convergence to equilibrium for the relaxation approximations of conservation laws we study the cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term special comparison and compactness properties are established by assuming the subcharacteristic condition therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to 0 this research was strongly motivated by the recent numerical investigations of s jin and z xin on the relaxation schemes for conservation laws 1996 john wiley sons inc

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

higher order properties of gmm and generalized empirical likelihood estimators in an effort to improve the small sample properties of generalized method of moments gmm estimators a number of alternative estimators have been suggested these include empirical likelihood el continuous updating and exponential tilting estimators we show that these estimators share a common structure being members of a class of generalized empirical likelihood gel estimators we use this structure to compare their higher order asymptotic properties we find that gel has no asymptotic bias due to correlation of the moment functions with their jacobian eliminating an important source of bias for gmm in models with endogeneity we also find that el has no asymptotic bias from estimating the optimal weight matrix eliminating a further important source of bias for gmm in panel data models we give bias corrected gmm and gel estimators we also show that bias corrected el inherits the higher order property of maximum likelihood that it is higher order asymptotically efficient relative to the other bias corrected estimators

identifying predictive markers for personalized treatment selection it is now well recognized that the effectiveness and potential risk of a treatment often vary by patient subgroups although trial and error and one size fits all approaches to treatment selection remain a common practice much recent focus has been placed on individualized treatment selection based on patient information la thangue and kerr 2011 ong et al 2012 genetic and molecular markers are becoming increasingly available to guide treatment selection for various diseases including hiv and breast cancer mallal et al 2008 zujewski and kamin 2008 in recent years many statistical procedures for developing individualized treatment rules itrs have been proposed however less focus has been given to efficient selection of predictive biomarkers for treatment selection the standard wald test for interactions between treatment and the set of markers of interest may not work well when the marker effects are nonlinear furthermore interaction based test is scale dependent and may fail to capture markers useful for predicting individualized treatment differences in this article we propose to overcome these difficulties by developing a kernel machine km score test that can efficiently identify markers predictive of treatment difference simulation studies show that our proposed km based score test is more powerful than the wald test when there is nonlinear effect among the predictors and when the outcome is binary with nonlinear link functions furthermore when there is high correlation among predictors and when the number of predictors is not small our method also over performs wald test the proposed method is illustrated with two randomized clinical trials

kernel machine approach to testing the significance of multiple genetic markers for risk prediction summary there is growing evidence that genomic and proteomic research holds great potentials for changing irrevocably the practice of medicine the ability to identify important genomic and biological markers for risk assessment can have a great impact in public health from disease prevention to detection to treatment selection however the potentially large number of markers and the complexity in the relationship between the markers and the outcome of interest impose a grand challenge in developing accurate risk prediction models the standard approach to identifying important markers often assesses the marginal effects of individual markers on a phenotype of interest when multiple markers relate to the phenotype simultaneously via a complex structure such a type of marginal analysis may not be effective to overcome such difficulties we employ a kernel machine cox regression f ramework and propose an efficient score test to assess the overall effect of a set of markers such as genes within a pathway or a network on survival outcomes the proposed test has the advantage of capturing the potentially non linear effects without explicitly specifying a particular non linear functional form to approximate the null distribution of the score statistic we propose a simple resampling procedure that can be easily implemented in practice numerical studies suggest that the test performs well with respect to both empirical size and power even when the number of variables in a gene set is not small compared to the sample size

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

kernel methods in machine learning we review machine learning methods employing positive definite kernels these methods formulate learning and estimation problems in a reproducing kernel hilbert space rkhs of functions defined on the data domain expanded in terms of a kernel working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions the latter include nonlinear functions as well as functions defined on nonvectorial data we cover a wide range of methods ranging from binary classifiers to sophisticated methods for estimation with structured data

model evaluation based on the sampling distribution of estimated absolute prediction error the construction of a reliable practically useful prediction rule for future responses is heavily dependent on the adequacy of the fitted regression model in this article we consider the absolute prediction error the expected value of the absolute difference between the future and predicted responses as the model evaluation criterion this prediction error is easier to interpret than the average squared error and is equivalent to the misclassification error for a binary outcome we show that the prediction error can be consistently estimated via the resubstitution and crossvalidation methods even when the fitted model is not correctly specified furthermore we show that the resulting estimators are asymptotically normal when the prediction rule is nonsmooth the variance of the above normal distribution can be estimated well with a perturbation resampling method with two real examples and an extensive simulation study we demonstrate that the interval estimates obtained from the above normal approximation for the prediction errors provide much more information about model adequacy than their point estimate counterparts

