identification of linear directions in multivariate adaptive spline models identifying linear directions in multivariate regression has been a statistical challenge and has attracted particular attention since projection pursuit was developed to this end we propose and investigate the use of linear discriminant analysis and projection hessian directions in conjunction with multivariate adaptive regression splines simulation studies in a variety of settings demonstrate the usefulness of our approach in revealing both the functional forms and the linear substructures based on the observed data mathematical results are also provided to support our approach comparisons are made between our approach and existing approaches and the improvements are evident depending on the circumstance the extent of improvement can be substantial

dimension reduction in a semiparametric regression model with errors in covariates we consider a semiparametric estimation method for general regression models when some of the predictors are measured with error the technique relies on a kernel regression of the true covariate on all the observed covariates and surrogates this requires a nonparametric regression in as many dimensions as there are covariates and surrogates the usual theory copes with such higher dimensional problems by using higher order kernels but this is unrealistic for most problems we show that the usual theory is essentially as good as one can do with this technique instead of regression with higher order kernels we propose the use of dimension reduction techniques we assume that the true covariate depends only on a linear combination of the observed covariates and surrogates if this linear combination were known we could apply the one dimensional versions of the semiparametric problem for which standard kernels are applicable we show that if one can estimate the linear directions at the root n rate then asymptotically the resulting estimator of the parameters in the main regression model behaves as if the linear combination were known simulations lend some credence to the asymptotic results

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

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat

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

efficient and adaptive linear regression in semi supervised settings we consider the linear regression problem under semi supervised settings wherein the available data typically consists of i a small or moderate sized labeled data and ii a much larger sized unlabeled data such data arises naturally from settings where the outcome unlike the covariates is expensive to obtain a frequent scenario in modern studies involving large databases like electronic medical records emr supervised estimators like the ordinary least squares ols estimator utilize only the labeled data it is often of interest to investigate if and when the unlabeled data can be exploited to improve estimation of the regression parameter in the adopted linear model in this paper we propose a class of efficient and adaptive semi supervised estimators ease to improve estimation efficiency the ease are two step estimators adaptive to model mis specification leading to improved optimal in some cases efficiency under model mis specification and equal optimal efficiency under a linear model this adaptive property often unaddressed in the existing literature is crucial for advocating safe use of the unlabeled data the construction of ease primarily involves a flexible semi nonparametric imputation including a smoothing step that works well even when the number of covariates is not small and a follow up refitting step along with a cross validation cv strategy both of which have useful practical as well as theoretical implications towards addressing two important issues under smoothing and over fitting we establish asymptotic results including consistency asymptotic normality and the adaptive properties of ease we also provide influence function expansions and a double cv strategy for inference the results are further validated through extensive simulations followed by application to an emr study on auto immunity

linear regression after spline transformation summary in a transformation model h y x 3 e for some smooth and usually monotone function h we are often interested in the direction of f without knowing the exact form of h we consider a projection of h onto a linear space of b spline functions which has the highest correlation with the design variable x as with the box cox transformation the transformed response may then be analysed by standard linear regression software the direction estimate from canonical correlation calculations agrees with the least squares estimate for the approximating model subject to an identifiability constraint this approach is also closely related to the slicing regression of duan li 1991 the dimensionality of the space of spline transformations can be determined by a model selection principle typically a very small number of b spline knots is needed a number of real and simulated data examples is presented to demonstrate the usefulness of this approach

the coexistence problem for the discrete mathieu operator we solve the coexistence problem for the periodic discrete mathieu operator in all parametric cases the main tool in the proof will be bezout s theorem for projective plane curves as an additional result we obtain the gap opening and gap growth powers for this operator

reweighting to achieve elliptically contoured covariates in regression abstract we investigate a method of constructing weights to induce elliptically contoured covariates in regression analyses much recent work in regression has identified various data analytic and model robustness advantages associated with such covariates in particular new estimation methods like sir sirii save and phd have been built around the assumption of elliptically contoured covariates finite samples of regression covariates may deviate from this ideal in practice and the method developed here termed voronoi weighting can be used to induce elliptical symmetry in such samples in a number of examples we show that reweighting cases by the voronoi method can substantially enhance various procedures for covariates that deviate from elliptical symmetry we show that voronoi weighting in conjunction with some trimming via the minimum volume ellipsoid method can be effective

