using the bootstrap to select one of a new class of dimension reduction methods dimension reduction in a regression analysis of response y given a p dimensional vector of predictors x reduces the dimension of x by replacing it with a lower dimensional linear combination x of the x s without specifying a parametric model and without loss of information about the conditional distribution of y given x we unify three existing methods sliced inverse regression sir sliced average variance estimate save and principal hessian directions phd into a larger class of methods each method estimates a particular candidate matrix essentially a matrix of parameters we introduce broad classes of dimension reduction candidate matrices and we distinguish estimators of the matrices from the matrices themselves given these classes of methods and several ways to estimate any matrix we now have the problem of selecting a particular matrix and estimation method we propose bootstrap methodology to select among candidate matrices estimators and dimension and in particular we investigate l

class field theory for arithmetic schemes let x be a regular arithmetic scheme i e a regular integral separated scheme flat and of finite type over spec mathbb z generalising classical class field theory for number fields we define a class group cx and show there is a natural surjective map c x rightarrow pi 1 ab x whose kernel is the connected component of 0

zeta functions of regular arithmetic schemes at s 0 lichtenbaum conjectured in 22 the existence of a weil etale cohomology in order to describe the vanishing order and the special value of the zeta function of an arithmetic scheme x at s 0 in terms of euler poincare characteristics assuming the conjectured finite generation of some motivic cohomology groups we construct such a cohomology theory for regular schemes proper over spec z in particular we compute uncon ditionally the right weil etale cohomology of number rings and projective spaces over number rings we state a precise version of lichtenbaum s con jecture which expresses the vanishing order resp the special value of the zeta function x s at s 0 as the rank resp the determinant of a single perfect complex of abelian groups r w c x z then we relate this conjecture to soule s conjecture and to the tamagawa number conjecture lichtenbaum s conjecture for projective spaces over the ring of integers of an abelian number field follows

likelihood based sufficient dimension reduction we obtain the maximum likelihood estimator of the central subspace under conditional normality of the predictors given the response analytically and in simulations we found that our new estimator can preform much better than sliced inverse regression sliced average variance estimation and directional regression and that it seems quite robust to deviations from normality

on the weil etale cohomology of number fields we give a direct description of the category of sheaves on lichtenbaum s weil etale site of a number ring then we apply this result to define a spectral sequence relating weil etale cohomology to artin verdier etale cohomology finally we construct complexes of etale sheaves computing the expected weil etale cohomology

the k theory of fields in characteristic p we show that for a field k of characteristic p h i k n is uniquely p divisible for i n we use higher chow groups as our definition of motivic cohomology this implies that the natural map k n m k k n k from milnor k theory to quillen k theory is an isomorphism up to uniquely p divisible groups and that k n m k and k n k are p torsion free as a consequence one can calculate the k theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de rham witt sheaves for example k n x p r 0 for n dimx another consequence is gersten s conjecture with finite coefficients for smooth varieties over discrete valuation rings with residue characteristic p as the last consequence bloch s cycle complexes localized at p satisfy all beilinson lichtenbaum milne axioms for motivic complexes except possibly the vanishing conjecture

polylogarithms regulators and arakelov motivic complexes we construct an explicit regulator map from bloch s higher chow group complex to the deligne complex of a complex algebraic variety x we define the arakelov motivic complex as the cone of this map shifted by one its last cohomology group is a version of the arakelov chow group defined by h gillet and c soule we relate the grassmannian n logarithms defined as in g5 to the geometry of symmetric space sln c su n for n 2 we recover lobachevsky s formula expressing the volume of an ideal geodesic simplex in the hyperbolic space via the dilogarithm using the relationship with symmetric spaces we construct the borel regulator on k2n 1 c via the grassmannian n logarithms we study the chow dilogarithm and prove a reciprocity law which strengthens suslin s reciprocity law for milnor s group k 3 on curves our note g5 can serve as an introduction to this paper

motivic cohomology over dedekind rings we study properties of bloch s higher chow groups on smooth varieties over dedekind rings we prove the vanishing of open image in new window for i n and the existence of a gersten resolution for open image in new window if the residue characteristic is p we also show that the bloch kato conjecture implies the beilinson lichtenbaum conjecture open image in new window an identification open image in new window for m invertible and a gersten resolution with arbitrary finite coefficients over a complete discrete valuation ring of mixed characteristic 0 p we construct a map from motivic cohomology to syntomic cohomology which is a quasi isomorphism provided the bloch kato conjecture holds

marginal tests with sliced average variance estimation summary we present a new computationally feasible test for the dimension of the central subspace in a regression problem based on sliced average variance estimation we also provide a marginal coordinate test under the null hypothesis both the test of dimension and the marginal coordinate test involve test statistics that asymptotically have chi squared distributions given normally distributed predictors and have a distribution that is a linear combination of chi squared distributions in general

on galois structure invariants associated to tate motives we establish the equivalence of two definitions of invariants measuring the galois module structure of k groups of rings of integers in number fields due to chinburg et al on the one hand and the authors on the other we also make some remarks concerning the possibility of yet another such definition via lichtenbaum complexes