using regression models to analyze randomized trials asymptotically valid hypothesis tests despite incorrectly specified models regression models are often used to test for cause effect relationships from data collected in randomized trials or experiments this practice has deservedly come under heavy scrutiny because commonly used models such as linear and logistic regression will often not capture the actual relationships between variables and incorrectly specified models potentially lead to incorrect conclusions in this article we focus on hypothesis tests of whether the treatment given in a randomized trial has any effect on the mean of the primary outcome within strata of baseline variables such as age sex and health status our primary concern is ensuring that such hypothesis tests have correct type i error for large samples our main result is that for a surprisingly large class of commonly used regression models standard regression based hypothesis tests but using robust variance estimators are guaranteed to have correct type i error for large samples even when the models are incorrectly specified to the best of our knowledge this robustness of such model based hypothesis tests to incorrectly specified models was previously unknown for poisson regression models and for other commonly used models we consider our results have practical implications for understanding the reliability of commonly used model based tests for analyzing randomized trials

model selection principles in misspecified models summary model selection is of fundamental importance to high dimensional modelling featured in many contemporary applications classical principles of model selection include the bayesian principle and the kullback leibler divergence principle which lead to the bayesian information criterion and akaike information criterion respectively when models are correctly specified yet model misspecification is unavoidable in practice we derive novel asymptotic expansions of the two well known principles in misspecified generalized linear models which give the generalized bayesian information criterion and generalized akaike information criterion a specific form of prior probabilities motivated by the kullback leibler divergence principle leads to the generalized bayesian information criterion with prior probability gbicp which can be naturally decomposed as the sum of the negative maximum quasi log likelihood a penalty on model dimensionality and a penalty on model misspecification directly numerical studies demonstrate the advantage of the new methods for model selection in both correctly specified and misspecified models

linear model selection by cross validation abstract we consider the problem of selecting a model having the best predictive ability among a class of linear models the popular leave one out cross validation method which is asymptotically equivalent to many other model selection methods such as the akaike information criterion aic the c p and the bootstrap is asymptotically inconsistent in the sense that the probability of selecting the model with the best predictive ability does not converge to 1 as the total number of observations n we show that the inconsistency of the leave one out cross validation can be rectified by using a leave n v out cross validation with n v the number of observations reserved for validation satisfying n v n 1 as n this is a somewhat shocking discovery because nv n 1 is totally opposite to the popular leave one out recipe in cross validation motivations justifications and discussions of some practical aspects of the use of the leave n v out cross validation method are provided and results

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

finite volume flows and morse theory in this paper we present a new approach to morse theory based on the de rham federer theory of currents the full classical theory is derived in a transparent way the methods carry over uniformly to the equivariant and the holomorphic settings moreover the methods are substantially stronger than the classical ones and have interesting applications to geometry they lead for example to formulas relating characteristic forms and singularities of bundle maps

estimating subject specific dependent competing risk profile with censored event time observations in a longitudinal study suppose that the primary endpoint is the time to a specific event this response variable however may be censored by an independent censoring variable or by the occurrence of one of several dependent competing events for each study subject a set of baseline covariates is collected the question is how to construct a reliable prediction rule for the future subject s profile of all competing risks of interest at a specific time point for risk benefit decision makings in this paper we propose a two stage procedure to make inferences about such subject specific profiles for the first step we use a parametric model to obtain a univariate risk index score system we then estimate consistently the average competing risks for subjects which have the same parametric index score via a nonparametric function estimation procedure we illustrate this new proposal with the data from a randomized clinical trial for evaluating the efficacy of a treatment for prostate cancer the primary endpoint for this study was the time to prostate cancer death but had two types of dependent competing events one from cardiovascular death and the other from death of other causes

