toward a deterministic polynomial time algorithm with optimal additive query complexity in this paper we study two combinatorial search problems the coin weighing problem with a spring scale also known as the vector reconstructing problem using additive queries and the problem of reconstructing weighted graphs using additive queries suppose we are given n identical looking coins suppose that m out of the n coins are counterfeit and the rest are authentic assume that we are allowed to weigh subsets of coins with a spring scale it is known that the optimal number of weighings for identifying the counterfeit coins and their weights is at least w mlognlogm we give a deterministic polynomial time adaptive algorithm for identifying the counterfeit coins and their weights using o mlognlogm mloglogm weighings assuming that the weight of the counterfeit coins are greater than the weight of the authentic coins this algorithm is optimal when email protected n c l o g l o g n where c is any constant also our weighing complexity is within loglogm times the optimal complexity for all m to obtain this result our algorithm makes use of search matrices the divide and conquer approach and the guess and check approach when combining these methods with the technique introduced in h mazzawi 2008 20 we get a similar positive result for the problem of reconstructing a hidden weighted graph using additive queries

space group clebsch gordan coefficients i special solutions of the multiplicity problem and dirl s criterion the wavevector selection rules wvsr occurring in the reduction of kronecker products of space group unirreps are classified for convenience into three types for wvsr of type i dirl 1979 has shown that special solutions of the multiplicity problem always exist for wvsr of type ii dirl has given a simple criterion for the existence for special solutions of the multiplicity problem and the authors show that for all 230 single and double space groups the miller and love matrix unirreps satisfy this criterion wvsr of type iii will be considered in a subsequent paper

one sided cross validation abstract a new method of selecting the smoothing parameters of nonparametric regression estimators is introduced the method termed one sided cross validation oscv has the objectivity of cross validation and statistical properties comparable to those of a plug in rule the new method may be viewed as an application of the prequential model selection method of dawid as such our results identify a situation in which the prequential method is a more efficient model selector than cross validation an example simulations and theoretical results demonstrate the utility of oscv when used with local linear and kernel estimators

empirical process approach in a two sample location scale model with censored data to compare two samples of possibly right censored failure times a location scale model without assuming the distribution form is considered for the log transformed data this new accelerated failure time model is introduced to accommodate the possible heterogeneity between within treatment variations of the two groups considered one distinct feature of this model is that with the presence of heterogeneity statistical inferences on both location and scale parameters based on log rank tests or linear rank tests will become inappropriate in this paper we propose the empirical process approach to construct a regression setup derived from the theory of strong approximation of the kaplan meier product limit empirical quantile process a generalized least squares gls estimator is obtained and shown to be semiparametric efficient also it is shown that this estimation is adaptive for a special case at the end results on the two sample setting are applied to the k sample problem

spline smoothing the equivalent variable kernel method on considere une approche spline pour la regression non parametrique et l estimation de courbe on montre que dans un certain sens le lissage spline correspond approximativement au lissage par une methode du noyau avec une largeur de bande dependant de la densite locale des points du plan de regression

on the efficiency of selection criteria in spline regression this paper concerns the cubic smoothing spline approach to nonparametric regres sion after first deriving sharp asymptotic formulas for the eigenvalues of the smoothing ma trix the paper uses these formulas to investigate the efficiency of different selection criteria for choosing the smoothing parameter special attention is paid to the generalized maximum likelihood gml cp and extended exponential ee criteria and their marginal bayesian interpretation it is shown that a when the bayesian model that motivates gml is true using cp to estimate the smoothing parameter would result in a loss of efficiency with a factor of 10 3 proving and strengthening a conjecture proposed in stein 1990 b when the data indeed come from the cp density using gml would result in a loss of efficiency of c the loss of efficiency of the ee criterion is at most 1 543 when the data are sampled from its consistent density family the paper not only studies equally spaced observations the setting of stein 1990 but also investigates general sampling scheme of the design points and shows that the efficiency results remain the same in both cases