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

asymptotics for kernel estimate of sliced inverse regression to explore nonlinear structures hidden in high dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression li and duan proposed the sliced inverse regression sir method which is simple to use in this paper the asymptotic properties of the kernel estimate of sliced inverse regression are investigated it turns out that regardless of the kernel function the asymptotic distribution remains the same for a wide range of smoothing parameters

a study of the ambiguity in the solutions to the diophantine equation for chern numbers the chern numbers for hofstadter models with rational flux 2 p q are partially determined by a diophantine equation a mod q ambiguity remains the resolution of this ambiguity is only known for the rectangular lattice with nearest neighbors hopping where it has the form of a window condition we study a hofstadter butterfly on the triangular lattice for which the resolution of ambiguity is open in the model many pairs p q satisfy a window condition which is shifted relative to the window of the square model however we also find pairs p q where the chern numbers do not belong to any contiguous window this shows that the rectangular model and the one we study on the triangular lattice are not adiabatically connected many gaps must close our results suggest the conjecture that the mod q ambiguity in the diophantine equation generically reduces to a sign ambiguity

principal hessian directions for regression with measurement error summary we consider a nonlinear regression problem with predictors with measurement error we assume that the response is related to unknown linear combinations of a p dimensional predictor vector through an unknown link function instead of observing the predictors we observe a surrogate vector with the property that its expectation is linearly related to the predictor vector with constant variance we use an important linear transformation of the surrogates based on the transformed variables we develop the modified principal hessian directions method for estimating the subspace of the effective dimension reduction space we derive the asymptotic variances of the modified principal hessian directions estimators several examples are reported and comparisons are made with the sliced inverse regression method of carroll li 1992

a robust and efficient approach to causal inference based on sparse sufficient dimension reduction a fundamental assumption used in causal inference with observational data is that treatment assignment is ignorable given measured confounding variables this assumption of no missing confounders is plausible if a large number of baseline covariates are included in the analysis as we often have no prior knowledge of which variables can be important confounders thus estimation of treatment effects with a large number of covariates has received considerable attention in recent years most existing methods require specifying certain parametric models involving the outcome treatment and confounding variables and employ a variable selection procedure to identify confounders however selection of a proper set of confounders depends on correct specification of the working models the bias due to model misspecification and incorrect selection of confounding variables can yield misleading results we propose a robust and efficient approach for inference about the average treatment effect via a flexible modeling strategy incorporating penalized variable selection specifically we consider an estimator constructed based on an efficient influence function that involves a propensity score and an outcome regression we then propose a new sparse sufficient dimension reduction method to estimate these two functions without making restrictive parametric modeling assumptions the proposed estimator of the average treatment effect is asymptotically normal and semiparametrically efficient without the need for variable selection consistency the proposed methods are illustrated via simulation studies and a biomedical application

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

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

interactive tree structured regression via principal hessian directions abstract an interactive approach to tree structured regression is introduced unlike other procedures driven by cost optimization this approach focuses on the exploration of geometric information in the data the procedure begins with finding a direction along which the regression surface bends the most this direction is used for splitting the data into two regions within each region another direction is found and the partition is made in the same manner the process continues recursively until the entire regressor domain is decomposed into regions wherein the surface no longer bends significantly and linear regression fit becomes appropriate for implementing the direction search the method of principal hessian directions is applied several simulation and empirical results are reported comparison with three methods cart support and mars is made the benefit of using geometric information is highlighted

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

inverse regression for longitudinal data sliced inverse regression is an appealing dimension reduction method for regression models with multivariate covariates it has been extended to functional covariates where the whole trajectories of random functional covariates are observed completely we aim at developing an inverse regression approach for intermittently and sparsely measured longitudinal covariates we show under some regularity conditions that the convergence rate of the estimated edr effective dimension reduction directions is a function of smoothing parameters sample size and the number of eigenfucntions used to reconstruct the empirical inverse of the covariance operator simulation studies and data analysis are also provided to demonstrate the performance of our method