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

graphical tools for quadratic discriminant analysis sufficient dimension reduction methods provide effective ways to visualize discriminant analysis problems for example cook and yin showed that the dimension reduction method of sliced average variance estimation save identifies variates that are equivalent to a quadratic discriminant analysis qda solution this article makes this connection explicit to motivate the use of save variates in exploratory graphics for discriminant analysis classification can then be based on the save variates using a suitable distance measure if the chosen measure is mahalanobis distance then classification is identical to qda using the original variables just as canonical variates provide a useful way to visualize linear discriminant analysis lda so do save variates help visualize qda this would appear to be particularly useful given the lack of graphical tools for qda in current software furthermore whereas lda and qda can be sensitive to nonnormality save is more robust

on motivic cohomology with z l coefficients in this paper we prove the conjecture of bloch and kato which relates milnor s k theory of a field with its galois cohomology as well as the related comparisons results for motivic cohomology with finite coefficients in the nisnevich and etale topologies

the weil etale topology for number rings there should be a grothendieck topology for an arithmetic scheme x such that the euler characteristic of the cohomology groups of the constant sheaf z with compact support at infinity gives up to sign the leading term of the zeta function of x at s 0 we construct a topology the weil etale topology for the ring of integers in a number field whose cohomology groups hl z determine such an euler characterstic if we restrict to 3

sufficient reductions in regressions with elliptically contoured inverse predictors there are two general approaches based on inverse regression for estimating the linear sufficient reductions for the regression of y on x the moment based approach such as sir pir save and dr and the likelihood based approach such as principal fitted components pfc and likelihood acquired directions lad when the inverse predictors x y are normal by construction these methods extract information from the first two conditional moments of x y they can only estimate linear reductions and thus form the linear sufficient dimension reduction sdr methodology when var x y is constant e x y contains the reduction and it can be estimated using pfc when var x y is nonconstant pfc misses the information in the variance and second moment based methods save dr lad are used instead resulting in efficiency loss in the estimation of the mean based reduction in this article we prove that a if x y is elliptically contoured with parameters y and density gy there is no linear nontrivial suf

duality via cycle complexes we show that bloch s complex of relative zero cycles can be used as a dualizing complex over perfect fields and number rings this leads to duality theorems for torsion sheaves on arbitrary separated schemes of finite type over algebraically closed fields finite fields local fields of mixed characteristic and rings of integers in number rings generalizing results which so far have only been known for smooth schemes or in low dimensions and unifying the p adic and l adic theory as an application we generalize rojtman s theorem to normal projective schemes

the bloch kato conjecture and a theorem of suslin voevodsky we give a new proof of the theorem of suslin voevodsky which shows that the bloch kato conjecture implies a portion of the beilinsonlichtenbaum conjectures our proof does not rely on resolution of singularities and thereby extends the suslin voevodsky theorem to positive characteristic

efficient integration of sufficient dimension reduction and prediction in discriminant analysis abstractsufficient dimension reduction sdr methods are popular model free tools for preprocessing and data visualization in regression problems where the number of variables is large unfortunate

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

diagnostic studies in sufficient dimension reduction sufficient dimension reduction in regression aims to reduce the predictor dimension by replacing the original predictors with some set of linear combinations of them without loss of information numerous dimension reduction methods have been developed based on this paradigm however little effort has been devoted to diagnostic studies within the context of dimension reduction in this paper we introduce methods to check goodness of fit for a given dimension reduction subspace the key idea is to extend the so called distance correlation to measure the conditional dependence relationship between the covariates and the response given a reduction subspace our methods require minimal assumptions which are usually much less restrictive than the conditions needed to justify the original methods asymptotic properties of the test statistic are studied numerical examples demonstrate the effectiveness of the proposed approach

weil etale cohomology over finite fields we calculate the derived functors r for the base change from the weil etale site to the etale site for a variety over a finite field for smooth and proper varieties we apply this to express tate s conjecture and lichtenbaum s conjecture on special values of functions in terms of weil etale cohomology of the motivic complex n

