction problem where both the response and the predictor are vectors existing estimators of this problem take one of the following routes 1 targeting the part of the dimension reduction space that is related to the conditional mean or moments of the response vector 2 pooling the estimates for the marginal dimension reduction spaces and 3 estimating the whole dimension reduction space directly by multivariate slicing however the first two approaches do not fully recover the dimension reduction space and the third is hampered by the fact that the accuracy of estimators based on multivariate slicing drops sharply as the dimension of response increases a phenomenon often called the curse of dimensionality we propose a new method that overcomes both difficulties in that it involves univariate slicing only and it is guaranteed to fully recover the dimension reduction space under reasonable conditions the method will be compared with the existing estimators by simula

sequential sufficient dimension reduction for large p small n problems summary we propose a new and simple framework for dimension reduction in the large p small n setting the framework decomposes the data into pieces thereby enabling existing approaches for n p to be adapted to n p problems estimating a large covariance matrix which is a very difficult task is avoided we propose two separate paths to implement the framework our paths provide sufficient procedures for identifying informative variables via a sequential approach we illustrate the paths by using sufficient dimension reduction approaches but the paths are very general empirical evidence demonstrates the efficacy of our paths additional simulations and applications are given in an on line supplementary file

a constructive approach to the estimation of dimension reduction directions in this paper we propose two new methods to estimate the dimension reduction directions of the central subspace cs by constructing a regression model such that the directions are all captured in the regression mean compared with the inverse regression estimation methods e g j amer statist assoc 86 1991 328 332 j amer statist assoc 86 1991 316 342 j amer statist assoc 87 1992 1025 1039 the new methods require no strong assumptions on the design of covariates or the functional relation between regressors and the response variable and have better performance than the inverse regression estimation methods for finite samples compared with the direct regression estimation methods e g j amer statist assoc 84 1989 986 995 ann statist 29 2001 1537 1566 j r stat soc ser b stat methodol 64 2002 363 410 which can only estimate the directions of cs in the regression mean the new methods can detect the directions of cs exhaustively consistency of the estimators and the convergence of corresponding algorithms are proved

efficient sparse estimate of sufficient dimension reduction in high dimension abstractin this article we propose a new efficient sparse estimate ese in sufficient dimension reduction using distance covariance our method is model free and does not need any kernel function or slicing selection moreover it can naturally deal with multivariate response scenarios making it appealing in a modified sequential algorithm that targets the large p small n problems compared with screening procedures that only use marginal utility our method can extract more useful information from the data and is capable of determining the size of the selected submodel automatically while most of screening procedures cannot under mild conditions based on manifold theories and techniques it can be shown that our method would perform asymptotically as well as if the true irrelevant predictors were known which is referred to as the oracle property extensive simulation studies and two real data examples demonstrate the effectiveness and efficiency of the proposed approach it is remarkable that the a

on sliced inverse regression with high dimensional covariates sliced inverse regression is a promising method for the estimation of the central dimension reduction subspace cdr space in semiparametric regression models it is particularly useful in tackling cases with high dimensional covariates in this article we study the asymptotic behavior of the estimate of the cdr space with high dimensional covariates that is when the dimension of the covariates goes to infinity as the sample size goes to infinity strong and weak convergence are obtained we also suggest an estimation procedure of the bayes information criterion type to ascertain the dimension of the cdr space and derive the consistency a simulation study is conducted

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

nonparametric estimation of composite functions we study the problem of nonparametric estimation of a multivariate function g d that can be represented as a composition of two unknown smooth functions f and g d we suppose that f and g belong to known smoothness classes of functions with smoothness and respectively we obtain the full description of minimax rates of estimation of g in terms of and and propose rate optimal estimators for the sup norm loss for the construction of such estimators we first prove an approximation result for composite functions that may have an independent interest and then a result on adaptation to the local structure interestingly the construction of rate optimal estimators for composite functions with given fixed smoothness needs adaptation but not in the traditional sense it is now adaptation to the local structure we prove that composition models generate only two types of local structures the local single index model and the local model with roughness isolated to a single dimension i e a model containing elements of both additive and single index structure we also find the zones of where no local structure is generated as well as the zones where the composition modeling leads to faster rates as compared to the classical nonparametric rates that depend only to the overall smoothness of g