convergence of a splitting scheme applied to the ruijgrok wu model of the boltzmann equation this paper deals with upwind splitting schemes for the ruijgrok wu model physica a 113 1982 401 416 of the kinetic theory of rare7ed gases in the 8uid dynamic scaling we prove the stability and the convergence for these schemes the relaxation limit is also investigated and the limit equation is proved to be a 7rst order quasi linear conservation law the loss of quasi monotonicity of the present model makes it necessary to give a more careful analysis of its structure we also obtain global error estimates in the spaces w s p for 16s61 p 16p6 and pointwise error estimates for the approximate solution the proof naturally uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 1505 1519 due to the convexity of the 8ux function c

flexible regression model selection for survival probabilities with application to aids clinicians are often interested in the effect of covariates on survival probabilities at prespecified study times because different factors can be associated with the risk of short and long term failure a flexible modeling strategy is pursued given a set of multiple candidate working models an objective methodology is proposed that aims to construct consistent and asymptotically normal estimators of regression coefficients and average prediction error for each working model that are free from the nuisance censoring variable it requires the conditional distribution of censoring given covariates to be modeled the model selection strategy uses stepup or stepdown multiple hypothesis testing procedures that control either the proportion of false positives or generalized familywise error rate when comparing models based on estimates of average prediction error the context can actually be cast as a missing data problem where augmented inverse probability weighted complete case estimators of regression coefficients and prediction error can be used tsiatis 2006 semiparametric theory and missing data a simulation study and an interesting analysis of a recent aids trial are provided

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

parsimonious conditional mean model selection with multiple covariates an analysis of infant mortality in the usa this paper proposes and evaluates an objective methodology to select a parsimonious conditional mean model when faced with multiple candidate predictor variables the methodology attempts to fine tune a well established covariate screening method such as iterative sure independence screening with smoothly clipped absolute deviation penalty by using the following i cross validated or bootstrap estimates of prediction error ii an objective model comparison strategy and iii multiple hypothesis testing the methods are analytically and numerically shown to work well in the sense that the probability that the final model selected contains one or more unimportant variables is asymptotically bounded at a preselected level for arbitrary data generating distributions this methodology is illustrated with a dataset consisting of birth certificate information and mortality records from year 2001 from the us department of health and human services on non hispanic african american female and male infants it is shown how the instantaneous daily mortality hazard can be modeled flexibly by allowing both the set of important predictors and their effect on the hazard to change arbitrarily thru time results indicate that once controlling for birth weight no other variables on the birth certificate are significantly associated with mortality furthermore time and sex modify the birth weight survival relationship with the strongest association at earliest days and low birth weight female infants having a better survival experience than male counterparts

morse theory for lagrangian intersections soit p une variete symplectique compacte et soit l p une sous variete lagrangienne avec 2 p l 0 pour un diffeomorphisme exact de p avec la propriete que l coupe l transversalement on demontre une inegalite de morse reliant l ensemble l l a la cohomologie de l

the little bootstrap and other methods for dimensionality selection in regression x fixed prediction error abstract when a regression problem contains many predictor variables it is rarely wise to try to fit the data by means of a least squares regression on all of the predictor variables usually a regression equation based on a few variables will be more accurate and certainly simpler there are various methods for picking good subsets of variables and programs that do such procedures are part of every widely used statistical package the most common methods are based on stepwise addition or deletion of variables and on best subsets the latter refers to a search method that given the number of variables to be in the equation say five locates that regression equation based on five variables that has the lowest residual sum of squares among all five variable equations all of these procedures generate a sequence of regression equations the first based on one variable the next based on two variables and so on each member of this sequence is called a submodel and the number of variables in the e

improvements on cross validation the 632 bootstrap method abstract a training set of data has been used to construct a rule for predicting future responses what is the error rate of this rule this is an important question both for comparing models and for assessing a final selected model the traditional answer to this question is given by cross validation the cross validation estimate of prediction error is nearly unbiased but can be highly variable here we discuss bootstrap estimates of prediction error which can be thought of as smoothed versions of cross validation we show that a particular bootstrap method the 632 rule substantially outperforms cross validation in a catalog of 24 simulation experiments besides providing point estimates we also consider estimating the variability of an error rate estimate all of the results here are nonparametric and apply to any possible prediction rule however we study only classification problems with 0 1 loss in detail our simulations include smooth prediction rules like fisher s linear discriminant fun