asymptotic theory of overparameterized structural models abstract a theory of overparameterized structural models is presented in such a model some redundant parameters are involved the parameter vector is not identified and the information matrix is not nonsingular the minimum discrepancy function mdf test statistic is shown to have an asymptotic chi squared distribution almost everywhere for a wide class of discrepancy functions asymptotic distribution properties of the mdf estimators are investigated the factor analysis model is discussed as an example

what you need to know to knead 1 introduction 2 the main kneading identity 3 milnor and thurston s definitions 4 an identity for the kneading determinant 5 the inverse of the kneading matrix 6 meromorphic functions in the unit disk 7 piecewise linear models 8 counting fixed points 9 total variation 10 an invariant measure with maximal entropy 11 appendix kneading theory for mappings with discontinuities

sufficient dimension reduction via inverse regression a family of dimension reduction methods the inverse regression ir family is developed by minimizing a quadratic objective function an optimal member of this family the inverse regression estimator ire is proposed along with inference methods and a computational algorithm the ire has at least three desirable properties 1 its estimated basis of the central dimension reduction subspace is asymptotically efficient 2 its test statistic for dimension has an asymptotic chi squared distribution and 3 it provides a chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of predictors given the remaining predictors current methods like sliced inverse regression belong to a suboptimal class of the ir family comparisons of these methods are reported through simulation studies the approach developed here also allows a relatively straightforward derivation of the asymptotic null distribution of the test statistic for dimension used in slic

an etale cohomology duality theorem for number fields with a real embedding abstract the restriction on 2 primary components in the artin verdier duality theorem 2 has been eliminated by zink 9 who has shown that the sheaf of units for the etale topology over the ring of integers of any number field acts as a dualizing sheaf for a modified cohomology of sheaves the present paper provides an alternate means of removing the 2 primary restriction like zink s it involves a topology which includes infinite primes but it avoids modified cohomology and will be more directly applicable in the proof of a theorem of lichtenbaum regarding zetaand l functions 4 5 related results including the cohomology of units sheaves the norm theorem and punctual duality theorem of mazur 6 are also affected by the use of a topology including the infinite primes the corresponding results in the new setting are included here

model checking for parametric single index models a dimension reduction model adaptive approach summary local smoothing testing based on multivariate non parametric regression estimation is one of the main model checking methodologies in the literature however the relevant tests suffer from the typical curse of dimensionality resulting in slow rates of convergence to their limits under the null hypothesis and less deviation from the null hypothesis under alternative hypotheses this problem prevents tests from maintaining the level of significance well and makes tests less sensitive to alternative hypotheses in the paper a model adaptation concept in lack of fit testing is introduced and a dimension reduction model adaptive test procedure is proposed for parametric single index models the test behaves like a local smoothing test as if the model were univariate it is consistent against any global alternative hypothesis and can detect local alternative hypotheses distinct from the null hypothesis at a fast rate that existing local smoothing tests can achieve only when the model is univariate simulations are conducted to examine the performance of our methodology an analysis of real data is shown for illustration the method can be readily extended to global smoothing methodology and other testing problems

fused estimators of the central subspace in sufficient dimension reduction when studying the regression of a univariate variable y on a vector x of predictors most existing sufficient dimension reduction sdr methods require the construction of slices of y to estimate moments of the conditional distribution of x given y but there is no widely accepted method for choosing the number of slices while a poorly chosen slicing scheme may produce miserable results we propose a novel and easily implemented fusing method that can mitigate the problem of choosing a slicing scheme and improve estimation efficiency at the same time we develop two fused estimators called fire and dire based on an optimal inverse regression estimator the asymptotic variance of fire is no larger than that of the original methods regardless of the choice of slicing scheme while dire is less computational intense and more robust simulation studies show that the fused estimators perform effectively the same as or substantially better than the parent methods fused estimators based on other methods can be

a semiparametric approach to dimension reduction we provide a novel and completely different approach to dimension reduction problems from the existing literature we cast the dimension reduction problem in a semiparametric estimation framework and derive estimating equations viewing this problem from the new angle allows us to derive a rich class of estimators and obtain the classical dimension reduction techniques as special cases in this class the semiparametric approach also reveals that in the inverse regression context while keeping the estimation structure intact the common assumption of linearity and or constant variance on the covariates can be removed at the cost of performing additional nonparametric regression the semiparametric estimators without these common assumptions are illustrated through simulation studies and a real data example this article has online supplementary material

on motivic decompositions arising from the method of bialynicki birula recently v chernousov s gille and a merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the bruhat decomposition using the method of a bialynicki birula and a corollary which is essentially due to s del bano i generalize this decomposition to the case of a possibly anisotropic smooth projective variety homogeneous under the action of an isotropic reductive group this answers a question of n karpenko

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