optimal sufficient dimension reduction in regressions with categorical predictors though partial sliced inverse regression partial sir chiaromonte et al 2002 sufficient dimension reduction in regressions with categorical predictors ann statist 30 475 497 extended the scope of sufficient dimension reduction to regressions with both continuous and categorical predictors its requirement of homogeneous predictor covariances across the subpopulations restricts its application in practice when this condition fails partial sir may provide misleading results in this article we propose a new estimation method via a minimum discrepancy approach without this restriction our method is optimal in terms of asymptotic efficiency and its test statistic for testing the dimension of the partial central subspace always has an asymptotic chi squared distribution it also gives us the ability to test predictor effects an asymptotic chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of the continuous predictors given the remaining predictors is obtained

direction estimation in single index regressions we propose a general dimension reduction method that combines the ideas of likelihood correlation inverse regression and information theory we do not require that the dependence be confined to particular conditional moments nor do we place restrictions on the predictors or on the regression that are necessary for methods like ordinary least squares and sliced inverse regression although we focus on single index regressions the underlying idea is applicable more generally illustrative examples are presented

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

sparse sliced inverse regression sliced inverse regression sir is an innovative and effective method for dimension reduction and data visualization of high dimensional problems it replaces the original variables with low dimensional linear combinations of predictors without any loss of regression information and without the need to prespecify a model or an error distribution however it suffers from the fact that each sir component is a linear combination of all the original predictors thus it is often difficult to interpret the extracted components by representing sir as a regression type optimization problem we propose in this article a new method called sparse sir that combines the shrinkage idea of the lasso with sir to produce both accurate and sparse solutions the efficacy of the proposed method is verified by simulation and a real data example is given

a convex formulation for high dimensional sparse sliced inverse regression summarysliced inverse regression is a popular tool for sufficient dimension reduction which replaces covariates with a minimal set of their linear combinations without loss of information on the conditional distribution of the response given the covariates the estimated linear combinations include all covariates making results difficult to interpret and perhaps unnecessarily variable particularly when the number of covariates is large in this paper we propose a convex formulation for fitting sparse sliced inverse regression in high dimensions our proposal estimates the subspace of the linear combinations of the covariates directly and performs variable selection simultaneously we solve the resulting convex optimization problem via the linearized alternating direction methods of multiplier algorithm and establish an upper bound on the subspace distance between the estimated and the true subspaces through numerical studies we show that our proposal is able to identify the correct covariates in the high dimensional setting

component identification and estimation in nonlinear high dimensional regression models by structural adaptation this article proposes a new method of analysis of a partially linear model whose nonlinear component is completely unknown the target of analysis is identification of the set of regressors that enter in a nonlinear way in the model function and complete estimation of the model including slope coefficients of the linear component and the link function of the nonlinear component the procedure also allows selection of the significant regression variables we also develop a test of linear hypothesis against a partially linear alternative or more generally a test that the nonlinear component is m dimensional for m 0 1 2 the approach proposed in this article is fully adaptive to the unknown model structure and applies under mild conditions on the model the only important assumption is that the dimensionality of nonlinear component is relatively small the theoretical results indicate that the procedure provides a prescribed level of the identification error and estimates the linear component with

critical behaviour of the 3d xy model a monte carlo study we present the results of a study of the three dimensional xy model on a simple cubic lattice using the single cluster updating algorithm combined with improved estimators we have measured the susceptibility and the correlation length for various couplings in the high temperature phase on lattices of size up to l 112 at the transition temperature we studied the fourth order cumulant and other cumulant like quantities on lattices of size up to l 64 from our numerical data we obtain for the critical coupling kc 0 45420 2 and for the static critical exponents v 1 976 6 and v 0 662 7

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

svm decision boundary based discriminative subspace induction we study the problem of linear dimension reduction for classification with a focus on sufficient dimension reduction i e finding subspaces without loss of discrimination power first we formulate the concept of sufficient subspace for classification in parallel terms as for regression then we present a new method to estimate the smallest sufficient subspace based on an improvement of decision boundary analysis dba the main idea is to combine dba with support vector machines svm to overcome the inherent difficulty of dba in small sample size situations while keeping dba s estimation simplicity the compact representation of svm boundary results in a significant gain in both speed and accuracy over previous dba implementations alternatively this technique can be viewed as a way to reduce the run time complexity of svm itself comparative experiments on one simulated and four real world benchmark datasets highlight the superior performance of the proposed approach