semiparametric box cox power transformation models for censored survival observations the accelerated failure time model specifies that the logarithm of the failure time is linearly related to the covariate vector without assuming a parametric error distribution in this paper we consider the semiparametric box cox transformation model which includes the above regression model as a special case to analyse possibly censored failure time observations inference procedures for the transformation and regression parameters are proposed via a resampling technique prediction of the survival function of future subjects with a specific covariate vector is also provided via pointwise and simultaneous interval estimates all the proposals are illustrated with datasets from two clinical studies

morse theory for lagrange multipliers and adiabatic limits abstract given two morse functions f on a compact manifold m we study the morse homology for the lagrange multiplier function on m r which sends x to f x x take a product metric on m r and rescale its r component by a factor 2 we show that generically for large the morse smale witten chain complex is isomorphic to the one for f and the metric restricted to 1 0 with grading shifted by one on the other hand in the limit 0 we obtain another chain complex which is geometrically quite different but has the same homology as the singular homology of 1 0 the isomorphism between the chain complexes is provided by the homotopy obtained by varying our proofs use both the implicit function theorem on banach manifolds and geometric singular perturbation theory

the lip stability and error estimates for a relaxation scheme we show the discrete lip stability for a relaxation scheme proposed by jin and xin comm pure appl math 48 1995 pp 235 277 to approximate convex conservation laws equipped with the lip stability we obtain global error estimates in the spaces w s p for 1 leq s leq 1 p 1 leq p leq infty and pointwise error estimates for the approximate solution obtained by the relaxation scheme the proof uses the framework introduced by nessyahu and tadmor siam j numer anal 29 1992 pp 1505 1519 we also show a maximum principle for the relaxation scheme when the initial data are in an equilibrium state

estimating subject specific survival functions under the accelerated failure time model summary we use the semiparametric accelerated failure time model to predict the survival function and its related quantities for future subjects with a given set of covariates we derive the large sample distribution for the subject specific cumulative hazard function estimate we then propose a simple resampling technique for constructing pointwise confidence intervals and simultaneous bands for the corresponding survival function and its quantile function over a properly selected time interval the new proposals are illustrated with the mayo primary biliary cirrhosis data

evaluating prediction rules for t year survivors with censored regression models suppose that we are interested in establishing simple but reliable rules for predicting future t year survivors through censored regression models in this article we present inference procedures for evaluating such binary classification rules based on various prediction precision measures quantified by the overall misclassification rate sensitivity and specificity and positive and negative predictive values specifically under various working models we derive consistent estimators for the above measures through substitution and cross validation estimation procedures furthermore we provide large sample approximations to the distributions of these nonsmooth estimators without assuming that the working model is correctly specified confidence intervals for example for the difference of the precision measures between two competing rules can then be constructed all of the proposals are illustrated with real examples and their finite sample properties are evaluated through a simulation study

a discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws abstract we present a new relaxation approximation to scalar conservation laws in several space variables by means of semilinear hyperbolic systems of equations with a finite number of velocities under a suitable multidimensional generalization of the whitham relaxation subcharacteristic condition we show the convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

lagrangian intersections and the serre spectral sequence for a transversal pair of closed lagrangian submanifolds l lof a sym plectic manifold m such that 1 l 1 l 0 c1 2 m 2 m and for a generic almost complex structure j we construct an invariant with a high homotopical content which consists in the pages of order 2o f as pec tral sequence whose differentials provide an algebraic measure of the high dimensional moduli spaces of pseudo holomorpic strips of finite energy that join l and l when l and lare hamiltonian isotopic we show that the pages of the spectral sequence coincide up to a horizontal translation with the terms of the serre spectral sequence of the path loop fibration l pl l and we deduce some applications

discussion of a risk based measure of time varying prognostic discrimination for survival models by c jason liang and patrick j heagerty we congratulate drs liang and heagerty l h for introducing a risk based measure of time varying discrimination for determining the value of predictors in time to event models their proposed hazard discrimination summary hds is intuitive and its statistical properties are appealing the local in time model framework provides additional flexibility for departures from the cox proportional hazards model our discussion focuses on use of the proposed method to determine the added value of intermediate event information and an alternative approach for confidence interval estimation

