bootstrapping a regression equation some empirical results abstract the bootstrap like the jackknife is a technique for estimating standard errors the idea is to use monte carlo simulation based on a nonparametric estimate of the underlying error distribution the main object of this article is to present the bootstrap in the context of an econometric equation describing the demand for energy by industry as it turns out the conventional asymptotic formulas for estimating standard errors are too optimistic by factors of nearly three when applied to a particular finite sample problem in a simpler context this finding can be given a mathematical proof

bootstrapping regression models the regression and correlation models are considered it is shown that the bootstrap approximation to the distribution of the least squares estimates is valid and some error bounds are given

plots and tests for goodness of fit with randomly censored data summary in this paper we consider nonparametric procedures for assessing goodness of fit when the data may be subject to random censoring we introduce two classes of large sample tests based respectively on the maximum and average weighted difference between the specified survival function and the kaplan meier estimate of the true survival function the procedures in the first class are inverted to obtain simultaneous confidence bands for the survival and cumulative hazard functions these bands provide simultaneous bands associated with the percentage p p quantile q q and hazard plots for censored data some key word8 hazard function nonparametric procedure probability plot shift function survival function

convergence rates for the bootstrapped product limit process on etablit des taux d approximation forte pour le processus limite produit du bootstrap et le processus quantile correspondant

hyperbolic polynomials and interior point methods for convex programming hyperbolic polynomials have their origins in partial differential equations we show in this paper that they have applications in interior point methods for convex programming each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone we give an explicit representation of this cone in terms of polynomial inequalities the function fx log px is a logarithmically homogeneous self concordant barrier function for the hyperbolicity cone with barrier parameter equal to the degree of p the function fx possesses striking additional properties that are useful in designing long step interior point methods for example we show that the long step primal potential reduction methods of nesterov and todd and the surface following methods of nesterov and nemirovskii extend to hyperbolic barrier functions we also show that there exists a hyperbolic barrier function on every homogeneous cone

lowner s operator and spectral functions in euclidean jordan algebras we study analyticity differentiability and semismoothness of lowner s operator and spectral functions under the framework of euclidean jordan algebras in particular we show that many optimization related classical results in the symmetric matrix space can be generalized within this framework for example the metric projection operator over any symmetric cone defined in a euclidean jordan algebra is shown to be strongly semismooth the research also raises several open questions whose answers would be of strong interest for optimization research

derivatives of spectral functions a spectral function of a hermitian matrix x is a function which depends only on the eigenvalues of x 1x 2x nx and hence may be written f 1x 2x nx for some symmetric function f such functions appear in a wide variety of matrix optimization problems we give a simple proof that this spectral function is differentiable at x if and only if the function f is differentiable at the vector x and we give a concise formula for the derivative we then apply this formula to deduce an analogous expression for the clarke generalized gradient of the spectral function a similar result holds for real symmetric matrices

self concordant barriers for hyperbolic means the geometric mean and the function det 1 m on the m by m positive definite matrices are examples of hyperbolic means functions of the form p 1 m where p is a hyperbolic polynomial of degree m a homogeneous polynomial p is hyperbolic with respect to a vector d if the polynomial t p x td has only real roots for every vector x any hyperbolic mean is positively homogeneous and concave on a suitable domain we present a self concordant barrier for its hypograph with barrier parameter o m2 our approach is direct and shows for example that the function mlog det 1 is an m2 self concordant barrier on a natural domain such barriers suggest novel interior point approaches to convex programs involving hyperbolic means

convex analysis on the hermitian matrices there is growing interest in optimization problems with real symmetric matrices as variables generally the matrix functions involved are spectral they depend only on the eigenvalues of the matrix it is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues a new approach to this characterization is given via a simple fenchel conjugacy formula we then apply this formula to derive expressions for subdifferentials and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables analogous results hold for hermitian matrices

on the behavior of some estimators from probability plots abstract probability plots are popular graphical methods used to assess distributional assumptions under a location scale model the plot tends to lie on a straight line a common practice in this situation is to fit a line through the plot and use the intercept and slope of the fitted line as estimates of the location and scale parameters what are the properties of these estimators estimators from weighted least squares lines are considered and their asymptotic finite sample robustness and optimality properties are discussed included among these are the ordinary least squares estimators and estimators from least squares lines fitted after trimming or winsorizing some of the extreme order statistics

