on projective resolutions of simple modules over the borel subalgebra s n r of the schur algebra s n r for n 3 we construct projective resolutions for simple modules over the borel subalgebras s 2 r and s 3 r of the schur algebras s 2 r and s 3 r over a field of positive characteristic in the case s 2 r we get a minimal projective resolution we also find projective resolutions of minimal possible length for weyl modules over the schur algebra s 2 r corresponding to the regular weights

denseness of the generalized eigenvectors of an h s discrete operator let t be a closed densely defined linear operator in a hilbert space h and assume there exists 0 p t such that r 0 t is a hilbert schmidt operator the operator t is a special type of discrete operator a so called h s discrete operator which is shown to be a fredholm operator in h with fredholm set equal to the whole complex plane let t i i 1 where the operator t i has ascent mi let pi i 1 2 be the projection of h onto the generalized eigenspace n t i mi along r t i mi and let s and m be the subspaces of h consisting of all x h such that x i 1 pix and such that pix 0 for i 1 2 respectively sufficient conditions are introduced which guarantee that s h and m 0 these conditions require that r t be bounded on certain rays in the complex plane and r t 0 as on at least one of the rays but specific decay rates for r t are not necessary

on modules arising from quantum groups at pr th roots of unity abstract this paper studies the reduction mod p method which constructs large classes of representations for a semisimple algebraic group g from representations for the corresponding lusztig quantum group u at a p r th root of unity the g modules arising in this way include the weyl modules the induced modules and various reduced versions of these modules we present a relation between ext g n v w and ext u n v w when v w are obtained from v w by reduction mod p since the dimensions of ext n spaces for u modules are known in many cases our result guarantees the existence of many new extension classes and homomorphisms between rational g modules one application is a new proof of james franklin s result on certain homomorphisms between two weyl modules we also provide some examples which show that the p th root of unity case and a general p r th root of unity case are essentially different

the global dimension of schur algebras for gl2 and gl3 abstract we first define the notion of good filtration dimension and weyl filtration dimension in a quasi hereditary algebra we calculate these dimensions explicitly for all irreducible modules in sl 2 and sl 3 we use these to show that the global dimension of a schur algebra for gl 2 and gl 3 is twice the good filtration dimension to do this for sl 3 we give an explicit filtration of the modules by modules of the form f l where is a dominant weight and is p restricted

on an unconditional basis of generalized eigenvectors of the nonself adjoint gribov operator in bargmann space abstract in this article we prove that the system of generalized eigenvectors of the gribov operator is an unconditional basis in bargmann space we also give a generalized diagonalization in the abel sense of the semigroups respectively associated to this operator and to its square root finally some open problems are pointed out

bounding and unbounding higher extensions for sl2 we analyse the recursive formula found for various ext groups for sl2 k k a field of characteristic p and derive various generating functions for these groups we use this to show that the growth rate for the cohomology of sl2 k is at least exponential in particular max dimextsl2 k i k a a i n has at least exponential growth for all p we also show that max dimextsl2 k i k a a n for a fixed i is bounded

the role of pseudo data for robust smoothing with application to wavelet regression summary we propose a robust curve and surface estimator based on m type estimators and penalty based smoothing this approach also includes an application to wavelet regression the concept of pseudo data a transformation of the robust additive model to the one with bounded errors is used to derive some theoretical properties and also motivate a computational algorithm the resulting algorithm termed the es algorithm is computationally fast and provides a simple way of choosing the amount of smoothing moreover it is easily described straightforwardly implemented and can be extended to other wavelet regression settings such as irregularly spaced data and image denoising results from a simulation study and real data examples demonstrate the promising empirical properties of the proposed approach

diagonalization of non selfadjoint analytic semigroups and application to the shape memory alloys operator abstract to a densely defined but not necessarily selfadjoint operator a on a hilbert space h we consider on r h the following abstract elliptic problem of dirichlet type formula then in this paper we establish that for every t 0 the solution formula can be expanded into a series of generalized eigenvectors of the operator a provided that its resolvent belongs to carleman class c p for some p 0 1 2 a similar result holds for t large enough if the inverse a 1 belongs to carleman class c p for every p 1 2 see theorem 3 1 and theorem 3 2 furthermore we apply these obtained results to the shape memory alloys non selfadjoint operator formula and d n n x n when acting on an appropriate hilbert space e of functions on the interval 0 1 by establishing that the inverse c 1 belongs to the carleman class c p for every p 1 2 so that we get in this case more regularity in the sense that the operatorial series k 1 e tc p k converges strongly in e to the analytic semigroup e tc for every t 0 the p k are the projectors into the root subspaces of c a similar result holds for formula provided that t is large enough see theorem 4 1 and theorem 4 2 in section 4 for the precise result