discrete kinetic schemes for multidimensional systems of conservation laws we present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations which includes the relaxation schemes by s jin and z xin these schemes have a simple formulation even in the multidimensional case and do not need the solution of the local riemann problems for these approximations we give a suitable multidimensional generalization of the whitham s stability subcharacteristic condition in the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

convergence rates for relaxation schemes approximating conservation laws in this paper we prove a global error estimate for a relaxation scheme approximating scalar conservation laws to this end we decompose the error into a relaxation error and a discretization error including an initial error omega ep we obtain the rate of convergence of sqrt ep in l1 for the relaxation step the estimate here is independent of the type of nonlinearity in the discretization step a convergence rate of sqrt del x in l1 is obtained these rates are independent of the choice of initial error omega ep thereby we obtain the order 1 2 for the total error

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

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

on measuring and correcting the effects of data mining and model selection abstract in the theory of linear models the concept of degrees of freedom plays an important role this concept is often used for measurement of model complexity for obtaining an unbiased estimate of the error variance and for comparison of different models i have developed a concept of generalized degrees of freedom gdf that is applicable to complex modeling procedures the definition is based on the sum of the sensitivity of each fitted value to perturbation in the corresponding observed value the concept is nonasymptotic in nature and does not require analytic knowledge of the modeling procedures the concept of gdf offers a unified framework under which complex and highly irregular modeling procedures can be analyzed in the same way as classical linear models by using this framework many difficult problems can be solved easily for example one can now measure the number of observations used in a variable selection process different modeling procedures such as a tree based regression and a

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

bootstrap model selection abstract in a regression problem typically there are p explanatory variables possibly related to a response variable and we wish to select a subset of the p explanatory variables to fit a model between these variables and the response a bootstrap variable model selection procedure is to select the subset of variables by minimizing bootstrap estimates of the prediction error where the bootstrap estimates are constructed based on a data set of size n although the bootstrap estimates have good properties this bootstrap selection procedure is inconsistent in the sense that the probability of selecting the optimal subset of variables does not converge to 1 as n this inconsistency can be rectified by modifying the sampling method used in drawing bootstrap observations for bootstrapping pairs response explanatory variable it is found that instead of drawing n bootstrap observations a customary bootstrap sampling plan much less bootstrap observations should be sampled the bootstrap selection p

on the existence of maximum likelihood estimators for the binomial response models summary necessary and sufficient conditions are given for the existence of maximum likelihood estimators of the linear regression parameter in binomial response this includes logit and probit models the question of existence of maximum likelihood estimators mle for logit models arose in an analysis of the relationship of psychiatric caseness to scores on a psychiatric screening questionnaire tennant 1977 administered the ghq general health questionnaire goldberg 1972 to 120 patients attending a general practitioner s surgery and also gave each one a standardized psychiatric interview from the interview patients were classified as psychiatric case non case in a secondary analysis of tennant s data duncan jones and henderson 1978 fitted a logit regression of caseness on ghq score and obtained a good fit with the model

estimation and testing for multiple regulation of multivariate mixed outcomes considerable interest has recently been focused on studying multiple phenotypes simultaneously in both epidemiological and genomic studies either to capture the multidimensionality of complex disorders or to understand shared etiology of related disorders we seek to identify multiple regulators or predictors that are associated with multiple outcomes when these outcomes may be measured on very different scales or composed of a mixture of continuous binary and not fully observed elements we first propose an estimation technique to put all effects on similar scales and we induce sparsity on the estimated effects we provide standard asymptotic results for this estimator and show that resampling can be used to quantify uncertainty in finite samples we finally provide a multiple testing procedure which can be geared specifically to the types of multiple regulators of interest and we establish that under standard regularity conditions the familywise error rate will approach 0 as sample size diverges simulation results indicate that our approach can improve over unregularized methods both in reducing bias in estimation and improving power for testing

on the regularity of approximate solutions to conservation laws with piecewise smooth solutions in this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted w1 1 space convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise error estimate is given in other words we are able to convert weak pointwise error estimates to optimal error bounds in a weighted w1 1 space for convex conservation laws the assumption of a weak pointwise error estimate is verified by tadmor siam j numer anal 28 1991 pp 891 906 therefore one immediate application of our w1 1 convergence theory is that for convex conservation laws we indeed have w1 1 error bounds for the approximate solutions to conservation laws furthermore the co en pointwise error estimates of tadmor and tang siam j numer anal 36 1999 pp 1739 1758 are recovered by the use of the w1 1 convergence result

morse bott homology we give a new proof of the morse homology theorem by constructing a chain complex associated to a morse bott smale function that reduces to the morse smale witten chain complex when the function is morse smale and to the chain complex of smooth singular n cube chains when the function is constant we show that the homology of the chain complex is independent of the morse bott smale function by using compactified moduli spaces of time dependent gradient flow lines to prove a floer type continuation theorem