structural adaptation via mathbb l _p norm oracle inequalities in this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption we propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function the problem of structural adaptation is stated as the problem of selection from a given collection of estimators we develop a general selection rule and establish for it global oracle inequalities under arbitrary mathbb l p losses these results are applied for adaptive estimation in the additive multi index model

pseudo ϵ expansion and critical exponents of superfluid helium abstract the pseudo expansions series for critical exponents of the three dimensional x y model describing transition in liquid helium are derived up to the 6 terms numerical estimates extracted from the series obtained using the pade borel resummation technique scaling relations and the seven loop 7 estimate for the fisher exponent are presented including those for the exponents and measured in experiments with a record accuracy for the exponent the procedure argued to be most reliable gives 0 0117 the number that is very close to the most accurate experimental values it signals that the pseudo expansion approach is a powerful tool robust enough to evaluate critical exponents with small absolute errors the arguments in favor of such a robustness are presented

model free feature screening for ultrahigh dimensional data with the recent explosion of scientific data of unprecedented size and complexity feature ranking and screening are playing an increasingly important role in many scientific studies in this article we propose a novel feature screening procedure under a unified model framework which covers a wide variety of commonly used parametric and semiparametric models the new method does not require imposing a specific model structure on regression functions and thus is particularly appealing to ultrahigh dimensional regressions where there are a huge number of candidate predictors but little information about the actual model forms we demonstrate that with the number of predictors growing at an exponential rate of the sample size the proposed procedure possesses consistency in ranking which is both useful in its own right and can lead to consistency in selection the new procedure is computationally efficient and simple and exhibits a competent empirical performance in our intensive simulations and real

direct semiparametric estimation of single index models with discrete covariates abstract others have developed average derivative estimators of the parameter in the model e y x x g x where g is an unknown function and x is a random vector these estimators are noniterative and easy to compute but require that x be continuously distributed this article develops a noniterative easily computed estimator of for models in which some components of x are discrete the estimator is n consistent and asymptotically normal an application to data on product innovation by german manufacturers illustrates the estimator s usefulness

trace pursuit a general framework for model free variable selection abstractwe propose trace pursuit for model free variable selection under the sufficient dimension reduction paradigm two distinct algorithms are proposed stepwise trace pursuit and forward trace pursuit stepwise trace pursuit achieves selection consistency with fixed p forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality the screening consistency property of forward trace pursuit based on sliced inverse regression is established finite sample performances of trace pursuit and other model free variable selection methods are compared through numerical studies supplementary materials for this article are available online

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

sufficient dimension reduction based on an ensemble of minimum average variance estimators we introduce a class of dimension reduction estimators based on an ensemble of the minimum average variance estimates of functions that characterize the central subspace such as the characteristic functions the box cox transformations and wavelet basis the ensemble estimators exhaustively estimate the central subspace without imposing restrictive conditions on the predictors and have the same convergence rate as the minimum average variance estimates they are flexible and easy to implement and allow repeated use of the available sample which enhances accuracy they are applicable to both univariate and multivariate responses in a unified form we establish the consistency and convergence rate of these estimators and the consistency of a cross validation criterion for order determination we compare the ensemble estimators with other estimators in a wide variety of models and establish their competent performance

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

model free variable selection computing power has encouraged the modelling of data sets of ever increasing size data mining applications in finance marketing and bioinformatics are obvious examples a limitation of nearly all existing variable selection methods is the need to specify the correct model before selection when the number of predictors is large model formulation and validation can be difficult or even infeasible on the basis of the theory of sufficient dimension reduction we propose a new class of model free variable selection approaches the methods proposed assume no model of any form require no nonparametric smoothing and allow for general predictor effects the efficacy of the methods proposed is demonstrated via simulation and an empirical example is given