moment inequalities via optimal design theory abstract a unified approach is used to rediscover a class of moment inequalities in particular complementary inequalities in respect to ratios and differences of means and also inequalities complementary to both holder s and minkowski s inequalities each of which have been established in the literature by ad hoc methods are rediscovered the impetus of the paper is that this unified method is simple and illuminating a class of optimization problems is considered in which optimization is with respect to a probability distribution moment inequalities are implied by the solutions equality occuring under the optimizing distribution however we do not claim to prove any fundamentally new inequalities at this juncture the method used is illuminating in that it first discovers the support points of the optimizing distribution it then proves to be simple to discover the optimizing probabilities the method employs the differential calculus approach developed to establish optimality conditions for optimal linear regression designs however the link with optimal design theory is not just this tenous one results in respect of optimal design criteria of themselves have implications concerning moments including the multivariate generalization of the nonnegativity of a variance

bandwidth selection for kernel density estimation the problem of automatic bandwidth selection for a kernel density estimator is considered it is well recognized that the bandwidth estimate selected by the least squares cross validation is subject to large sample variation this difficulty limits the application of the cross validation estimate based on characteristic functions an important expression for the cross validation bandwidth estimate is obtained the expression clearly points out the source of variation to stabilize the variation a simple bandwidth selection procedure is proposed it is shown that the stabilized bandwidth selector gives a strongly consistent estimate of the optimal bandwidth under commonly used smoothness conditions the stabilized bandwidth estimate has a faster convergence rate than the convergence rate of the cross validation estimate for sufficiently smooth density functions it is shown that the stabilized bandwidth estimate is asymptotically normal with a relative convergence rate n 1 2 instead of the rate n 1 10 of the cross validation estimate a plug in estimate and an adjusted plug in estimate are also proposed and their asymptotic distributions are obtained it is noted that the plug in estimate is asymptotically efficient the adjusted plug in bandwidth estimate and the stabilized bandwidth estimate are shown to be asymptotically equivalent the simulation results verify that the proposed procedures perform much better than the cross validation for finite samples

a cross validatory choice of smoothing parameter in adaptive location estimation abstract this article proposes a new data driven method for selecting the smoothing parameter involved in constructing kernel based adaptive location estimators the method consists of minimizing a cross validatory criterion with respect to the bandwidth occurring in the kernel type estimators of the efficient score function it is shown that the location estimator with a data driven bandwidth selector is indeed an adaptive estimator a simulation study reveals that the method is also practicable showing that our estimator performs well in comparison with some other well known location estimators it also shows that our method has comparable finite sample performance with the bootstrap method of selecting the smoothing parameter and yet has great computational advantages

on optimal data based bandwidth selection in kernel density estimation a bandwidth selection method is proposed for kernel density estimation this is based on the straightforward idea of plugging estimates into the usual asymptotic representation for the optimal bandwidth but with two important modifications the result is a bandwidth selector with the by nonparametric standards extremely fast asymptotic rate of convergence of n where n denotes sample size comparison is given to other bandwidth selection methods and small sample impact is investigated

do validation for kernel density estimation bandwidth selection in kernel density estimation is one of the fundamental model selection problems of mathematical statistics the study of this problem took major steps forward with the articles of hall and marron 1987 and hall and johnstone 1992 since then the focus seems to have been on various versions of implementing the so called plug in method aimed at estimating the minimum mean integrated squared error mise the most successful of these efforts still seems to be the plug in method of sheather and jones 1991 or park and marron 1990 that we also use as a benchmark in this article in this article we derive a new theorem deriving the asymptotic theory for linear combinations of bandwidths obtained from different selectors as for example direct and indirect cross validation and plug in where we take advantage of recent advances in the study of indirect cross validation see hart and yi 1998 hart and lee 2005 and savchuk hart and sheather 2008 2010 we conclude that the slow

bayes and likelihood calculations from confidence intervals recently there has been considerable progress on setting good approximate confidence intervals for a single parameter in a multi parameter family here we use these frequentist results as a convenient device for making bayes empirical bayes and likelihood inferences about a simple formula is given that produces an approximate likelihood function l x for with all nuisance parameters eliminated based on any system of approximate confidence intervals the statistician can then modify l x with bayes or empirical bayes information for without worrying about nuisance parameters