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

semi supervised approaches to efficient evaluation of model prediction performance in many modern machine learning applications the outcome is expensive or time consuming to collect whereas the predictor information is easy to obtain semi supervised ss learning aims at utilizing large amounts of unlabelled data along with small amounts of labelled data to improve the efficiency of a classical supervised approach though numerous ss learning classification and prediction procedures have been proposed in recent years no methods currently exist to evaluate the prediction performance of a working regression model in the context of developing phenotyping algorithms derived from electronic medical records we present an efficient two step estimation procedure for evaluating a binary classifier based on various prediction performance measures in the ss setting in step i the labelled data are used to obtain a non parametrically calibrated estimate of the conditional risk function in step ii ss estimates of the prediction accuracy parameters are constructed based on the estimated conditional risk function and the unlabelled data we demonstrate that under mild regularity conditions the estimators proposed are consistent and asymptotically normal importantly the asymptotic variance of the ss estimators is always smaller than that of the supervised counterparts under correct model specification we also correct for potential overfitting bias in the ss estimators in finite samples with cross validation and we develop a perturbation resampling procedure to approximate their distributions our proposals are evaluated through extensive simulation studies and illustrated with two real electronic medical record studies aiming to develop phenotyping algorithms for rheumatoid arthritis and multiple sclerosis

comparison of maximum statistics for hypothesis testing when a nuisance parameter is present only under the alternative in many practical problems a hypothesis testing involves a nuisance parameter which appears only under the alternative hypothesis davies 1977 biometrika 64 247 254 proposed the maximum of the score statistics over the whole range of the nuisance parameter as a test statistic for this type of hypothesis testing freidlin podgor and gastwirth 1999 biometrics 55 883 886 studied two other simpler maximum test statistics the maximum of the score statistics at two extreme points of the nuisance parameter and the maximum of the score statistics at three points of the nuisance parameter including the two extreme points in this article we compare the powers of these three maximum type statistics in the context of three genetic problems

maximum likelihood estimation in semiparametric regression models with censored data summary semiparametric regression models play a central role in formulating the effects of covariates on potentially censored failure times and in the joint modelling of incomplete repeated measures and failure times in longitudinal studies the presence of infinite dimensional parameters poses considerable theoretical and computational challenges in the statistical analysis of such models we present several classes of semiparametric regression models which extend the existing models in important directions we construct appropriate likelihood functions involving both finite dimensional and infinite dimensional parameters the maximum likelihood estimators are consistent and asymptotically normal with efficient variances we develop simple and stable numerical techniques to implement the corresponding inference procedures extensive simulation experiments demonstrate that the inferential and computational methods proposed perform well in practical settings applications to three medical studies yield important new insights we conclude that there is no reason theoretical or numerical not to use maximum likelihood estimation for semiparametric regression models we discuss several areas that need further research

predicting future responses based on possibly mis specified working models summary under a general regression setting we propose an optimal unconditional prediction procedure for future responses the resulting prediction intervals or regions have a desirable average cov erage level over a set of covariate vectors of interest when the working model is not correctly specified the traditional conditional prediction method is generally invalid on the other hand one can empirically calibrate the above unconditional procedure and also obtain its crossvalidated counterpart various large and small sample properties of these unconditional methods are ex amined analytically and numerically we find that the ic fold crossvalidated procedure performs exceptionally well even for cases with rather small sample sizes the new proposals are illustrated with two real examples one with a continuous response and the other with a binary outcome

more comments on c p we discuss the interpretation of c p plots and show how they can be calibrated in several ways we comment on the practice of using the display as a basis for formal selection of a subset regression model and extend the range of application of the device to encompass arbitrary linear estimates of the regression coefficients for example ridge estimates