robust nonparametric regression with simultaneous scale curve estimation on considere l estimation non parametrique simultanee par des lissages de type m l approche est etroitement liee a l estimation m simultanee de position et d echelle

a data dependent approach to density estimation a data dependent approach to density estimation is proposed here the proposed method requires boundedness and some weak integrability condition on the unknown density but not any assumption of smoothness applications to histogram kernel spline and orthogonal series methods are discussed

semiparametric inference in a partial linear model in a partial linear model the dependence of a response variate y on covariates w x is given by y w x e where e is independent of w x with densities g and f respectively in this paper an asymptotically efficient estimator of is constructed solely under mild smoothness assumptions on the unknown f and g thereby removing the assumption of finite residual variance on which all least squares type estimators available in the literature are based

how many variables should be entered in a regression equation abstract the optimal number of regressors is determined to minimize mean squared prediction error and is shown to be a small fraction of the number of data points as the number of regressors grows large the sp criterion provides an asymptotically optimal rule for the number of variables to enter

representation type of schur algebras we give a complete classification of the classical schur algebras and the infinitesimal schur algebras which have tame representation type in combination with earlier work of some of the authors on semisimplicity and finiteness this completes the classification of representation type of all classical and infinitesimal schur algebras in all characteristics

a robbins monro procedure for estimation in semiparametric regression models this paper is devoted to the parametric estimation of a shift together with the nonparametric estimation of a regression function in a semiparametric regression model we implement a robbins monro procedure very efficient and easy to handle on the one hand we propose a stochastic algorithm similar to that of robbins monro in order to estimate the shift parameter a preliminary evaluation of the regression function is not necessary for estimating the shift parameter on the other hand we make use of a recursive nadaraya watson estimator for the estimation of the regression function this kernel estimator takes in account the previous estimation of the shift parameter we establish the almost sure convergence for both robbins monro and nadaraya watson estimators the asymptotic normality of our estimates is also provided

densite des vecteurs propres generalises d une classe d operateurs compacts non auto adjoints et applications we consider a closed densely defined linear operatort in a hilbert spacee and assume the existence of 0 t such thatk t 0i 1 is compact and the existence ofp 0 such thats n k o n 1 p wheres n k denotes the sequence of non zero eigenvalues of the compact hermitian operator sqrt k k in this work sufficient conditions announced in 1 are introduced to assure that the closed subspace ofe spanned by the generalized eigenvectors oft coincides withe these conditions are in particular verified by a family of non self adjoint operators arising in reggeon s field theory

bandwidth choice for nonparametric classification it is shown that for kernel based classification with univariate distributions and two populations optimal bandwidth choice has a dichotomous character if the two densities cross at just one point where their curvatures have the same signs then minimum bayes risk is achieved using bandwidths which are an order of magnitude larger than those which minimize pointwise estimation error on the other hand if the curvature signs are different or if there are multiple crossing points then bandwidths of conventional size are generally appropriate the range of different modes of behavior is narrower in multivariate settings there the optimal size of bandwidth is generally the same as that which is appropriate for pointwise density estimation these properties motivate empirical rules for bandwidth choice

panel data discrete choice models with lagged dependent variables in this paper we consider identification and estimation in panel data discrete choice models when the explanatory variable set includes strictly exogenous variables lags of the endogenous dependent variable as well as unobservable individual specific effects for the binary logit model with the dependent variable lagged only once chamberlain 1993 gave conditions under which the model is not identified we present a stronger set of conditions under which the parameters of the model are identified the identification result suggests estimators of the model and we show that these are consistent and asymptotically normal although their rate of convergence is slower than the inverse of the square root of the sample size we also consider identification in the semiparametric case where the logit assumption is relaxed we propose an estimator in the spirit of the conditional maximum score estimator manski 1987 and we show that it is consistent in addition we discuss an extension of the identification result to multinomial discrete choice models and to the case where the dependent variable is lagged twice finally we present some monte carlo evidence on the small sample performance of the proposed estimators for the binary response model