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

generating relations for reducing matrices i ordinary representations auxiliary groups are constructed that make it possible to reduce the multiplicity problem and to derive consistent generating relations for the elements of reducing matrices three examples are worked out to illustrate the general scheme

optimal inference for instrumental variables regression with non gaussian errors abstract this paper is concerned with inference on the coefficient on the endogenous regressor in a linear instrumental variables model with a single endogenous regressor nonrandom exogenous regressors and instruments and i i d errors whose distribution is unknown it is shown that under mild smoothness conditions on the error distribution it is possible to develop tests which are nearly efficient in the sense of andrews et al 2006 when identification is weak and consistent and asymptotically optimal when identification is strong in addition an estimator is presented which can be used in the usual way to construct valid indeed optimal confidence intervals when identification is strong the estimator is of the two stage least squares variety and is asymptotically efficient under strong identification whether or not the errors are normal

optimal query complexity bounds for finding graphs we consider the problem of finding an unknown graph by using queries with an additive property this problem was partially motivated by dna shotgun sequencing and linkage discovery problems of artificial intelligence given a graph an additive query asks the number of edges in a set of vertices while a cross additive query asks the number of edges crossing between two disjoint sets of vertices the queries ask the sum of weights for weighted graphs for a graph g with n vertices and at most m edges we prove that there exists an algorithm to find the edges of g using o mlogn 2mlog m 1 queries of both types for all m the bound is best possible up to a constant factor for a weighted graph with a mild condition on weights it is shown that o mlognlogm queries are enough provided m logn a for a sufficiently large constant a which is best possible up to a constant factor if m 0 this settles in particular a conjecture of grebinski v grebinski on the power of additive combinatorial search model in proceedings of the 4th annual international conference on computing and combinatorics cocoon 1998 taipei taiwan 1998 pp 194 203 for finding an unweighted graph using additive queries we also consider the problem of finding the fourier coefficients of a certain class of pseudo boolean functions as well as a similar coin weighing problem

learning a hidden matching we consider the problem of learning a matching i e a graph in which all vertices have degree 0 or 1 in a model where the only allowed operation is to query whether a set of vertices induces an edge this is motivated by a problem that arises in molecular biology in the deterministic nonadaptive setting we prove a frac 1 2 o 1 n choose 2 upper bound and a nearly matching 0 32 n choose 2 lower bound for the minimum possible number of queries in contrast if we allow randomness then we obtain by a randomized nonadaptive algorithm a much lower o n log n upper bound which is best possible even for randomized fully adaptive algorithms

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

smoothing in adaptive estimation an adaptive maximum likelihood estimator based on the estimation of the log density by b splines is introduced a data driven method of selecting the smoothing paramieter involved in the consequent density estimation is demonstrated a monte carlo study is conducted to evaluate the small sample performance of the estimator in a location and a regression problem the adaptive estimator is seen to compare favorably to some standard estimates we show that the estimator is asymptotically efficient 1 introduction the problem of adaptive estimation was introduced by stein 1956 one wishes to estimate a euclidean parameter 0 in the presence of an infinite dimensional shape parameter g usually the density an adaptive estimate performs asymptotically as well in the sense that the limiting distributions are the same with g unknown as any estimate which utilizes knowledge of g note that the term adaptive estimation has been used elsewhere in the literature in the lesser sense of adapting to the data in some way the estimates considered here are adaptive in a much stronger sense an adaptive estimator of the center of symmetry of an unknown distribution was constructed by stone 1975 bickel 1982 dealt with the multiple regression problem and simplified stein s conditions for the circumstances under which adaptive estimation is possible however all the aforementioned work pertains to large sample behavior problems arise when one tries to apply these procedures in a practical small sample situation the adaptive estimates proposed all depend on nonparametric density estimation and specifically the use of kernel density estimation essentially one replaces the true density used in a one step maximum likelihood estimate by an estimate of that density a problem which pervades nonparametric density estimation is the choice of the smoothing parameters much work has been done on this subject and various schemes for the optimal choice of smoothing parameters have been proposed mostly for the mean integrated square error criteria unfortunately this is of little help in selecting the optimal smoothing parameters for the estimation of the parameters in the location or regression problem since these methods may give a good estimate of the density but that is secondary to the problem at hand we need to estimate the score not the density and our criterion is to minimize the mse say of the estimate of 0 in fact experimentation reveals that the optimal choice of smoothing parameters as regards the estimation of the location parameter tends to produce an under