pre and post selection weak values and contextuality by analysing the concept of contextuality bell kochen specker in terms of pre and post selection it is possible to assign definite values to observables in a new and surprising way physical reasons are presented for restrictions on these assignments when measurements are performed which do not disturb the pre and post selection i e weak measurements then novel experimental aspects of contextuality can be demonstrated we also prove that every pps paradox with definite predictions directly implies quantum contextuality which is introduced as the analogue of contextuality at the level of quantum mechanics rather than at the level of hidden variable theories finally we argue that certain results of these measurements e g eccentric weak values outside the eigenvalue spectrum cannot be explained by a classical like hidden variable theory

statistical methods for estimating the minimum thickness along a pipeline pipeline integrity is important because leaks can result in serious economic or environmental losses inspection information from a sample of locations along the pipeline can be used to estimate corrosion levels the traditional parametric model method for this problem is to estimate parameters of a specified corrosion distribution and then to use these parameters to estimate the minimum thickness in a pipeline inferences using this method are however highly sensitive to the distributional assumption extreme value modeling provides a more robust method of estimation if a sufficient amount of data is available for example the block minima method produces a more robust method to estimate the minimum thickness in a pipeline to use the block minima method however one must carefully choose the size of the blocks to be used in the analysis in this article we use simulation to compare the properties of different models for estimating minimum pipeline thickness investigate the effect of using differen

nonsmooth analysis of lorentz invariant functions a real valued function g x t on mathbb r n times mathbb r is called a lorentz invariant if g x t g ux t for all n times n orthogonal matrices u and all x t in the domain of g in other words g is invariant under the linear orthogonal transformations preserving the lorentz cone x t in mathbb r n times mathbb r t ge x it is easy to see that every lorentz invariant function can be decomposed as g f circ beta where f mathbb r 2 rightarrow mathbb r is a symmetric function and beta is the root map of the hyperbolic polynomial p x t t 2 x 1 2 cdots x n 2 we investigate a variety of important variational and nonsmooth properties of g and characterize them in terms of the symmetric function f

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

nonsmooth variants of powell s bfgs convergence theorem the popular bfgs quasi newton minimization algorithm under reasonable conditions converges globally on smooth convex functions this result was proved by powell in a landmark 1976 paper we consider its implications for functions that are not smooth in particular an analogous convergence result holds for functions like the euclidean norm whose minimizers are isolated nonsmooth points

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

classical three box paradox a simple classical probabilistic system a simple card game classically exemplifies aharonov and vaidman s three box paradox 1991 j phys a math gen 24 2315 implying that the three box example is neither quantal nor a paradox and leaving one with less difficulty to busy the interpreters of quantum mechanics an ambiguity in the usual expression of the retrodiction formula is shown to have misled albert et al 1985 phys rev lett 54 5 to a result not in fact curious the discussion illustrates how to avoid this ambiguity

three box paradox and cheshire cat grin the case of spin 1 atoms in this work we propose a definite theoretical implementation of the three box paradox a scheme in which a single quantum particle appears to be present with certainty in two separate boxes with spin 1 atoms we further show how our setup can give rise to a cheshire cat grin type of situation in which an atom can apparently be found with certainty in one of the boxes while one of its properties the angular momentum projection along a specifically chosen axis appears to be in a different box the significance of our findings is discussed relative to the status of the properties of a system obtained from weak measurements

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

goodness of fit tests for censored survival data the problem of testing the underlying distribution in a general regression model is considered when the data are right censored it is proposed that tests be based on efficient scores from a certain class of parametric alternatives the connection between tests based on the scores and more traditional approaches using empirical distribution functions is explored neyman s smooth test is extended to allow for censored data

quadratic expansions of spectral functions a function f on the space of n n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument that is f a f uau t for every orthogonal u and symmetric a in its domain spectral functions are in one to one correspondence with the symmetric functions on r n those that are invariant under arbitrary swapping of their arguments in this paper we show that a spectral function has a quadratic expansion around a point a if and only if its corresponding symmetric function has quadratic expansion around a the vector of eigenvalues we also give a concise and easy to use formula for the hessian of the spectral function in the case of convex functions we show that a positive definite hessian of f implies positive definiteness of the hessian of f 2002 elsevier science inc all rights reserved