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

using stopping rules to bound the mean integrated squared error in density estimation suppose x 1 x 2 ldots x n are i i d with unknown density f there is a well known expression for the asymptotic mean integrated squared error mise in estimating f by a kernel estimate hat f n under certain conditions on f the kernel and the bandwidth suppose that one would like to choose a sample size so that the mise is smaller than some preassigned positive number w based on the asymptotic expression for the mise one can identify an appropriate sample size to solve this problem however the appropriate sample size depends on a functional of the density that typically is unknown in this paper a stopping rule is proposed for the purpose of bounding the mise and this rule is shown to be asymptotically efficient in a certain sense as w approaches zero these results are obtained for data driven bandwidths that are asymptotically optimal as n goes to infinity

adaptive m estimation in nonparametric regression a method for robust nonparametric regression is discussed a method for robust nonparametric regression is discussed we consider kernel m estimates of the regression function using huber s function and extend results of hardle and gasser to the case of random designs a practical adaptive procedure is proposed consisting of simultaneously minimising a cross validatory criterion with respect to both the smoothing parameter and a robustness parameter occurring in the function this method is shown to possess a theoretical asymptotic optimality property while some simulated examples confirm that the approach is practicable

on the use of nonparametric regression techniques for fitting parametric regression models a new method for fitting parametric regression models is proposed it consists of applying the least squares ls principle on the pairs xi rh xi i 1 n where x denotes the explanatory variable and mn x is a location estimate of the conditional distribution of the response variable y given that x x consistency and asymptotic normality of the estimators are established under general conditions these conditions are shown to be satisfied when the data are incomplete due to random censoring or truncation usable expressions for the asymptotic variance covariance matrix of the parameters are provided in these incomplete data cases as an extra bonus this regression method allows the use of ordinary residual plots as a data analytic aide this is illustrated on two real data sets a simulation study examines the small sample behavior of the estimators and their estimated asymptotic variance extension of the method to more complicated models is discussed

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

sheaf cohomology on gb and tensor products of weyl modulus let k be an algebraically closed field it is assumed that char k p 0 unless otherwise indicated g is a simply connected semisimple algebraic group over k t is a fixed maximal torus and b a bore1 subgroup of g containing t let qi g t be the root system of g d the set of all simple roots and and the sets of all positive and negative roots relative to d respectively we assume that the set a is chosen such that b corresponds to 0 let x be the weight lattice of sp which may be considered as the character group of t or b therefore if 1 e x by abuse of notation we also denote by l the one dimensional tor b module determined by a let x be the set of all dominant weights and w the fundamental weight corresponding to a simple root a e a we denote by 6 the half sum of all positive roots and we know that 6 is the sum of all fundamental weights let w be the weyl group of 0 the reflection with respect to a root a is denoted by s the longest element of w by w there are two ways of w acting on x i the ordinary action if 1 e x m e w the image of i under w is denoted by 2 ii dot action w a 5 2 6 s in particular let a w a the dual root system of is denoted by and the dual root of a e by a i e u 2a a u where is the inner product invariant under the ordinary action of w of the euclidean space spanned by for 1 e x a e let a a 1 a all modules of an algebraic group we consider in this paper are assumed to be rational in particular for an infinite dimensional module the term rational module means this module is a direct limit of finite dimensional rational modules let h be a closed subgroup of g containing t such as g b or t m an h module we define for l e x

finitistic dimension of properly stratified algebras abstract we prove that the finitistic dimension of a properly stratified algebra having a simple preserving duality and for which every tilting module is cotilting equals twice the projective dimension of the characteristic tilting module as a corollary we get that the global dimension of a quasi hereditary algebra with duality equals twice the projective dimension of the characteristic tilting module as another corollary we obtain an affirmative answer to the conjecture of erdmann and parker finally we calculate the finitistic dimension of the blocks of certain parabolic generalizations of the category o

nonparametric semiparametric estimation and testing of econometric models with data dependent smoothing parameters we consider nonparametric semiparametric estimation and testing of econometric models with data dependent smoothing parameters most of the existing works on asymptotic distributions of a nonparametric semiparametric estimator or a test statistic are based on some deterministic smoothing parameters while in practice it is important to use data driven methods to select the smoothing parameters in this paper we give a simple sufficient condition that can be used to establish the first order asymptotic equivalence of a nonparametric estimator or a test statistic with stochastic smoothing parameters to those using deterministic smoothing parameters we also allow for general weakly dependent data

sur une valeur propre d un operateur the studied eigenvalue is the smallest one of the operator h mu mu a i lambda a a a a in the orthogonal complement of the vacuum wherea anda are the creation and annihilation operators call this eigenvaluee as attention is focused on the dependence on this eigenvalue exists and is a positive number for positive values of using the fact that the inverse ofh is a positive operator it is proved thate extends to a positive increasing analytic function on the whole real line in particular e 0 0 contrary to what might have been expected from the fact thata a a a is formally self adjoint