bulk and surface properties in the critical phase of the two dimensional xy model monte carlo simulations of the two dimensional xy model are performed in a square geometry with various boundary conditions bc using conformal mappings we deduce the exponent t of the order parameter correlation function and its surface analogue t as a function of the temperature in the critical low temperature phase of the model the temperature dependence of both exponents is obtained numerically with a good accuracy up to the kosterlitz thouless transition temperature the bulk exponent follows from simulations of correlation functions with periodic boundary conditions or order parameter profiles with open boundary conditions and with symmetry breaking surface fields at very low temperatures t is found in pretty good agreement with the linear temperature dependence of berezinskii s spin wave approximation we also show some evidence that there are no noticeable logarithmic corrections to the behaviour of the order parameter density profile at the kosterlitz thouless kt transition temperature while these corrections exist for the correlation function at the kt transition the value tkt 1 4 is accurately recovered the exponent associated with the surface correlations is similarly obtained after a slight modification of the boundary conditions the correlation function is computed with free bc and the profile with mixed fixed free bc it exhibits a monotonic behaviour with temperature starting linearly according to the spin wave approximation and increasing up to a value tkt 1 2 at the kosterlitz thouless transition temperature the thermal exponent e t is also computed and we give some evidence that it keeps a constant value in agreement with the marginality condition of the temperature field below the kt transition

sliced regression for dimension reduction a new dimension reduction method involving slicing the region of the response and applying local kernel regression to each slice is proposed compared with the traditional inverse regression methods e g sliced inverse regression sir the new method is free of the linearity condition and has much better estimation accuracy compared with the direct estimation methods e g mave the new method is much more robust against extreme values and can capture the entire central subspace cs exhaustively to determine the cs dimension a consistent cross validation criterion is developed extensive numerical studies including a real example confirm our theoretical findings

on the numerical integration of nonlinear two point boundary value problems using iterated deferred corrections part 2 the development and analysis of highly stable deferred correction formulae iterated deferred correction methods have been very widely used for the numerical solution of general nonlinear two point boundary value problems in ordinary differential equations however there

feature screening via distance correlation learning this article is concerned with screening features in ultrahigh dimensional data analysis which has become increasingly important in diverse scientific fields we develop a sure independence screening procedure based on the distance correlation dc sis the dc sis can be implemented as easily as the sure independence screening sis procedure based on the pearson correlation proposed by fan and lv however the dc sis can significantly improve the sis fan and lv established the sure screening property for the sis based on linear models but the sure screening property is valid for the dc sis under more general settings including linear models furthermore the implementation of the dc sis does not require model specification e g linear model or generalized linear model for responses or predictors this is a very appealing property in ultrahigh dimensional data analysis moreover the dc sis can be used directly to screen grouped predictor variables and multivariate response variables we establish

model checking in regression via dimension reduction lack of fit checking for parametric and semiparametric models is essential in reducing misspecification the efficiency of most existing model checking methods drops rapidly as the dimension of the covariates increases we propose to check a model by projecting the fitted residuals along a direction that adapts to the systematic departure of the residuals from the desired pattern consistency of the method is proved for parametric and semiparametric regression models a bootstrap implementation is also discussed simulation comparisons with several existing methods are made suggesting that the proposed methods are more efficient than the existing methods when the dimension increases air pollution data from chicago are used to illustrate the procedure

test for random number generators schwinger dyson equations for the ising model we use a set of schwinger dyson equations for the ising model to check several random number generators for the model in two and three dimensions it is shown that the equations are sensitive tests of bias originating with the random numbers the method is almost costless in computer time when added to any simulation

asymptotic properties of bridge estimators in sparse high dimensional regression models we study the asymptotic properties of bridge estimators in sparse high dimensional linear regression models when the number of covariates may increase to infinity with the sample size we are particularly interested in the use of bridge estimators to distinguish between covariates whose coefficients are zero and covariates whose coefficients are nonzero we show that under appropriate conditions bridge estimators correctly select covariates with nonzero coefficients with probability converging to one and that the estimators of nonzero coefficients have the same asymptotic distribution that they would have if the zero coefficients were known in advance thus bridge estimators have an oracle property in the sense of fan and li j amer statist assoc 96 2001 1348 1360 and fan and peng ann statist 32 2004 928 961 in general the oracle property holds only if the number of covariates is smaller than the sample size however under a partial orthogonality condition in which the covariates of the zero coefficients are uncorrelated or weakly correlated with the covariates of nonzero coefficients we show that marginal bridge estimators can correctly distinguish between covariates with nonzero and zero coefficients with probability converging to one even when the number of covariates is greater than the sample size