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

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

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

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

discriminant adaptive nearest neighbor classification nearest neighbour classification expects the class conditional probabilities to be locally constant and suffers from bias in high dimensions we propose a locally adaptive form of nearest neighbour classification to try to ameliorate this curse of dimensionality we use a local linear discriminant analysis to estimate an effective metric for computing neighbourhoods we determine the local decision boundaries from centroid information and then shrink neighbourhoods in directions orthogonal to these local decision boundaries and elongate them parallel to the boundaries thereafter any neighbourhood based classifier can be employed using the modified neighbourhoods the posterior probabilities tend to be more homogeneous in the modified neighbourhoods we also propose a method for global dimension reduction that combines local dimension information in a number of examples the methods demonstrate the potential for substantial improvements over nearest neighbour classification

higher arithmetic intersection theory abstract we give a new definition of higher arithmetic chow groups for smooth projective varieties defined over an arithmetic field which is similar to gillet and soule s definition of arithmetic chow groups we also give a compact description of the intersection theory of such groups a consequence of this theory is the definition of a heigh pairing between two higher algebraic cycles of complementary dimensions whose real regulator class is zero this description agrees with beilinson s height pairing for the classical arithmetic chow groups we also give examples of higher arithmetic intersection pairing in dimension zero

flexible parsimonious smoothing and additive modeling a simple method is presented for fitting regression models that are nonlinear in the explanatory variables despite its simplicity or perhaps because of it the method has some powerful characteristics that cause it to be competitive with and often superior to more sophisticated techniques especially for small data sets in the presence of high noise

on the crossing rule abstract the paper gives conditions on a family of matrices which guarantee that some matrix in the family will have a multiple eigenvalue in particular the main theorem states exactly which dimensions admit k dimensional subspaces of matrices for which all nonzero elements have distinct eigenvalues this question arises naturally in the theory of first order hyperbolic systems of partial differential equations the main theorem in this context tells exactly for which integers n an n x n system in k space variables may be strictly hyperbolic author

on surrogate dimension reduction for measurement error regression an invariance law we consider a general nonlinear regression problem where the predictors contain measurement error it has been recently discovered that several well known dimension reduction methods such as ols sir and phd can be performed on the surrogate regression problem to produce consistent estimates for the original regression problem involving the unobserved true predictor in this paper we establish a general invariance law between the surrogate and the original dimension reduction spaces which implies that at least at the population level the two dimension reduction problems are in fact equivalent consequently we can apply all existing dimension reduction methods to measurement error regression problems the equivalence holds exactly for multivariate normal predictors and approximately for arbitrary predictors we also characterize the rate of convergence for the surrogate dimension reduction estimators finally we apply several dimension reduction methods to real and simulated data sets involving measurement error to compare their performances

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

an rkhs formulation of the inverse regression dimension reduction problem suppose that y is a scalar and x is a second order stochastic process where y and x are conditionally independent given the random variables 1 p which belong to the closed span l2 of x this paper investigates a unified framework for the inverse regression dimension reduction problem it is found that the identification of l2 with the reproducing kernel hilbert space of x provides a platform for a seamless extension from the finite to infinite dimensional settings it also facilitates convenient computational algorithms that can be applied to a variety of models

sensitivity analysis for box cox power transformation model contrast parameters summary sensitivity analysis is an important tool for the evaluation of scientific models we consider the box cox power transformation model and study the sensitivity for the estimated slope vector when the transformation parameter is perturbed this is a key issue in the debate among bickel doksum 1981 box cox 1982 and hinkley runger 1984 on the application of the power transformation model hinkley runger 1984 conjectured that box cox s 1964 z transformation would eliminate asymptotically the sensitivity for the estimated slope vector we establish this conjecture under appropriate symmetry conditions on the joint distribution for the regressors x when the true transformation is logarithmic the conjecture holds if the distribution for x is axially symmetric for other power transformations the conjecture holds if x is multivariate normal we also consider a weaker version of hinkley runger s conjecture namely the z transformation would achieve the best possible reduction in the sensitivity we establish this weaker conjecture under less restrictive symmetry conditions which only involve the marginal distribution for xfl both conjectures might fail when those symmetry conditions are not satisfied two examples are given to illustrate the limitations of the conjectures sensitivity analysis is an important tool for the evaluation of scientific models many scientific models include some nuisance parameters which are not known precisely it is usually desirable for the model output to be stable with respect to those parameters the performance of the model can be questionable if the model output is highly sensitive to those parameters it is therefore useful to carry out a sensitivity analysis in which those parameters are perturbed the model output is derived under the perturbed parameters and compared with the model output prior to the perturbation we will study the sensitivity analysis for the box cox power transformation model with respect to the specific transformation taken let yi xi i 1 n be the observed data where yi denotes a positive scalar outcome and xi denotes a p dimensional row vector of regressors let the transformed outcome be

