the hirsch conjecture for dual transportation polyhedra an algorithm is given that joins any pair of extreme points of a dual transportation polyhedron by a path of at most m 1 n 1 extreme edges

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

a relaxation column signature method for assignment problems abstract a recent computational study 7 on algorithms for assignment problems revealed that the so called modified hung rom algorithm mhra performs very well in practice the mhra is a row signature method and its effectiveness comes mainly from its capability to reduce the level by more than one unit per cycle we present an o n 3 column signature method which has the above mentioned capability the algorithm solves a sequence of relaxed transportation problems and generates strong dual feasible bases it is shown that the average number of iterations is bounded by n log n a result on balinski s strong dual feasible bases is also presented

steinberg symbols modulo the trace class holonomy and limit theorems for toeplitz determinants suppose that z where z and lip 1 2 1 and the toeplitz operator t is invertible let d n t be the determinant of the toeplitz matrix i j i j 0 i j n where k 1 2 2 0 e ik d let pn be the orthogonal projection onto ker s n 1 1 e i e 2i e in where s t z set qn 1 p n let h denote the hankel operator associated to and set w t 1 t for t t for the wiener hopf factorization fg where f g and 1 f 1 g lip h t 1 2 1 put e exp k 1 k log f k log g k g exp log 0 theorem a d n t 1 n 1 g n 1 e g g f det t f g z n 1 1 h g f q n h f g 1 z 1 z 1 1 o n 1 2 let h 2 t y be a decomposition into t t 1 invariant subspaces x n 1 ran t t 1 n and y n 1 ker t 1 n so that restricted to x is invertible y is finite dimensional and t t 1 restricted to y is nilpotent let w 1 be the basis t f z 1 0 for the null space of t t 1 and let u be the top vector in a jordan root vector chain of length m 1 lying over 1 m w i e t t 1 m u 1 m w where m max m z x so that t t 1 m x w 1 theorem b e g g f t t 1 0 det u t 1 z 1 g f g f z t the holonomy of a deligne bundle with connection defined by the factorization fgz note that the generalizations of the szego limit theorem for d n t which have appeared in the literature with 1 instead of 1 h g f q n h f g 1 have the defect that the limit of d n t 1 n 1 g n 1 det t f g z n 1 z 1 z 1 does not exist in general an example is given with d n t 0 yet d 1 t f g z n 1 0 for infinitely many n

efficiency for regularization parameter selection in penalized likelihood estimation of misspecified models it has been shown that akaike information criterion aic type criteria are asymptotically efficient selectors of the tuning parameter in nonconcave penalized regression methods under the assumption that the population variance is known or that a consistent estimator is available we relax this assumption to prove that aic itself is asymptotically efficient and we study its performance in finite samples in classical regression it is known that aic tends to select overly complex models when the dimension of the maximum candidate model is large relative to the sample size simulation studies suggest that aic suffers from the same shortcomings when used in penalized regression we therefore propose the use of the classical corrected aic aic c as an alternative and prove that it maintains the desired asymptotic properties to broaden our results we further prove the efficiency of aic for penalized likelihood methods in the context of generalized linear models with no dispersion parameter similar result

nonparametric prediction in measurement error models predicting the value of a variable y corresponding to a future value of an explanatory variable x based on a sample of previously observed independent data pairs x1 y1 xn yn distributed like x y is very important in statistics in the error free case where x is observed accurately this problem is strongly related to that of standard regression estimation since prediction of y can be achieved via estimation of the regression curve e y x when the observed xis and the future observation of x are measured with error prediction is of a quite different nature here if t denotes the future contaminated available version of x prediction of y can be achieved via estimation of e y t in practice estimating e y t can be quite challenging as data may be collected under different conditions making the measurement errors on xi and x nonidentically distributed we take up this problem in the nonparametric setting and introduce estimators which allow a highly adaptive approach to smoothing

a strongly polynomial algorithm for the transportation problem for the linear transportation problem withm supply nodes n demand nodes andk feasible arcs we describe an algorithm which runs in time proportional tom logm k n logn assuming w l o g m n the algorithm uses excess scaling the complexity bound is a slight improvement over the bound achieved by an application of a min cost flow algorithm of orlin to the transportation problem