an efficient semiparametric estimator for binary response models this paper proposes an estimator for discrete choice models that makes no assumption concerning the functional form of the choice probability function where this function can be characterized by an index the estimator is shown to be consistent asymptotically normally distributed and to achieve the semiparametric efficiency bound monte carlo evidence indicates that there may be only modest efficiency losses relative to maximum likelihood estimation when the distribution of the disturbances is known and that the small sample behavior of the estimator in other cases is good

exploring regression structure using nonparametric functional estimation abstract average derivative functionals of regression are proposed for nonparametric model selection and diagnostics the functionals are of the integral type which under certain conditions allows their estimation at the usual parametric rate of n 1 2 we analyze asymptotic properties of the estimators of these functionals based on kernel regression these estimators can then be used for assessing the validity of various restrictions imposed on the form of regression in particular we show how they could be used to reduce the dimensionality of the model assess the relative importance of predictors measure the extent of nonlinearity and nonadditivity and under certain conditions help identify projection directions in projection pursuit models and decide on the number of these directions

sparse sufficient dimension reduction existing sufficient dimension reduction methods suffer from the fact that each dimension reduction component is a linear combination of all the original predictors so that it is difficult to interpret the resulting estimates we propose a unified estimation strategy which combines a regression type formulation of sufficient dimension reduction methods and shrinkage estimation to produce sparse and accurate solutions the method can be applied to most existing sufficient dimension reduction methods such as sliced inverse regression sliced average variance estimation and principal hessian directions we demonstrate the effectiveness of the proposed method by both simulations and real data analysis

fourier methods for estimating the central subspace and the central mean subspace in regression in regression with a high dimensional predictor vector it is important to estimate the central and central mean subspaces that preserve sufficient information about the response and the mean response using the fourier transform we have derived the candidate matrices whose column spaces recover the central and central mean subspaces exhaustively under the normality assumption of the predictors explicit estimates of the central and central mean subspaces are derived bootstrap procedures are used for determining dimensionality and choosing tuning parameters simulation results and an application to a real data are reported our methods demonstrate competitive performance compared with sir save and other existing methods the approach proposed in the article provides a novel view on sufficient dimension reduction and may lead to more powerful tools in the future

dimension reduction in regression without matrix inversion summary regressions in which the fixed number of predictors p exceeds the number of independent observational units n occur in a variety of scientific fields sufficient dimension reduction provides a promising approach to such problems by restricting attention to d n linear combinations of the original p predictors however standard methods of sufficient dimension reduction require inversion of the sample predictor covariance matrix we propose a method for estimating the central subspace that eliminates the need for such inversion and is applicable regardless of the n p relationship simulations show that our method compares favourably with standard large sample techniques when the latter are applicable we illustrate our method with a genomics application

on the interpretation of regression plots abstract a framework is developed for the interpretation of regression plots including plots of the response against selected covariates residual plots added variable plots and detrended added variable plots it is shown that many of the common interpretations associated with these plots can be misleading the framework also allows for the generalization of standard plots and the development of new plotting paradigms a paradigm for graphical exploration of regression problems is sketched

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

a new algorithm for estimating the effective dimension reduction subspace the statistical problem of estimating the effective dimension reduction edr subspace in the multi index regression model with deterministic design and additive noise is considered a new procedure for recovering the directions of the edr subspace is proposed many methods for estimating the edr subspace perform principal component analysis on a family of vectors say 1 l nearly lying in the edr subspace this is in particular the case for the structure adaptive approach proposed by hristache et al 2001a in the present work we propose to estimate the projector onto the edr subspace by the solution to the optimization problem minimize maxl 1 l lt i a l subject to a am where am is the set of all symmetric matrices with eigenvalues in 0 1 and trace less than or equal to m with m being the true structural dimension under mild assumptions n consistency of the proposed procedure is proved up to a logarithmic factor in the case when the structural dimension is not larger than 4 moreover the stochastic error of the estimator of the projector onto the edr subspace is shown to depend on l logarithmically this enables us to use a large number of vectors l for estimating the edr subspace the empirical behavior of the algorithm is studied through numerical simulations