algorithm to calculate a large number of roots of the cross product of bessel functions this paper describes an algorithm to calculate a large number of roots of the cross product of bessel functions and of their first derivatives the algorithm initially finds the roots of the zeroth order using an auxiliary function that exhibits the same roots as the original cross products but with better behavior for numerical root search with the newton raphson algorithm in order to find the roots for higher orders the algorithm follows a pyramidal scheme using the interlacing property of the cross product of bessel functions the algorithm shows globally convergent behavior for a large range of values of the argument and of the order of the bessel functions the roots can be computed to any precision limited only by the computer implementation and the convergence is attained in six iterations per root in average showing a much better performance than previous works for the calculation of these roots

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

learning a hidden hypergraph we consider the problem of learning a hypergraph using edge detecting queries in this model the learner may query whether a set of vertices induces an edge of the hidden hypergraph or not we show that an r uniform hypergraph with m edges and n vertices is learnable with o 24rm poly r logn queries with high probability the queries can be made in o min 2r log m r 2 log m r 3 rounds we also give an algorithm that learns an almost uniform hypergraph of dimension r using o 2o 1 2 r m1 2 poly log n queries with high probability where is the difference between the maximum and the minimum edge sizes this upper bound matches our lower bound of m 1 2 1 2 for this class of hypergraphs in terms of dependence on m the queries can also be made in o 1 min 2r log m r 2 log m r 3 rounds

a study of methods of choosing the smoothing parameter in image restoration by regularization the method of regularization is portrayed as providing a compromise between fidelity to the data and smoothness with the tradeoff being determined by a scalar smoothing parameter various ways of choosing this parameter are discussed in the case of quadratic regularization criteria they are compared algebraically and their statistical properties are comparatively assessed from the results of all extensive simulation study based on simple images

similarity transformations of irreducible corepresentations in wigner canonical form the general form of the matrices that transform an irreducible corepresentation coirrep into an equivalent one where both representations are assumed to be in the form proposed by wigner is analyzed for the three types of coirreps in addition the relation of these transformations to inner automorphisms of the corresponding group algebras is clarified

generating relations for reducing matrices ii corepresentations the auxiliary group approach developed in paper 1 r dirl p kasperkovitz m i aroyo j n kotzev and m angelova tjurkedjieva j math phys 27 37 1986 is generalized for the case of corepresentations of antiunitary groups it allows us to reduce the multiplicity problem and to derive consistent generating relations for the elements of the reducing matrices for coreps two examples are worked out to illustrate the general scheme

nonparametric estimation of global functionals and a measure of the explanatory power of covariates in regression in a nonparametric regression setting with multiple random predictor variables we give the asymptotic distributions of estimators of global integral functionals of the regression surface we apply the results to the problem of obtaining reliable estimators for the nonparametric coefficient of determination this coefficient which is also called pearson s correlation ratio gives the fraction of the total variability of a response that can be explained by a given set of covariates it can be used to construct measures of nonlinearity of regression and relative importance of subsets of regressors and to assess the validity of other model restrictions in addition to providing asymptotic results we propose several data based bandwidth selection rules and carry out a monte carlo simulation study of finite sample properties of these rules and associated estimators of explanatory power we also provide two real data examples