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

projections of a general binary model on a logistic regression abstract we consider a general binary model for which conditional probability of success given vector of predictors x equals q 1 t x k t x and a family of possibly misspecified logistic regressions fitted to it in the case when x satisfies linearity condition we show that their algebraic structure is uniquely determined and that the vector corresponding to kullback leibler projection on this family is a linear combination of 1 k this generalizes the known result proved by p ruud for k 1 which says that the true and projected vectors are collinear it also follows that the projected vector has the same direction as the first canonical vector which justifies frequent observations that logistic fit yields well performing classifiers even if misspecification is expected in the special case of additive binary model with multivariate normal predictors and when response function q is a convex combination of univariate responses we show that the variance of t x is not larger than the maximal variance of the projected linear combinations for the corresponding univariate problems in the case of balanced additive logistic model it follows that the contribution of i to is bounded by the corresponding coefficient in the convex representation of response function q

robust nonparametric methods the terms nonparametric statistics and distributionfree methods have historically referred to a collection of statistical tests whose null distributions do not depend on the underlying distribution of the data the area has been expanded to include estimates and confidence intervals derived from the tests and includes asymptotically distribution free tests for complex models as well it is now generally recognized that these statistical methods are highly efficient over large sets of possible models and are robust as well this was not always the case the earliest work was by hotelling and pabst in 1936 references to works in this paragraph can be found in the books cited in the next paragraph on rank correlation and by friedman in 1937 on rank tests in a two way design followed in 1945 with the introduction of the signed rank and rank sum tests by wilcoxon in 1947 mann and whitney extended the ideas of wilcoxon all of these rank tests were considered quick and dirty but not competitive with the presumably more efficient t tests this perspective was about to change in 1948 pitman developed efficiency concepts in a set of lecture notes that although never published were widely disseminated during the 1950s and 1960s lehmann working with hodges and with his students at berkeley showed that rank tests are surprisingly efficient and robust in fact the wilcoxon rank tests are essentially efficient at the normal model and can be much more efficient than least squares methods when the underlying distribution of the data has heavy tails in the early 1960s hodges and lehmann derived estimates and confidence intervals from rank test statistics and also introduced aligned rank tests for use in the regression model in 1967 hajek and sid published their seminal book on the theory of rank tests it is impossible to mention all of the contributions to this early development hence we cite several key mono

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

gauss polynomials and the rotation algebra newton s binomial theorem is extended to an interesting noncommutative setting as follows if in a ring ba ab with commuting witha andb then the generalized binomial coefficient left begin array 20 c n k end array right r arising in the expansion left a b right n sum limits k 0 n left begin array 20 c n k end array right gamma a n k b k resulting from these relations is equal to the value at of the gaussian polynomial left begin array 20 c n k end array right frac left n right left k right left n k right where m 1 x m 1 x m 1 1 x this is of course known in the case 1 from this it is deduced that in the universal c algebraa gq generated by unitariesu andv such thatvu e 2 i uv the spectrum of the self adjoint element u v u v has all the gaps that have been predicted to exist provided that either is rational or is a liouville number in the latter case the gaps are labelled in the natural way viak theory by the set of all non zero integers and the spectrum is a cantor set

dimension reduction for multivariate response data this article concerns the analysis of multivariate response data with multivariate regressors methods for reducing the dimensionality of response variables are developed with the goal of preserving as much regression information as possible we note parallels between this goal and the goal of sliced inverse regression which intends to reduce the regressor dimension in a univariate regression while preserving as much regression information as possible a detailed discussion is given for the case where the response is a curve measured at fixed points the problem in this setting is to select basis functions for fitting an aggregate of curves we propose that instead of focusing on goodness of fit attention should be shifted to the problem of explaining the variation of the curves in terms of the regressor variables a data adaptive basis searching method based on dimension reduction theory is proposed simulation results and an application to a climatology problem are given

