cross validation in survival analysis the predictive value of a statistical model is conceptually different from the explained variation in this paper we construct a measure of the predictive value of the cox proportional hazards model computed from the leave one out regression coefficients these coefficients can also be used to calculate a shrinkage factor which can be applied to improve the predictions and that can be used in r2 type measures of the proportion of explained variation our methods are illustrated by a study of chemotherapy for advanced ovarian cancer

model selection criteria in multivariate models with multiple structural changes abstract this paper considers the issue of selecting the number of regressors and the number of structural breaks in multivariate regression models in the possible presence of multiple structural changes we develop a modified akaike information criterion aic a modified mallows c p criterion and a modified bayesian information criterion bic the penalty terms in these criteria are shown to be different from the usual terms we prove that the modified bic consistently selects the regressors and the number of breaks whereas the modified aic and the modified c p criterion tend to overfit with positive probability the finite sample performance of these criteria is investigated through monte carlo simulations and it turns out that our modification is successful in comparison to the classical model selection criteria and the sequential testing procedure robust to heteroskedasticity and autocorrelation

modelling multivariate binary data with alternating logistic regressions marginal models for multivariate binary data permit separate modelling of the relationship of the response with explanatory variables and the association between pairs of responses when the former is the scientific focus a first order generalized estimating equation method liang zeger 1986 is easy to implement and gives efficient estimates of regression coefficients although estimates of the association among the binary outcomes can be inefficient when the association model is a focus simultaneous modelling of the responses and all pairwise products prentice 1988 using second order estimating equations gives more efficient estimates of association parameters as well however this procedure can become computationally infeasible as the cluster size gets large this paper proposes an alternative approach alternating logistic regressions for simultaneously regressing the response on explanatory variables as well as modelling the association among responses in terms of pairwise odds ratios this algorithm iterates between a logistic regression using first order generalized estimating equations to estimate regression coefficients and a logistic regression of each response on others from the same cluster using an appropriate offset to update the odds ratio parameters for clusters of size n alternating logistic regression involves evaluation and inversion of matrices of order n2 rather than n4 as required for second order generalized estimating equations the alternating logistic regression estimates are shown to be reasonably efficient relative to solutions of second order equations in a few problems the new method is illustrated with an analysis of neuropsychological tests on patients with epileptic seizures

variable selection in semiparametric linear regression with censored data summary we describe two procedures for selecting variables in the semiparametric linear regression model for censored data one procedure penalizes a vector of estimating equations and simultaneously estimates regression coefficients and selects submodels a second proce dure controls systematically the proportion of unimportant variables through forward selection and the addition of pseudorandom variables we explore both rank based statistics and buckley james statistics in the setting proposed and evaluate the performance of all methods through extensive simulation studies and one real data set

data driven calibration of penalties for least squares regression penalization procedures often suffer from their dependence on multiplying factors whose optimal values are either unknown or hard to estimate from data we propose a completely data driven calibration algorithm for these parameters in the least squares regression framework without assuming a particular shape for the penalty our algorithm relies on the concept of minimal penalty recently introduced by birge and massart 2007 in the context of penalized least squares for gaussian homoscedastic regression on the positive side the minimal penalty can be evaluated from the data themselves leading to a data driven estimation of an optimal penalty which can be used in practice on the negative side their approach heavily relies on the homoscedastic gaussian nature of their stochastic framework the purpose of this paper is twofold stating a more general heuristics for designing a data driven penalty the slope heuristics and proving that it works for penalized least squares regression with a random design even for heteroscedastic non gaussian data for technical reasons some exact mathematical results will be proved only for regressogram bin width selection this is at least a first step towards further results since the approach and the method that we use are indeed general