high dimensional classification using features annealed independence rules classification using high dimensional features arises frequently in many contemporary statistical studies such as tumor classification using microarray or other high throughput data the impact of dimensionality on classifications is largely poorly understood in a seminal paper bickel and levina 2004 show that the fisher discriminant performs poorly due to diverging spectra and they propose to use the independence rule to overcome the problem we first demonstrate that even for the independence classification rule classification using all the features can be as bad as the random guessing due to noise accumulation in estimating population centroids in high dimensional feature space in fact we demonstrate further that almost all linear discriminants can perform as bad as the random guessing thus it is paramountly important to select a subset of important features for high dimensional classification resulting in features annealed independence rules fair the conditions under which all the important features can be selected by the two sample t statistic are established the choice of the optimal number of features or equivalently the threshold value of the test statistics are proposed based on an upper bound of the classification error simulation studies and real data analysis support our theoretical results and demonstrate convincingly the advantage of our new classification procedure

non analyticity of the callan symanzik β function of two dimensional o n models we discuss the analytic properties of the callan symanzik function g associated with the zero momentum four point coupling g in the two dimensional 4 model with o n symmetry using renormalization group arguments we derive the asymptotic behaviour of g at the fixed point g we argue that g g o g g 1 7 for n 1 and g g o 1 log g g for n 3 our claim is supported by an explicit calculation in the ising lattice model and by a 1 n calculation for the two dimensional 4 theory we discuss how these non analytic corrections may give rise to a slow convergence of the perturbative expansion in powers of g

nonparametric estimation of an additive quantile regression model this article is concerned with estimating the additive components of a nonparametric additive quantile regression model we develop an estimator that is asymptotically normally distributed with a rate of convergence in probability of n r 2r 1 when the additive components are r times continuously differentiable for some r 2 this result holds regardless of the dimension of the covariates and thus the new estimator has no curse of dimensionality in addition the estimator has an oracle property and is easily extended to a generalized additive quantile regression model with a link function the numerical performance and usefulness of the estimator are illustrated by monte carlo experiments and an empirical example

dimension reduction based on constrained canonical correlation and variable filtering the curse of dimensionality has remained a challenge for high dimensional data analysis in statistics the sliced inverse regression sir and canonical correlation cancor methods aim to reduce the dimensionality of data by replacing the explanatory variables with a small number of composite directions without losing much information however the estimated composite directions generally involve all of the variables making their interpretation difficult to simplify the direction estimates ni cook and tsai biometrika 92 2005 242 247 proposed the shrinkage sliced inverse regression ssir based on sir in this paper we propose the constrained canonical correlation c 3 method based on cancor followed by a simple variable filtering method as a result each composite direction consists of a subset of the variables for interpretability as well as predictive power the proposed method aims to identify simple structures without sacrificing the desirable properties of the unconstrained cancor estimates the simulation studies demonstrate the performance advantage of the proposed c 3 method over the ssir method we also use the proposed method in two examples for illustration

iterated defect correction for the solution of singular initial value problems we investigate the convergence properties of the iterated defect correction idec method based on the implicit euler rule for the solution of singular initial value problems with a singularity of the first kind we show that the method retains its classical order of convergence which means that the sequence of approximations obtained during the iteration shows gradually growing order of convergence limited by the smoothness of the data and technical details of the procedure

semiparametric models with single index nuisance parameters abstract in many semiparametric models the parameter of interest is identified through conditional expectations where the conditioning variable involves a single index that is estimated in the first step among the examples are sample selection models and propensity score matching estimators when the first step estimator follows cube root asymptotics no method of analyzing the asymptotic variance of the second step estimator exists in the literature this paper provides nontrivial sufficient conditions under which the asymptotic variance is not affected by the first step single index estimator regardless of whether it is root n or cube root consistent the finding opens a way to simple inference procedures in these models results from monte carlo simulations show that the procedures perform well in finite samples

kernel dimension reduction in regression we present a new methodology for sufficient dimension reduction sdr our methodology derives directly from the formulation of sdr in terms of the conditional independence of the covariate x from the response y given the projection of x on the central subspace cf j amer statist assoc 86 1991 316 342 and regression graphics 1998 wiley we show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel hilbert spaces and we show how this characterization leads to an m estimator for the central subspace the resulting estimator is shown to be consistent under weak conditions in particular we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for sdr methods we also present empirical results showing that the new methodology is competitive in practice