trace formulas for a class of truncated block toeplitz operators abstract the strong szeg limit theorem may be formulated in terms of finite dimensional operators of the form p n g p n n p n g n p n for n 1 2 where g denotes the operator of multiplication by a suitably restricted d d mvf matrix valued function acting on the space of d 1 vvf s vector valued functions f that meet the constraint 0 2 f e i e i f e i d where e i i d and p n denotes the orthogonal projection onto the space of trigonometric vector polynomials of degree at most n that are subject to the same summability constraint in this paper we study these operators for a class of mvf s which admit factorizations e i q e i q e i r e i r e i where q 1 r 1 belong to the wiener plus algebra of d d mvf s on the unit circle we show that n g def lim n trace p n g p n n p n g n p n exists and is independent of when the commutativity conditions g q q g and r g r g are in force the space of trigonometric vector polynomials of degree at most n is identified as a de branges reproducing kernel hilbert space of vector polynomials of degree at most n and weighted analogs of the strong szeg limit theorem are established if q 1 and r 1 are matrix polynomials then the inverse of the block toeplitz matrix corresponding to is of the band type explicit formulas for trace p n g n p n are obtained in this case

toeplitz matrices with an exponential growth of entries and the first szego limit theorem the toeplitz or block toeplitz matrices s r sj k rk j 1 generated by the taylor coefficients at zero of analytic functions s02 p 1 s p p and s02 p 1 sp p are considered a method is proposed for removing the poles of and or in other words for replacing s whose entries grow exponentially by a matrix j k k j 1 with better behaviour and the same asymptotics of r det r r as the sequence r det s r a szego type limit formula for the case when s r s r r n0 have a fixed number of negative eigenvalues is obtained

model averaging based on leave subject out cross validation for vector autoregressions abstract the vector autoregressive var model is a useful tool for economic evaluation and prediction this paper develops a leave subject out cross validation model averaging lsoma method to average predictions from var models the approximate unbiasedness of lsoma and its asymptotic optimality in terms of obtaining the lowest possible quadratic errors are established the rate of the lsoma based weights converging to the optimal weights minimizing the expected quadratic errors is also derived simulation experiments show that our method is generally more efficient than the other frequently used model selection and averaging methods two empirical applications further illustrate that the proposed method is promising

regularization parameter selections via generalized information criterion we apply the nonconcave penalized likelihood approach to obtain variable selections as well as shrinkage estimators this approach relies heavily on the choice of regularization parameter which controls the model complexity in this paper we propose employing the generalized information criterion encompassing the commonly used akaike information criterion aic and bayesian information criterion bic for selecting the regularization parameter our proposal makes a connection between the classical variable selection criteria and the regularization parameter selections for the nonconcave penalized likelihood approaches we show that the bic type selector enables identification of the true model consistently and the resulting estimator possesses the oracle property in the terminology of fan and li 2001 in contrast however the aic type selector tends to overfit with positive probability we further show that the aic type selector is asymptotically loss efficient while the bic type selector is not

automatic context sensitive karyotyping of human chromosomes based on elliptically symmetric statistical distributions we introduce a statistical model of a metaphase cell consisting of independent chromosomes with elliptically symmetric feature vectors from this model we derive the ml classifier for classification in the 24 chromosomal classes taking into account the correct number of chromosomes in each class experimental results show that error rates of the best of these classifiers are less than 2 with respect to chromosomes if applied to the large copenhagen data set cpr simulation studies suggest that there should be even more information contained in the features of this data set

szego limits for infinite toeplitz matrices determined by the taylor series of two rational functions explicit expressions for the well known szego limits are obtained for the cases when the upper and lower triangular parts of the toeplitz matrix are determined separately by the taylor series of rational matrix functions and the exponential growth of the entries is permissible

the estimation of prediction error having constructed a data based estimation rule perhaps a logistic regression or a classification tree the statistician would like to know its performance as a predictor of future cases there are two main theories concerning prediction error 1 penalty methods such as cp akaike s information criterion and stein s unbiased risk estimate that depend on the covariance between data points and their corresponding predictions and 2 cross validation and related nonparametric bootstrap techniques this article concerns the connection between the two theories a rao blackwell type of relation is derived in which nonparametric methods such as cross validation are seen to be randomized versions of their covariance penalty counterparts the model based penalty methods offer substantially better accuracy assuming that the model is believable