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

akaike s information criterion in generalized estimating equations correlated response data are common in biomedical studies regression analysis based on the generalized estimating equations gee is an increasingly important method for such data however there seem to be few model selection criteria available in gee the well known akaike information criterion aic cannot be directly applied since aic is based on maximum likelihood estimation while gee is nonlikelihood based we propose a modification to aic where the likelihood is replaced by the quasi likelihood and a proper adjustment is made for the penalty term its performance is investigated through simulation studies for illustration the method is applied to a real data set

jackknife bias reduction for polychotomous logistic regression despite theoretical and empirical evidence that the usual mles can be misleading in finite samples and some evidence that bias reduced estimates are less biased and more efficient they have not seen a wide application in practice one can obtain bias reduced estimates by jackknife methods with or without full iteration or by use of higher order terms in a taylor series expansion of the log likelihood to approximate asymptotic bias we provide details of these methods for polychotomous logistic regression with a nominal categorical response we conducted a monte carlo comparison of the jackknife and taylor series estimates in moderate sample sizes in a general logistic regression setting to investigate dichotomous and trichotomous responses and a mixture of correlated and uncorrelated binary and normal covariates we found an approximate two step jackknife and the taylor series methods useful when the ratio of the number of observations to the number of parameters is greater than 15 but we cannot recommend the two step and the fully iterated jackknife estimates when this ratio is less than 20 especially when there are large effects binary covariates or multicollinearity in the covariates

penalized least squares regression methods and applications to neuroimaging the goals of this paper are to review the most popular methods of predictor selection in regression models to explain why some fail when the number p of explanatory variables exceeds the number n of participants and to discuss alternative statistical methods that can be employed in this case we focus on penalized least squares methods in regression models and discuss in detail two such methods that are well established in the statistical literature the lasso and elastic net we introduce bootstrap enhancements of these methods the be lasso and be enet that allow the user to attach a measure of uncertainty to each variable selected our work is motivated by a multimodal neuroimaging dataset that consists of morphometric measures volumes at several anatomical regions of interest white matter integrity measures from diffusion weighted data fractional anisotropy mean diffusivity axial diffusivity and radial diffusivity and clinical and demographic variables age education alcohol and drug history in this dataset the number p of explanatory variables exceeds the number n of participants we use the be lasso and be enet to provide the first statistical analysis that allows the assessment of neurocognitive performance from high dimensional neuroimaging and clinical predictors including their interactions the major novelty of this analysis is that biomarker selection and dimension reduction are accomplished with a view towards obtaining good predictions for the outcome of interest i e the neurocognitive indices unlike principal component analysis that are performed only on the predictors space independently of the outcome of interest

small sample adjustments in using the sandwich variance estimator in generalized estimating equations the generalized estimating equation gee approach is widely used in regression analyses with correlated response data under mild conditions the resulting regression coefficient estimator is consistent and asymptotically normal with its variance being consistently estimated by the so called sandwich estimator statistical inference is thus accomplished by using the asymptotic wald chi squared test however it has been noted in the literature that for small samples the sandwich estimator may not perform well and may lead to much inflated type i errors for the wald chi squared test here we propose using an approximate t or f test that takes account of the variability of the sandwich estimator the level of type i error of the proposed t or f test is guaranteed to be no larger than that of the wald chi squared test the satisfactory performance of the proposed new tests is confirmed in a simulation study our proposal also has some advantages when compared with other new approaches based on direct modifications of the sandwich estimator including the one that corrects the downward bias of the sandwich estimator in addition to hypothesis testing our result has a clear implication on constructing wald type confidence intervals or regions

least squares forecast averaging abstract this paper proposes forecast combination based on the method of mallows model averaging mma the method selects forecast weights by minimizing a mallows criterion this criterion is an asymptotically unbiased estimate of both the in sample mean squared error mse and the out of sample one step ahead mean squared forecast error msfe furthermore the mma weights are asymptotically mean square optimal in the absence of time series dependence we show how to compute mma weights in forecasting settings and investigate the performance of the method in simple but illustrative simulation environments we find that the mma forecasts have low msfe and have much lower maximum regret than other feasible forecasting methods including equal weighting bic selection weighted bic aic selection weighted aic bates granger combination predictive least squares and granger ramanathan combination

inverse groundwater modeling for hydraulic conductivity estimation using bayesian model averaging and variance window 1 this study proposes a bayesian model averaging bma method to address parameter estimation uncertainty arising from nonuniqueness in parameterization methods bma is able to incorporate multiple parameterization methods for prediction through the law of total probability and to obtain an ensemble average of hydraulic conductivity estimates two major issues in applying bma to hydraulic conductivity estimation are discussed the first problem is using occam s window in usual bma applications to measure approximated posterior model probabilities occam s window only accepts models in a very narrow range tending to single out the best method and discard other good methods we propose a variance window to replace occam s window to cope with this problem the second problem is the kashyap information criterion kic in the approximated posterior model probabilities which tends to prefer highly uncertain parameterization methods by considering the fisher information matrix with sufficient amounts of observation data the bayesian information criterion bic is a good approximation and is able to avoid controversial results from using kic this study adopts multiple generalized parameterization gp methods such as the bma models to estimate spatially correlated hydraulic conductivity numerical examples illustrate the issues of using kic and occam s window and show the advantages of using bic and the variance window in bma application finally we apply bma to the hydraulic conductivity estimation of the 1500 foot sand in east baton rouge parish louisiana