economical graph discovery we consider the problem of graph discovery in settings where the graph topology is known and the edge weights are hidden the setting consists of a weighted graph g with n vertices and m edges and with designated source s and destination t we study two different discovery problems namely i edge weight discovery where the goal is to discover all edge weights and ii shortest path discovery where the goal is to discover a shortest s t path this discovery is done by means of agents that traverse different s t paths in multiple rounds and report back the total cost they incurred three cost models are considered differing from each other in their approach to congestion effects in particular we consider congestion free models as well as models with positive and negative congestion effects we seek bounds on the number of rounds and the number of agents required to complete the edge weights or the shortest path discovery several results concerning such bounds for both directed and undirected graphs are established among these results we show that 1 for undirected graphs all edge weights can be discovered within a single round consisting of m agents 2 discovering a shortest path in either undirected or directed acyclic graphs requires at least m n 1 agents and 3 the edge weights in a directed acyclic graph can be discovered in m rounds with m n 2 agents under congestion aware cost models our study introduces a new setting of graph discovery under uncertainty and provides fundamental understanding of the problem

non complex representations and their relation to antiunitary symmetry equivalent eigenvalue problems in real complex and quaternionic hilbert spaces are discussed in all cases a compact symmetry group is assumed to exist in the complex case also an antiunitary operator commuting with both the self adjoint operator and the elements of the symmetry group and satisfying 2 1 it is shown that the extra degeneracies caused by the existence of can also be obtained by consideration of an equivalent eigenvalue problem in a noncomplex hilbert space

empirical smoothing parameter selection in adaptive estimation we provide a solution to the smoothing parameter selection problem involved in the construction of adaptive estimates for the symmetric location model and the general linear model linear b splines are used to give a simple form of the estimate of the score function of the underlying density new empirical methods are proposed to locate the knots optimally and to select the number of knots we also give asymptotic bounds for the empirical selection method and show that an estimate with an empirically selected smoothing parameter is adaptive our estimates are easy to compute and possess useful computational features simulation studies reveal that our estimates perform well in comparison with some well known estimates

generating relations for reducing matrices iii kronecker products in the first two of this series of papers r dirl et al j math phys 27 37 1986 m i aroyo et al ibid 27 2236 1986 a systematic method to calculate the elements of reducing matrices has been developed in this paper this auxiliary group approach is adapted to multiple kronecker products three examples are worked out to illustrate the efficiency of generating relations and the proposed reduction of the multiplicity problem

efficient independent component analysis independent component analysis ica has been widely used for blind source separation in many fields such as brain imaging analysis signal processing and telecommunication many statistical techniques based on m estimates have been proposed for estimating the mixing matrix recently several nonparametric methods have been developed but in depth analysis of asymptotic efficiency has not been available we analyze ica using semiparametric theories and propose a straightforward estimate based on the efficient score function by using b spline approximations the estimate is asymptotically efficient under moderate conditions and exhibits better performance than standard ica methods in a variety of simulations

a flexible and fast method for automatic smoothing abstract the choice of smoothing parameter or bandwidth is crucial when applying nonparametric regression estimators such as kernel estimators the optimal choice depends on the data at hand a data driven bandwidth selection close to the optimal one would make these curve estimators objective more reliable and easier to use minimizing residual mean squared error criteria such as cross validation have been frequently proposed to estimate the optimal smoothing parameter empirical and theoretical evidence indicates that cross validation rules and related methods lead to rather variable estimated optimal bandwidths the method presented here builds on estimating the asymptotically optimal bandwidth from the data since estimators for the residual variance and for an asymptotic expression for the bias are plugged into the asymptotic formula such selection rules are called plug in estimators the functional that quantifies bias is approximated by the integrated squared second derivative of the regre

type adapted subduction matrices if an irreducible representation is restricted to a subgroup it becomes reducible in general the matrices transforming this reducible representation into a direct sum of irreducible constituents are called subduction matrices their structure is discussed for real complex and quaternionic representations where all these representations are assumed to show a peculiar structure characteristic for the type of this representation character test 0 the choice of these type adapted representations a convention possible for all compact groups considerably reduces the number of parameters needed to fix a subduction matrix

on the existence of real clebsch gordan coefficients the question of the possible general existence of real clebsch gordan coefficients coupling coefficients for compact groups is considered criteria are established for a group to be susceptible to the classical rotation group approach in which a choice of standard irreducible matrix representations is made such that there is a fixed inner automorphism of the group carrying all standard representations into their complex conjugate in connection with a generalization of this approach the concept of quasi ambivalence of a group is shown to be relevant