on the good filtration dimension of weyl modules for a linear algebraic group let g be a linear algebraic group over an algebraically closed field of characteristic p whose corresponding root system is irreducible in this paper we calculate the weyl filtration di mension of the induced g modules and the simple g modules l fora regular weight we use this to calculate some ext groups of the form ext extl l and ext where are regular and is the weyl module of highest weight we then deduce the projective dimensions and injective dimensions for l and for a regular weight in associated generalised schur algebras we also deduce the global dimension of the schur algebras for gln s n r when p n and for s mp p with m an integer

cross validation and the estimation of conditional probability densities many practical problems especially some connected with forecasting require nonparametric estimation of conditional densities from mixed data for example given an explanatory data vector x for a prospective customer with components that could include the customer s salary occupation age sex marital status and address a company might wish to estimate the density of the expenditure y that could be made by that person basing the inference on observations of x y for previous clients choosing appropriate smoothing parameters for this problem can be tricky not in the least because plug in rules take a particularly complex form in the case of mixed data an obvious difficulty is that there exists no general formula for the optimal smoothing parameters more insidiously and more seriously it can be difficult to determine which components of x are relevant to the problem of conditional inference for example if the jth component of x is independent of y then that component is irrelevant to es

quelques nouvelles proprietes de regularite de l operateur de gribov in this work we establish new regularity properties for gribov s operator h a a i a a a a 2 wherea anda are the creation and annihilation operators particularly we prove that for all e 0 h 1 is in the class of carleman s operatorl1 e

on schur algebras and related algebras ii this paper is a continuation of 7 the purpose of 7 was to set up in the context of reductive algebraic groups a theory of generalized schur algebras a special case of our construction gives the schur algebras s n r studied extensively in lo we concentrate here on the schur algebras themselves and show that our construction yields new results in their representation theory by means of the schur functor see 10 chap 6 we obtain new results in the representation theory of the symmetric groups the main results are as follows the schur algebras s n r defined over a principal ideal domain r have finite global dimension see also 1 i over a field or complete valuation ring r every finitely generated projective s n r module has a weyl module filtration we prove the irreducibility of the socle of certain weyl modules see also 15 we describe a connection between weights of projective indecomposable modules for the schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation we show that over a complete valuation ring or a field every young module for a symmetric group has a specht series in particular the projective indecomposables have specht series finally we show that for r n and a complete valuation ring or field r the blocks of the schur algebra s n r are given by the rule embodied in nakayama s conjecture the notations and conventions are as in the first part of this work 7

empirical functionals and efficient smoothing parameter selection a striking feature of curve estimation is that the smoothing parameter h 0 which minimizes the squared error of a kernel or smoothing spline estimator is very difficult to estimate this is manifest both in slow rates of convergence and in high variability of standard methods such as cross validation we quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of h 0 that attain the bounds the efficient estimators are substantially less variable than cross validation and other current procedures and simulations suggest that they may offer improvements at moderate sample sizes at least in terms of minimizing the squared error

nonparametric estimation of optimal performance criteria in quality engineering abstract box 1987 and leon et al 1987 discuss the problem of closeness to target in quality engineering if the mean response f x z depends on x z the variance function is a permia if it is g z i e depends only on z the goal is to find x sub o z sub o which minimizes variance while achieving a target mean value the authors pose and answer the question for given smoothness assumptions about f and g how accurately can we estimate x sub o and z sub o as part of the investigation they also find optimal rates of convergence for estimating f g and their derivatives keywords nonparametric regression performance measure quality control variance function estimation quality engineering

an asymptotically efficient solution to the bandwidth problem of kernel density estimation a data driven method of choosing the bandwidth h of a kernel density estimator is heuristically motivated by considering modifications of the kullback leibler or pseudo likelihood cross validation function it is seen that this means of choosing h is asymptotically equivalent to taking the h that minimizes some compelling error criteria such as the average squared error and the integrated squared error thus for a given kernel function the bandwidth can be chosen optimally without making precise smoothness assumptions on the underlying density