inverse spectral problem for differential operators with rational scattering matrix functions in this paper we study the inverse scattering problem for linear canonical differential equations of a special type in the case when the scattering function is a rational matrix valued function the main result here is the form of the potential in terms of a realisation of the scattering function the difference between this publication and the previous one 3 consists in the fact that here is used the marchenko method instead of kre n s method for rational scattering functions we also prove the equivalence of the two methods

transportation problems which can be solved by the use of hirsch paths for the dual problems balinski uses his signature method for the proof of the hirsch conjecture for dual transportation polyhedra to obtain an efficient algorithm for the assignment problem we will show how to extend this method to other primal transportation problems including transportation problems with unit demands we then prove that balinski s assignment algorithm is equivalent cycle by cycle to that of hung and rom we demonstrate that under some assumptions for our probability model a modification of the latter algorithm has an average complexity of o n2logn and present some computational results confirming this we also present results that indicate that this modification compares favorably with balinski s algorithm and other codes

szego kac achiezer formulas in terms of realizations of the symbol abstract for rational and analytic matrix functions new formulas are obtained for the limits in the szego kac achiezer limit theorems in the rational case the new expressions are given in terms of finite matrices which come from special representations of the matrix functions these representations are known as realizations in mathematical systems theory

efficient regression calibration for logistic regression in main study internal validation study designs with an imperfect reference instrument an extension to the version of the regression calibration estimator proposed by rosner et al for logistic and other generalized linear regression models is given for main study internal validation study designs this estimator combines the information about the parameter of interest contained in the internal validation study with rosner et al s regression calibration estimate using a generalized inverse variance weighted average it is shown that the validation study selection model can be ignored as long as this model is jointly independent of the outcome and the incompletely observed covariates conditional at most upon the surrogates and other completely observed covariates in an extensive simulation study designed to follow a complex multivariate setting in nutritional epidemiology it is shown that with validation study sizes of 340 or more this estimator appears to be asymptotically optimal in the sense that it is nearly unbiased and nearly as efficient as a properly specified maximum likelihood estimator a modification to the regression calibration variance estimator which replaces the standard uncorrected logistic regression coefficient variance with the sandwich estimator to account for the possible misspecification of the logistic regression fit to the surrogate covariates in the main study was also studied in this same simulation experiment in this study the alternative variance formula yielded results virtually identical to the original formula a version of the proposed estimator is also derived for the case where the reference instrument available only in the validation study is imperfect but unbiased at the individual level and contains error that is uncorrelated with other covariates and with error in the surrogate instrument replicate measures are obtained in a subset of study participants in this case it is shown that the validation study selection model can be ignored when sampling into the validation study depends at most only upon perfectly measured covariates two data sets a study of fever in relation to occupational exposure to antineoplastics among hospital pharmacists and a study of breast cancer incidence in relation to dietary intakes of alcohol and vitamin a adjusted for total energy intake from the nurses health study were analysed using these new methods in these data because the validation studies contained less than 200 observations and the events of interest were relatively rare as is typical the potential improvements offered by this new estimator were not apparent

tradeoffs in linear time varying systems an analogue of bode s sensitivity integral bode s integral characterizes some of the tradeoffs imposed by the open loop dynamics of a linear time invariant system here we present an analogous time domain characterization applicable to time varying systems

on the degrees of freedom of the lasso we study the effective degrees of freedom of the lasso in the framework of stein s unbiased risk estimation sure we show that the number of nonzero coefficients is an unbiased estimate forthe degrees of freedom of the lasso a conclusion that requires no special assumption on the predictors in addition the unbiased estimator is shown to be asymptotically consistent with these results on hand various model selection criteria cp aic and bic are available which along with the lars algorithm provide a principled and efficient approach to obtaining the optimal lasso fit with the computational effort of a single ordinary least squares fit

an infeasible exterior point simplex algorithm for assignment problems the so called modified hung rom algorithm based upon theoretical considerations of hirsch paths seems to be one of the most efficient algorithms for assignment problems since any two basic feasible solutions to a linear problem can always be connected with a short simplex path passing through the infeasible region development of algorithms based upon theoretical considerations on infeasible paths seems to be of great practical interest this paper presents an algorithm of this kind for assignment problems