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

bootstrap choice of the smoothing parameter in kernel density estimation summary cross validation based on integrated squared error has already been applied to the choice of smoothing parameter in the kernel method of density estimation in this paper an alternative resampling plan based on the bootstrap is proposed in an attempt to estimate mean integrated squared error this leads to a further data based choice of smoothing parameter the two methods are compared and some simulations and examples demonstrate the relative merits for large samples the bootstrap performs better than cross validation for many distributions

comparison of data driven bandwidth selectors abstract this article compares several promising data driven methods for selecting the bandwidth of a kernel density estimator the methods compared are least squares cross validation biased cross validation and a plug in rule the comparison is done by asymptotic rate of convergence to the optimum and a simulation study it is seen that the plug in bandwidth is usually most efficient when the underlying density is sufficiently smooth but is less robust when there is not enough smoothness present we believe the plug in rule is the best of those currently available but there is still room for improvement

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

extent to which least squares cross validation minimises integrated square error in nonparametric density estimation let ho o and c be the windows which minimise mean integrated square error integrated square error and the least squares cross validatory criterion respectively for kernel density estimates it is argued that o not ho should be the benchmark for comparing different data driven approaches to the determination of window size asymptotic properties of ho o and c o and of differences between integrated square errors evaluated at these windows are derived it is shown that in comparison to the benchmark o the observable window c performs as well as the so called optimal but unattainable window ho to both first and second order

monte carlo evidence on adaptive maximum likelihood estimation of a regression on examine la performance des estimateurs du maximum de vraisemblance adaptatifs lorsque ces parametres sont preselectionnes ou alternativement determines par une methode du bootstrap basee sur les donnees

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

permutational properties of the generalized clebsch gordan coefficients the properties of the permutational symmetry of the 3 symbols for an arbitrary compact group have been generalized to a multiple kronecker product of irreducible representations it has been shown that these properties of the corresponding n symbols under a permutation of their columns are related to a choice of the coupling schemes for the appropriate polyads 1 n the simplest permutational symmetry i e associated with a young diagram for the symmetric group n is in general incompatible with schemes having definite intermediate representations and in the case of mixed symmetry does not preserve the absolute value of a n symbol the case of n 4 for su 2 has been studied in detail

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

coupling subduction and induction transformations for group representations the basis of a representation space of any group may be chosen according to its decomposition into representations of the group and a subgroup chain coupling recoupling subduction and resubduction factors are each shown to arise as a result of schur s lemmas and a particular group subgroup structure when induction and mackey s subgroup theorem are considered new transformation factors occur those of induction and reinduction key properties of these factors are given together with their relationship with double coset matrix elements

the amount of noise inherent in bandwidth selection for a kernel density estimator abstract any practical method of constructing a bandwidth must depend only on a statistical sample and should produce some sort of estimate of this bandwidth the purpose of this paper is to show that there a well defined limits to the accuracy of all data driven bandwidth estimates put another way there is an unbridgeable gap between the minimum integrated square error attained using a optimal bandwidth and the minimum achievable integrated square error using a data driven bandwidth estimate additional keywords stochastic processes cross validation and random variables

learning boolean halfspaces with small weights from membership queries abstract we consider the problem of proper learning of a boolean halfspace with integer weights 0 1 t from membership queries only the best known algorithm for this problem is an adaptive algorithm that asks n o t 5 membership queries while the best lower bound for the number of membership queries is n t 1 in this paper we close this gap and give an adaptive proper learning algorithm with two rounds and asking n o t membership queries we also give a non adaptive proper learning algorithm that asks n o t 3 membership queries

the character table for the corepresentations of magnetic groups a square character table is shown to exist for all finite magnetic groups the table possesses row and column orthogonality properties similar to the character table for linear groups

simultaneity and reality of u n and su n 3jm and 6j symbols a simple proof is given of the simultaneous existence of a real set of 3jm and 6j symbols for the unitary groups u n and su n including the case of mixed tensor representations we observe that simultaneity is incompatible with the conventional permutation symmetries for u n with n 3 the relevance of these results to the schur weyl duality of u n with the symmetric groups sl is discussed