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approximation of the semigroup generated by the hamiltonian of reggeon field theory in bargmann space abstract the reggeon field theory is governed by a non self adjoint operator constructed as a polynomial in a a the standard bose annihilation and creation operators in zero transverse dimension this hamiltonian acting in bargmann space is defined by h a 2 a 2 a a i a a a a where i 2 1 and are real numbers and the operators a a satisfy the commutation relation a a i as the quantum mechanical system described by h has a velocity dependent potential containing powers of momentum up to the fourth the problem of existence of hamiltonian path integral for the evolution operator e t h of this theory is of interest on its own in particular can we express e t h as a limit of integral operators in this article one considerably reduces the difficulty by studying the trotter product formula of h to reach two objectives the first objective is to prove a very specific error estimate for the error in a trotter product formula in trace norm for h viewed as the sum of the operators a 2 a 2 and a a i a a a a the second objective of this work is to give a approximation of the semigroup generated by h when h is split in the sum of a 2 a 2 a a and i a a a a we note that this case is entirely different in fact the usual trotter product formula is not defined because the interaction operator a a a a is not the infinitesimal generator of a semigroup on bargmann space for 0 and 0 we choose an approximation operator i i a a a a e a 2 a 2 a a and we give a connection between and e h this choice allows us to give in a intissar note on the path integral formulation of reggeon field theory preprint a generalized trotter product formula for t a a i a a a a i e for limit case as 0 and answers to the above question

bandwidth selection for indirect density estimation based on corrupted histogram data abstract motivated by a number of practical applications we consider a class of indirect nonparametric density estimation problems in which the observed data consist of a histogram of binned empirically corrected counts due to variability in the process of correction the histogram cannot be modeled in terms of simple scaled poisson statistics this departure necessitates the development of a new methodology for bandwidth selection a variant of the method of unbiased risk estimation is proposed the methodology is studied using numerical simulations and asymptotic analysis tools in the context of a class of idealized density deconvolution problems the methodology is adapted for application to the practical reconstruction problem of positron emission tomography pet realistic numerical simulations and physical phantom data are presented to validate the approach in this setting some illustrations with cerebral glucose utilization and myocardial blood flow studies from some actual patient data sets

cross validation in nonparametric estimation of probabilities and probability densities summary we examine large sample properties of cross validation for estimating cell probabilities starting from a completely general measure of loss necessary and sufficient conditions on the loss function are derived for the resulting estimator to be consistent or to minimize expected loss these results reveal that cross validation is extremely sensitive to the shape of the loss function nevertheless when the loss function is chosen correctly cross validation can be relied on to perform well for large samples we provide a simple method of generating loss functions with optimal properties extension to the estimation of univariate probability density functions is discussed at a heuristic level

an optimal selection of regression variables summary an asymptotically optimal selection of regression variables is proposed the key assumption is that the number of control variables is infinite or increases with the sample size it is also shown that mallows s qp akaike s fpe and aic methods are all asymptotically equivalent to this method

graphics for assessing the adequacy of regression models abstract graphical paradigms for assessing the adequacy of nearly any regression model are discussed the fundamental idea is to examine the fit of a model using a sequence of marginal model plots on each plot nonparametric estimates of fit derived from the model are compared to nonparametric fits based on the observed data several examples are given

robust estimation for ordinary differential equation models summary applied scientists often like to use ordinary differential equations odes to model complex dynamic processes that arise in biology engineering medicine and many other areas it is interesting but challenging to estimate ode parameters from noisy data especially when the data have some outliers we propose a robust method to address this problem the dynamic process is represented with a nonparametric function which is a linear combination of basis functions the nonparametric function is estimated by a robust penalized smoothing method the penalty term is defined with the parametric ode model which controls the roughness of the nonparametric function and maintains the fidelity of the nonparametric function to the ode model the basis coefficients and ode parameters are estimated in two nested levels of optimization the coefficient estimates are treated as an implicit function of ode parameters which enables one to derive the analytic gradients for optimization using the implicit function theorem simulation studies show that the robust method gives satisfactory estimates for the ode parameters from noisy data with outliers the robust method is demonstrated by estimating a predator prey ode model from real ecological data

biased and unbiased cross validation in density estimation abstract nonparametric density estimation requires the specification of smoothing parameters the demands of statistical objectivity make it highly desirable to base the choice on properties of the data set in this article we introduce some biased cross validation criteria for selection of smoothing parameters for kernel and histogram density estimators closely related to one investigated in scott and factor 1981 these criteria are obtained by estimating l 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared l 2 error or mean integrated squared error these criteria are roughly the analog of wahba s 1981 generalized cross validation procedure for orthogonal series density estimators we present the relationship of the biased cross validation procedure to the least squares cross validation procedure which provides unbiased estimates of the mean integrated squared error both methods are shown to be based on u statistics we compare the two metho