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

assessing the number of linear components in a general regression problem abstract this article presents a method for testing the dimension of a general regression model without assuming any specific model for the distribution of the regressors examples of application and comparisons with existing methods are also studied

on directional regression for dimension reduction we introduce directional regression dr as a method for dimension reduction like contour regression dr is derived from empirical directions but achieves higher accuracy and requires substantially less computation dr naturally synthesizes the dimension reduction estimators based on conditional moments such as sliced inverse regression and sliced average variance estimation and in doing so combines the advantages of these methods under mild conditions it provides exhaustive and consistent estimate of the dimension reduction space we develop the asymptotic distribution of the dr estimator and from that a sequential test procedure to determine the dimension of the central space we compare the performance of dr with that of existing methods by simulation and find strong evidence of its advantage over a wide range of models finally we apply dr to analyze a data set concerning the identification of hand written digits

on average derivative quantile regression for fixed 0 1 the quantile regression function gives the th quantile x in the conditional distribution of a response variable y given the value x x of a vector of covariates it can be used to measure the effect of covariates not only in the center of a population but also in the upper and lower tails a functional that summarizes key features of the quantile specific relationship between x and y is the vector of weighted expected values of the vector of partial derivatives of the quantile function x in a nonparametric setting can be regarded as a vector of quantile specific nonparametric regression coefficients in survival analysis models e g cox s proportional hazard model proportional odds rate model accelerated failure time model and in monotone transformation models used in regression analysis gives the direction of the parameter vector in the parametric part of the model can also be used to estimate the direction of the parameter vector in semiparametric single index models popular in econometrics we show that under suitable regularity conditions the estimate of obtained by using the locally polynomial quantile estimate of chaudhuri 1991a is n 1 2 consistent and asymptotically normal with asymptotic variance equal to the variance of the influence function of the functional we discuss how the estimate of can be used for model diagnostics and in the construction of a link function estimate in general single index models

on dimension folding of matrix or array valued statistical objects we consider dimension reduction for regression or classification in which the predictors are matrix or array valued this type of predictor arises when measurements are obtained for each combination of two or more underlying variables for example the voltage measured at different channels and times in electroencephalography data for these applications it is desirable to preserve the array structure of the reduced predictor e g time versus channel but this cannot be achieved within the conventional dimension reduction formulation in this paper we introduce a dimension reduction method to be called dimension folding for matrix and array valued predictors that preserves the array structure in an application of dimension folding to an electroencephalography data set we correctly classify 97 out of 122 subjects as alcoholic or nonalcoholic based on their electroencephalography in a cross validation sample

prospects of nonparametric modeling this article brie y summarizes key developments in nonparametric function estimation over the past three decades it highlights some potentially fruitful areas of future research on nonparametric modeling and its applications

score tests for the single index model the single index model is a generalization of the linear regression model with e y x g x where g is an unknown function the model provides a flexible alternative to the linear regression model while providing more structure than a fully nonparametric approach although the fitting of single index models does not require distributional assumptions on the error term the properties of the estimates depend on such assumptions as does practical application of the model in this article score tests are derived for three potential misspecifications of the single index model heteroscedasticity in the errors autocorrelation in the errors and the omission of an important variable in the linear index these tests have a similar structure to corresponding tests for nonlinear regression models monte carlo simulations demonstrate that the first two tests hold their nominal size well and have good power properties in identifying model violations often outperforming other tests testing for the need for ad

adjusted least squares estimates for the scaled regression coefficients with censored data abstract the ordinary least squares ols method is popular for analyzing linear regression models because of its simplicity in computation suppose that the regressor variables are stochastic and the dependent observations are censored we can prove that under very general design conditions the least squares ls method can still be useful in estimating the scaled regression coefficients of the general regression model y q xi i i 1 2 n provided that the censored response observations are properly weighted here is a constant is a 1 p row vector x i are p 1 column vectors of explanatory variables i are unobserved random errors and q is an arbitrary unknown function particularly we shall see that under stronger design conditions such as assuming that the regressor variables have elliptically symmetric distribution the ols estimator consistently estimates the scaled when the response observations are complete the model discussed here is not the usual nonlinear re