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

why bandwidth selectors tend to choose smaller bandwidths and a remedy summary this note is concerned with the problem of automatic data driven bandwidth selectors for nonparametric regression though some selectors were shown to be consistent and asymptotically unbiased by rice 1984 and hardle hall marron 1988 it is often observed in simulation studies that most selectors are biased toward undersmoothing an explanation for this is given the source of the large sample variation in the bandwidth estimates is also pointed out this leads to the consideration of a new procedure which is a simple modification of a classical selector a simulation study suggests that the proposed selector is much more consistent than the classical one

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

comparison of two bandwidth selectors with dependent errors for nonparametric regression in the case of dependent observations cross validation is known to be severely affected by dependence this effect is precisely quantified through a limiting distribution for the cross validated bandwidth the performance of two methods the leave 2e 1 out version of cross validation and partitioned cross validation which adjust for the dependence effect on bandwidth selection is investigated the bandwidths produced by these two methods are analyzed by further limiting distributions which reveal significantly different characteristics simulations demonstrate that the asymptotic effects hold for reasonable sample sizes

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

clebsch gordan coefficients for the corepresentations of shubnikov point groups iv groups of hexagonal and trigonal systems for pt iii see ibid vol 15 no 3 p 711 24 1982 the clebsch gordan coefficients for single and double valued corepresentations for even and odd basis functions of 35 anti unitary shubnikov magnetic point groups of hexagonal and trigonal systems have been calculated basis functions multiplication tables and compatibility tables are also presented a method based on the racah lemma has been applied the coefficients are necessary in solving problems of crystal field theory and solid state spectroscopy selection rules wigner eckart theorem etc

representation theory of compact groups over fields of characteristic zero the group algebras over the reals the complex numbers and the quaternions are discussed and compared for a fixed compact group special emphasis is put on the properties of ring bases of the minimal two sided ideals and their relation to the matrix representation of the group irreducible over the fields cited above

the auxiliary group approach to the reduction of kronecker products bijective mappings of matrix representations are used to derive generating relations for clebsch gordon coefficients and to reduce the multiplicity problem

asymptotic minimax estimation in semiparametric models we give several conditions on the estimator of efficient score function for estimating the parametric component of semiparametric models a semiparametric version of the one step mle using an estimator of efficient score function which fulfills the conditions is shown to converge to the normal distribution with minimum variance locally uniformly over a fairly large neighborhood around the assumed semiparametric model consequently it is shown to be asymptotically minimax with bounded subconvex loss functions a few examples are also considered

additional symmetry of clebsch gordan coefficients for corepresentations the concept of associated representations is generalised for the case of corepresentations symmetry relations for the clebsch gordan coefficients based on the association of corepresentations are derived this additional symmetry not only reduces the volume of the calculations and the tables of coupling coefficients but it also improves the correlation between the coefficients a simple method for obtaining the clebsch gordan coefficients for odd under space inversion basis functions is proposed

adaptive detection of known signals in additive noise by means of kernel density estimators we consider the problem of detecting known signals contaminated by additive noise with a completely unknown probability density function f to this end we propose a new adaptive detection rule it is defined by plugging a kernel density estimator f spl circ of f into the maximum a posteriori map detector the estimate f spl circ can either be computed off line from a training sequence or on line simultaneously with the detection for the off line detector we prove that the asymptotic error probability for weak signals converges to the minimal error probability of the map detector as the number of training data tends to infinity and we also establish rates of convergence and the optimal choice of bandwidth order for a certain class of noise densities in a monte carlo study the off line plug in map detectors are compared with the l sup 1 and l sup 2 detectors for various noise distributions when the training sequence is long enough the plug in detectors have excellent performance for a wide range of distributions whereas the l sup 2 detector breaks down for heavy tailed distributions and the l sup 1 detector for distributions with little mass around the origin