an abstract version of a limit theorem of szego abstract the theory of factorization with respect to chains of orthogonal projections is used to deduce an abstract version of a well known limit theorem of szego the theory is then specialized to matrix fredholm operators and finally to matrix wiener hopf operators the well known continuous analogues of kac and achiezer emerge as special cases

variable selection for partially linear models with measurement errors this article focuses on variable selection for partially linear models when the covariates are measured with additive errors we propose two classes of variable selection procedures penalized least squares and penalized quantile regression using the nonconvex penalized principle the first procedure corrects the bias in the loss function caused by the measurement error by applying the so called correction for attenuation approach whereas the second procedure corrects the bias by using orthogonal regression the sampling properties for the two procedures are investigated the rate of convergence and the asymptotic normality of the resulting estimates are established we further demonstrate that with proper choices of the penalty functions and the regularization parameter the resulting estimates perform asymptotically as well as an oracle procedure as proposed by fan and li choice of smoothing parameters is also discussed finite sample performance of the proposed variable selection procedures is asse

logarithmic integrals and system dynamics an analogue of bode s sensitivity integral for continuous time time varying systems a new time domain interpretation of bode s integral is presented this allows for a generalization to the class of time varying systems which possess an exponential dichotomy it is shown that the sensitivity function is constrained on average by the spectral values in the dichotomy spectrum of the antistable component of the open loop dynamics

efficient dual simplex algorithms for the assignment problem efficient algorithms based upon balinski s signature method are described for solving then n assignment problem these algorithms are special variants of the dual simplex method and are shown to have computational bounds of o n3 variants for solving sparse assignment problems withm arcs that require o m space and o mn n2 logn time in the worst case are also presented

szego via jacobi abstract at present there exist numerous different approaches to results on toeplitz determinants of the type of szego s strong limit theorem the intention of this paper is to show that jacobi s theorem on the minors of the inverse matrix remains one of the most comfortable tools for tackling the matter we repeat a known proof of the borodin okounkov formula and thus of the strong szego limit theorem that is based on jacobi s theorem we then use jacobi s theorem to derive exact and asymptotic formulas for toeplitz determinants generated by functions with nonzero winding number this derivation is new and completely elementary

signature classes of transportation polytopes signature algorithms solve certain classes of transportation problems in a number of steps bounded by the diameter of the dual polyhedron the class of problems to which signature algorithms may be applied the signature classes of the title are characterized and the monotonic hirsch conjecture is shown to hold for them in addition certain more precise results are given for different cases finally it is explained why a supposed proof of the hirsch conjecture for all transportation polytopes is incomplete and apparently irremedial

an operator theoretic approach to the brickwork ising model with second neighbor interactions abstract using results on block toeplitz matrices in particular the szeg widom limit theorem and gohberg and feldman s convergence theorem for the finite section method we compute the spontaneous magnetization of an ising model in this way the transition from the ferromagnetic phase to the paramagnetic phase may be understood as the transition from invertibility to non invertibility of a certain block toeplitz operator associated with the model

a note on conditional aic for linear mixed effects models the conventional model selection criterion aic has been applied to choose candidate models in mixed effects models by the consideration of marginal likelihood vaida and blanchard 2005 demonstrated that such a marginal aic and its small sample correction are inappropriate when the research focus is on clusters correspondingly these authors suggested to use conditional aic the conditional aic is derived under the assumptions of the variance covariance matrix or scaled variance covariance matrix of random effects being known we develop a general conditional aic but without these strong assumptions this allows vaida and blanchard s conditional aic to be applied in a wide range simulation studies show that the proposed method is promising

signature methods for the assignment problem the signature of a dual feasible basis of the assignment problem is an n vector whose ith component is the number of nonbasic activities of type i j this paper uses signatures to describe a method for finding optimal assignments that terminates in at most n 1n 2 2 pivot steps and takes at most on3 work

a competitive dual simplex method for the assignment problem where there is abundance of mystery and confusion in every direction the truth seldom remains hidden for long it s a matter of having plenty of angles to go at it from only the utterly simple crimes the simplex crimes you may say have the trick of remaining baffling sir john from michael innes the open house a sir john appleby mystery penguin books 1974