quantal phase factors accompanying adiabatic changes a quantal system in an eigenstate slowly transported round a circuit c by varying parameters r in its hamiltonian r will acquire a geometrical phase factor exp i c in addition to the familiar dynamical phase factor an explicit general formula for c is derived in terms of the spectrum and eigenstates of r over a surface spanning c if c lies near a degeneracy of c takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy as an illustration c is calculated for spinning particles in slowly changing magnetic fields although the sign reversal of spinors on rotation is a special case the effect is predicted to occur for bosons as well as fermions and a method for observing it is proposed it is shown that the aharonov bohm effect can be interpreted as a geometrical phase factor

a semiparametric correction for attenuation abstract a correction method is proposed for models including the generalized linear model when the covariate is measured with error the method requires a separate validation data set that consists of the surrogate w and the true covariate x or an unbiased estimate x of x we do not require the classical additive measurement error model in which the surrogate is unbiased for the true covariates we first obtain an estimate of e x w by using nonparametric kernel regression of x or x on w based on the validation data then we perform a standard analysis with the unknown x replaced by the estimate of e x w the asymptotic distribution of the resulting regression parameter estimator is obtained generalizations to include components of x measured without error are also discussed

learning gradients predictive models that infer geometry and statistical dependence the problems of dimension reduction and inference of statistical dependence are addressed by the modeling framework of learning gradients the models we propose hold for euclidean spaces as well as the manifold setting the central quantity in this approach is an estimate of the gradient of the regression or classification function two quadratic forms are constructed from gradient estimates the gradient outer product and gradient based diffusion maps the first quantity can be used for supervised dimension reduction on manifolds as well as inference of a graphical model encoding dependencies that are predictive of a response variable the second quantity can be used for nonlinear projections that incorporate both the geometric structure of the manifold as well as variation of the response variable on the manifold we relate the gradient outer product to standard statistical quantities such as covariances and provide a simple and precise comparison of a variety of supervised dimensionality reduction methods we provide rates of convergence for both inference of informative directions as well as inference of a graphical model of variable dependencies

the adjoint projection pursuit regression abstract consider a projection pursuit regression model y g x e with an arbitrary monotonic link function g we assume the link function is unknown thus we can only estimate the direction of that is the ratios among the components of say 1 2 the direction of is a useful estimand that measures say how many units of x 2 is equivalent to one unit of x 1 in terms of potency in affecting the outcome y projection pursuit regression usually solves the equation cov x y e y x 0 as an alternative we propose to solve the adjoint equation cov x e x x y 0 the adjoint equation has the advantage that e x x might be easier to estimate than e y x we establish two main results for the population case first is the unique solution up to a multiplicative scalar to the adjoint equation second satisfies a fixed point property based on a modified least squares regression derived from the adjoint equation we apply the population results to two important

partial least squares estimator for single index models the partial least squares pls approach first constructs new explanatory variables known as factors or components which are linear combinations of available predictor variables a small subset of these factors is then chosen and retained for prediction we study the performance of pls in estimating single index models especially when the predictor variables exhibit high collinearity we show that pls estimates are consistent up to a constant of proportionality we present three simulation studies that compare the performance of pls in estimating single index models with that of sliced inverse regression sir in the first two studies we find that pls performs better than sir when collinearity exists in the third study we learn that pls performs well even when there are multiple dependent variables the link function is non linear and the shape of the functional form is not known

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

dimension reduction for censored regression data without parametric assumptions high dimensional regression analysis is already complex this is made even harder when data are subject to censoring in this article we seek ways of reducing the dimensionality of the regressor before applying nonparametric smoothing techniques if the censoring time is independent of the lifetime then the method of sliced inverse regression can be applied directly otherwise modification is needed to adjust for the censoring bias a key identity leading to the bias correction is derived and the root n consistency of the modified estimate is established patterns of censoring can also be studied under a similar dimension reduction framework some simulation results and an application to a real data set are reported 1 introduction survival data are often subject to censoring when this occurs the incompleteness of the observed data may induce a substantial bias in the sample several approaches have been suggested to overcome the associated difficulties in regression including the accelerated failure time model censored linear regression the cox proportional hazard model and many others survival analysis becomes even more intricate when the dimension of the regressor increases to apply any of the aforementioned methods users are required to specify a functional form which relates the outcome variables to the input ones however in reality knowledge needed for an appropriate model specification is often inadequate as a matter of fact the acquisition of such information may well turn out to be one of the primary goals of the study itself under such circumstances it seems preferable to have exploratory tools that rely less on such model specification this is the issue to be addressed in this article the dimension reduction approach of li 1991 will be extended to settings which allow for censoring in the data we shall offer methods of finding low dimensional projections of the data for visually examining the censoring pattern we shall show how censored regression data can still be analyzed without assuming the functional form a priori