reconstructing weighted graphs with minimal query complexity in this paper we consider the problem of reconstructing a hidden weighted graph using additive queries we prove the following let g be a weighted hidden graph with n vertices and m edges such that the weights on the edges are bounded between n a and n b for any positive constants a and b for any m there exists a non adaptive algorithm that finds the edges of the graph using o mlognlogm additive queries this solves the open problem in s choi j h kim optimal query complexity bounds for finding graphs in stoc 2008 pp 749 758 choi and kim s proof holds for m logn a for a sufficiently large constant a and uses the graph theory we use the algebraic approach for the problem our proof is simple and holds for any m

lower bounds for bandwidth selection in density estimation this paper establishes asymptotic lower bounds which specify in a variety of contexts how well in terms of relative rate of convergence one may select the bandwidth of a kernel density estimator these results provide important new insights concerning how the bandwidth selection problem should be considered in particular it is shown that if the error criterion is integrated squared error ise then even under very strong assumptions on the underlying density relative error of the selected bandwidth cannot be reduced below ordern 1 10 as the sample size grows this very large error indicates that any technique which aims specifically to minimize ise will be subject to serious practical difficulties arising from sampling fluctuations cross validation exhibits this very slow convergence rate and does suffer from unacceptably large sampling variation on the other hand if the error criterion is mean integrated squared error mise then relative error of bandwidth selection can be reduced to ordern 1 2 when enough smoothness is assumed therefore bandwidth selection techniques which aim to minimize mise can be much more stable and less sensitive to small sampling fluctuations than those which try to minimize ise we feel this indicates that performance in minimizing mise rather than ise should become the benchmark for measuring performance of bandwidth selection methods

generating relations for reducing matrices iv subduced representations a subduced representation da gb is a co representation of the group gb obtained by restricting the co representation da of a supergroup ga to the elements of gb a systematic and efficient method for calculating the matrices needed for the decomposition of da gb into irreducible constituents is discussed in this paper where the auxiliary group approach developed previously j math phys 27 37 2236 1986 28 1947 1987 is adopted to the present problem

a fast monte carlo cross validation procedure for large least squares problems with noisy data we propose a fast monte carlo algorithm for calculating reliable estimates of the trace of the influence matrixa involved in regularization of linear equations or data smoothing problems where is the regularization or smoothing parameter this general algorithm is simply as follows i generaten pseudo random valuesw 1 w n from the standard normal distribution wheren is the number of data points and letw w 1 w n t ii compute the residual vectorw a w iii take the normalized inner product w t w a w w t w as an approximation to 1 n tr i a we show both by theoretical bounds and by numerical simulations on some typical problems that the expected relative precision of these estimates is very good whenn is large enough and that they can be used in practice for the minimization with respect to of the well known generalized cross validation gcv function this permits the use of the gcv method for choosing in any particular large scale application with only a similar amount of work as the standard residual method numerical applications of this procedure to optimal spline smoothing in one or two dimensions show its efficiency

smoothers and the cp generalized maximum likelihood and extended exponential criteria nonparametric regression often called smoothing is a widely used data analysis method the use of a smoother requires the choice of a smoothing parameter that by balancing fidelity and roughness controls how much smoothing is done two popular selection criteria for choosing the smoothing parameter are cp and generalized maximum likelihood gml each of these has its own problems forcp the problem is its high variability whereas for gml the problem is its potentially large bias by studying the geometry of selection criteria we give an intuitive explanation of the strength and weakness of cp and gml the geometry then motivates a new selection method the extended exponential ee criterion which combines the strength of cp and gml but mitigates their weaknesses in terms of variability bias and undersmoothing

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