linear model selection when covariates contain errors prediction precision is arguably the most relevant criterion of a model in practice and is often a sought after property a common difficulty with covariates measured with errors is the impossibility of performing prediction evaluation on the data even if a model is completely given without any unknown parameters we bypass this inherent difficulty by using special properties on moment relations in linear regression models with measurement errors the end product is a model selection procedure that achieves the same optimality properties that are achieved in classical linear regression models without covariate measurement error asymptotically the procedure selects the model with the minimum prediction error in general and selects the smallest correct model if the regression relation is indeed linear our model selection procedure is useful in prediction when future covariates without measurement error become available e g due to improved technology or better management and design of data collection procedures

tuning parameter selectors for the smoothly clipped absolute deviation method summary the penalized least squares approach with smoothly clipped absolute deviation penalty has been consistently demonstrated to be an attractive regression shrinkage and selection method it not only automatically and consistently selects the important variables but also produces estimators which are as efficient as the oracle estimator however these attractive features depend on appropriate choice of the tuning parameter we show that the commonly used generalized crossvalidation cannot select the tuning parameter satisfactorily with a nonignorable overfitting effect in the resulting model in addition we propose a bic tuning parameter selector which is shown to be able to identify the true model consistently simulation studies are presented to support theoretical findings and an empirical example is given to illustrate its use in the female labor supply data

on the degrees of freedom of reduced rank estimators in multivariate regression we study the effective degrees of freedom of a general class of reduced rank estimators for multivariate regression in the framework of stein s unbiased risk estimation a finite sample exact unbiased estimator is derived that admits a closed form expression in terms of the thresholded singular values of the least squares solution and hence is readily computable the results continue to hold in the high dimensional setting where both the predictor and the response dimensions may be larger than the sample size the derived analytical form facilitates the investigation of theoretical properties and provides new insights into the empirical behaviour of the degrees of freedom in particular we examine the differences and connections between the proposed estimator and a commonly used naive estimator the use of the proposed estimator leads to efficient and accurate prediction risk estimation and model selection as demonstrated by simulation studies and a data example

a semiparametric transformation approach to estimating usual daily intake distributions abstract the distribution of usual intakes of dietary components is important to individuals formulating food policy and to persons designing nutrition education programs the usual intake of a dietary component for a person is the long run average of daily intakes of that component for that person because it is impossible to directly observe usual intake for an individual it is necessary to develop an estimator of the distribution of usual intakes based on a sample of individuals with a small number of daily observations on a subsample of the individuals daily intake data for individuals are nonnegative and often very skewed also there is large day to day variation relative to the individual to individual variation and the within individual variance is correlated with the individual means we suggest a methodology for estimating usual intake distributions that allows for varying degrees of departure from normality and recognizes the measurement error associated with one day dietary intakes the est

the d step conjecture and its relatives the d step conjecture arose from an attempt to understand the computational complexity of edge following algorithms for linear programming such as the simplex algorithm it can be stated in terms of diameters of graphs of convex polytopes in terms of the existence of nonrevisiting paths in such graphs in terms of an exchange process for simplicial bases of a vector space and in terms of matrix pivot operations first formulated by w m hirsch in 1957 the conjecture remains unsettled though it has been proved in many special cases and counterexamples have been found for slightly stronger conjectures if the conjecture is false as we believe to be the case then finding a counterexample will be merely a small first step in the line of investigation related to the conjecture this report summarizes what is known about the d step conjecture and its relatives a considerable amount of new material is included but it does not seem to come close to settling the conjecture of special interest is the first example of a polytope that is not vertex decomposable showing that a certain natural approach to the conjecture will not work also significant are the quantitative relations among the lengths of paths associated with various forms of the conjecture

asymptotic optimality for c_p c_l cross validation and generalized cross validation discrete index set c p c l cross validation and generalized cross validation are useful data driven techniques for selecting a good estimate from a proposed class of linear estimates the asymptotic behaviors of these procedures are studied some easily interpretable conditions are derived to demonstrate the asymptotic optimality it is argued that cross validation and generalized cross validation can be viewed as some special ways of applying c l applications in nearest neighbor nonparametric regression and in model selection are discussed in detail