effective dimension reduction for sparse functional data we propose a method of effective dimension reduction for functional data emphasizing the sparse design where one observes only a few noisy and irregular measurements for some or all of the subjects the proposed method borrows strength across the entire sample and provides a way to characterize the effective dimension reduction space via functional cumulative slicing our theoretical study reveals a bias variance trade off associated with the regularizing truncation and decaying structures of the predictor process and the effective dimension reduction space a simulation study and an application illustrate the superior finite sample performance of the method

prediction by supervised principal components in regression problems where the number of predictors greatly exceeds the number of observations conventional regression techniques may produce unsatisfactory results we describe a technique called supervised principal components that can be applied to this type of problem supervised principal components is similar to conventional principal components analysis except that it uses a subset of the predictors selected based on their association with the outcome supervised principal components can be applied to regression and generalized regression problems such as survival analysis it compares favorably to other techniques for this type of problem and can also account for the effects of other covariates and help identify which predictor variables are most important we also provide asymptotic consistency results to help support our empirical findings these methods could become important tools for dna microarray data where they may be used to more accurately diagnose and treat cancer

measurement error regression with unknown link dimension reduction and data visualization abstract a general nonlinear regression problem is considered with measurement error in the predictors we assume that the response is related to an unknown linear combination of a multidimensional predictor through an unknown link function instead of observing the predictor we instead observe a surrogate with the property that its expectation is linearly related to the true predictor with constant variance we identify an important transformation of the surrogate variable using this transformed variable we show that if one proceeds with the usual analysis ignoring measurement error then both ordinary least squares and sliced inverse regression yield estimates which consistently estimate the true regression parameter up to a constant of proportionality we derive the asymptotic distribution of the estimates a simulation study is conducted applying sliced inverse regression in this context

jackknife variance estimation for nearest neighbor imputation nearest neighbor imputation is a popular hot deck imputation method used to compensate for nonresponse in sample surveys although this method has a long history of application the problem of variance estimation after nearest neighbor imputation has not been fully investigated because nearest neighbor imputation is a nonparametric method a nonparametric variance estimation technique such as the jackknife is desired we show that the naive jackknife that treats imputed values as observed data produces serious underestimation we also show that rao and shao s adjusted jackknife or the jackknife with each pseudoreplicate reimputed which produces asymptotically unbiased and consistent jackknife variance estimators for other imputation methods such as mean imputation random hot deck imputation and ratio or regression imputation produces serious overestimation in the case of nearest neighbor imputation two partially reimputed and a partially adjusted jackknife variance estimators are proposed and s

estimation of network parameters in semiparametric stochastic perceptron it was reported kabashima and shinomoto 1992 that estimators of a binary decision boundary show asymptotically strange behaviors when the probability model is ill posed or semiparametric we give a rigorous analysis of this phenomenon in a stochastic perceptron by using the estimating function method a stochastic perceptron consists of a neuron that is excited depending on the weighted sum of inputs but its probability distribution form is unknown here it is shown that there exists no n consistent estimator of the threshold value h that is no estimator h that converges to h in the order of 1 n as the number n of observations increases therefore the accuracy of estimation is much worse in this semiparametric case with an unspecified probability function than in the ordinary case on the other hand it is shown that there is a n consistent estimator of the synaptic weight vector these results elucidate strange behaviors of learning curves in a semiparametric statistical model

analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods the authors consider an electron on a triangular lattice with two magnetic fluxes using a c algebra formalism a perturbation theory around rational fluxes is developed this semiclassical type of analysis leads to the wilkinson rammal formula for the energy eigenvalues the analytic result is verified numerically with high accuracy up to second order crossing of eigenvalues near band edges is shown analytically as well as numerically