sparse generalized eigenvalue problem optimal statistical rates via truncated rayleigh flow the sparse generalized eigenvalue problem gep plays a pivotal role in a large family of high dimensional statistical models including sparse fisher s discriminant analysis canonical correlation analysis and sufficient dimension reduction the sparse gep involves solving a non convex optimization problem most existing methods and theory in the context of specific statistical models that are special cases of the sparse gep require restrictive structural assumptions on the input matrices we propose a two stage computational framework to solve the sparse gep at the first stage we solve a convex relaxation of the sparse gep taking the solution as an initial value we then exploit a non convex optimization perspective and propose the truncated rayleigh flow method which we call rifle to estimate the leading generalized eigenvector we show that rifle converges linearly to a solution with the optimal statistical rate of convergence theoretically our method significantly improves on the existing literature by eliminating structural assumptions on the input matrices to achieve this our analysis involves two key ingredients a new analysis of the gradient based method on non convex objective functions and a fine grained characterization of the evolution of sparsity patterns along the solution path thorough numerical studies are provided to validate the theoretical results

slicing regression a link free regression method slicing regression a link free regression m e t h o d author s naihua duan and k e r c h a u l i s o u r c e the annals of statistics v o l 19 n o 2 t u n 1991 p p 5 0 5 5 3 0 p u b l i s h e d b y institute of mathematical statistics s t a b l e u r l http www jstor org stable 2242072 accessed 16 05 2011 your use of the jstor archive indicates your acceptance of jstor s terms and conditions of use available at http www jstor org page info about policies terms jsp jstor s terms and conditions of use provides in part that unless you have obtained prior permission you may not download an entire issue of a journal or multiple copies of articles and you may use content in the jstor archive only for your personal non commercial use please contact the publisher regarding any further use of this work publisher contact information may be obtained at http www jstor org action showpublisher publishercode ims each copy of any part of a jstor transmission must contain the same copyright notice that appears on the screen or printed page of such transmission jstor is a not for profit service that helps scholars researchers and students discover use and build upon a wide range of content in a trusted digital archive we use information technology and tools to increase productivity and facilitate new forms of scholarship for more information about jstor please contact support jstor org http www jstor org

high precision monte carlo study of the 3d xy universality class we present a monte carlo study of the two component 4 model on the simple cubic lattice in three dimensions by suitable tuning of the coupling constant we eliminate leading order corrections to scaling high statistics simulations using finite size scaling techniques yield 0 6723 3 8 and 0 0381 2 2 where the statistical and systematical errors are given in the first and second bracket respectively these results are more precise than any previous theoretical estimate of the critical exponents for the 3d xy universality class

coordinate independent sparse sufficient dimension reduction and variable selection sufficient dimension reduction sdr in regression which reduces the dimension by replacing original predictors with a minimal set of their linear combinations without loss of information is very helpful when the number of predictors is large the standard sdr methods suffer because the estimated linear combinations usually consist of all original predictors making it difficult to interpret in this paper we propose a unified method coordinate independent sparse estimation cise that can simultaneously achieve sparse sufficient dimension reduction and screen out irrelevant and redundant variables efficiently cise is subspace oriented in the sense that it incorporates a coordinate independent penalty term with a broad series of model based and model free sdr approaches this results in a grassmann manifold optimization problem and a fast algorithm is suggested under mild conditions based on manifold theories and techniques it can be shown that cise would perform asymptotically as well as if the true irrelevant predictors were known which is referred to as the oracle property simulation studies and a real data example demonstrate the effectiveness and efficiency of the proposed approach

estimating sufficient reductions of the predictors in abundant high dimensional regressions we study the asymptotic behavior of a class of methods for sufficient dimension reduction in high dimension regressions as the sample size and number of predictors grow in various alignments it is demonstrated that these methods are consistent in a variety of settings particularly in abundant regressions where most predictors contribute some information on the response and oracle rates are possible simulation results are presented to support the theoretical conclusion

on the numerical solution of a class of singular two point boundary value problems abstract mono implicit runge kutta mirk formulae used in a deferred correction framework have proved to be a very effective way of solving first order systems of nonlinear two point boundary value problems we examine this general approach in detail and question whether mirk formulae are the most efficient class of runge kutta methods to use we also consider the extension of this deferred correction approach to deal with problems having a certain type of singularity at the end of the range of integration we consider the questions whether it is possible to derive mirk formulae which are applicable to such problems and whether it is in general possible to solve such problems automatically we will answer the first question in the affirmative however we will show that despite claims to the contrary appearing in the literature the answer to the second question is no