confidence bands for survival functions with censored data a comparative study consider the problem of obtaining simultaneous confidence bands for the survival function s x when the data are arbitrarily right censored the usual pointwise confidence intervals based on greenwood s variance formula can be adapted to yield a large sample confidence band this band has in a certain sense equal precision at each point of s x it is compared with the censored versions of the kolmogorov band and the renyi band the comparisons are made in terms of the widths and the adequacy of large sample approximations and are carried out under various censoring models and degrees of censoring the bands are illustrated by applying them to data from a mechanical switch life test

nonsmooth analysis of eigenvalues the eigenvalues of a symmetric matrix depend on the matrix nonsmoothly this paper describes the nonsmooth analysis of these eigenvalues in particular i present a simple formula for the approximate limiting frechet subdifferential of an arbitrary function of the eigenvalues subsuming earlier results on convex and clarke subgradients as an example i compute the subdifferential of the k th largest eigenvalue

regression modelling strategies for improved prognostic prediction regression models such as the cox proportional hazards model have had increasing use in modelling and estimating the prognosis of patients with a variety of diseases many applications involve a large number of variables to be modelled using a relatively small patient sample problems of overfitting and of identifying important covariates are exacerbated in analysing prognosis because the accuracy of a model is more a function of the number of events than of the sample size we used a general index of predictive discrimination to measure the ability of a model developed on training samples of varying sizes to predict survival in an independent test sample of patients suspected of having coronary artery disease we compared three methods of model fitting 1 standard step up variable selection 2 incomplete principal components regression and 3 cox model regression after developing clinical indices from variable clusters we found regression using principal components to offer superior predictions in the test sample whereas regression using indices offers easily interpretable models nearly as good as the principal components models standard variable selection has a number of deficiencies

the bootstrap and identification of prognostic factors via cox s proportional hazards regression model this paper describes the use of the bootstrap 1 2 a new computer based statistical methodology to help validate a regression model resulting from the fitting of cox s proportional hazards model3 to a set of censored survival data as an example we define a prognostic model for outcome in childhood acute lymphocytic leukemia with the cox model and use of a training set of 224 patients to validate the accuracy of the model we use a bootstrap resampling technique to mimic the population under study in two stages first we select the important prognostic factors via a stepwise regression procedure with 100 bootstrap samples secondly we estimate the corresponding regression parameters for these important factors with 400 bootstrap samples the bootstrap result suggests that the model constructed from the training set is reasonable

complete description of a quantum system at a given time a generalization of the description of a quantum system in the time interval between two measurements is presented a new concept of a generalized state is introduced generalized states yield a complete description of a quantum system when information about the system is available both from the past snd from the future the formalism of generalized states provides a natural language for describing many peculiar situations in particular situations in which one can ascertain the result of a measurement of any one of several non commuting variables are analysed weak measurements on quantum systems described by generalized states are discussed the relation belween weak and strong measurements is investigated

semismooth homeomorphisms and strong stability of semidefinite and lorentz complementarity problems based on an inverse function theorem for a system of semismooth equations this paper establishes several necessary and sufficient conditions for an isolated solution of a complementarity problem defined on the cone of symmetric positive semidefinite matrices to be strongly regular stable we show further that for a parametric complementarity problem of this kind if a solution corresponding to a base parameter is strongly stable then a semismooth implicit solution function exists whose directional derivatives can be computed by solving certain affine problems on the critical cone at the base solution similar results are also derived for a complementarity problem defined on the lorentz cone the analysis relies on some new properties of the directional derivatives of the projector onto the semidefinite cone and the lorentz cone

a correlation goodness of fit test for randomly censored data summary for testing a composite goodness of fit hypothesis with randomly censored data a correlation statistic which is an analogue of the shapiro francia statistic is presented the test for exponentiality in case of light censoring is asymptotically robust against departures from the koziol green model of random censorship the power of the test is compared with the total time on test procedure and the new better than used test using monte carlo simulation as an example the test is applied to some data with prostate cancer