adaptive estimation of the intensity of inhomogeneous poisson processes via concentration inequalities in this paper we establish oracle inequalities for penalized projection estimators of the intensity of an inhomogeneous poisson process we study consequently the adaptive properties of penalized projection estimators at first we provide lower bounds for the minimax risk over various sets of smoothness for the intensity and then we prove that our estimators achieve these lower bounds up to some constants the crucial tools to obtain the oracle inequalities are new concentration inequalities for suprema of integral functionals of poisson processes which are analogous to talagrand s inequalities for empirical processes

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

testing the order of a model this paper deals with order identification for nested models in the i i d framework we study the asymptotic efficiency of two generalized likelihood ratio tests of the order they are based on two estimators which are proved to be strongly consistent a version of stein s lemma yields an optimal underestimation error exponent the lemma also implies that the overestimation error exponent is necessarily trivial our tests admit non trivial underestimation error exponents the optimal underestimation error exponent is achieved in some situations the overestimation error can decay exponentially fast with respect to a positive power of the number of observations these results are proved under mild assumptions by relating the underestimation resp overestimation error to large resp moderate deviations of the log likelihood process in particular it is not necessary that the classical cramer condition be satisfied namely the log densities are not required to admit every exponential moment three benchmark examples with specific difficulties location mixture of normal distributions abrupt changes and various regressions are detailed so as to illustrate the generality of our results

single index model selections we derive a new model selection criterion for single index models aicc by minimising the expected kullback leibler distance between the true and candidate models the proposed criterion selects not only relevant variables but also the smoothing parameter for an unknown link function thus it is a general selection criterion that provides a unified approach to model selection across both parametric and nonparametric functions monte carlo studies demonstrate that aicc performs satisfactorily in most situations we illustrate the practical use of aicc with an empirical example for modelling the hedonic price function for cars in addition we extend the applicability of aicc to partially linear and additive single index models language en

range image segmentation using surface selection criterion in this paper we address the problem of recovering the true underlying model of a surface while performing the segmentation first and in order to solve the model selection problem we introduce a novel criterion which is based on minimising strain energy of fitted surfaces we then evaluate its performance and compare it with many other existing model selection techniques using this criterion we then present a robust range data segmentation algorithm capable of segmenting complex objects with planar and curved surfaces the presented algorithm simultaneously identifies the type order and geometric shape of each surface and separates all the points that are part of that surface this paper includes the segmentation results of a large collection of range images obtained from objects with planar and curved surfaces the resulting segmentation algorithm successfully segments various possible types of curved objects more importantly the new technique is capable of detecting the association between separated parts of a surface which has the same cartesian equation while segmenting a scene this aspect is very useful in some industrial applications of range data analysis

bayesian methods of analysis for cluster randomized trials with binary outcome data we explore the potential of bayesian hierarchical modelling for the analysis of cluster randomized trials with binary outcome data and apply the methods to a trial randomized by general practice an approximate relationship is derived between the intracluster correlation coefficient icc and the between cluster variance used in a hierarchical logistic regression model by constructing an informative prior for the icc on the basis of available information we are thus able implicitly to specify an informative prior for the between cluster variance the approach also provides us with a credible interval for the icc for binary outcome data several approaches to constructing informative priors from empirical icc values are described we investigate the sensitivity of results to the prior specified and find that the estimate of intervention effect changes very little in this data set while its interval estimate is more sensitive the bayesian approach allows us to assume distributions other than normality for the random effects used to model the clustering this enables us to gain insight into the robustness of our parameter estimates to the classical normality assumption in a model with a more complex variance structure bayesian methods can provide credible intervals for a difference between two variance components in order for example to investigate whether the effect of intervention varies across clusters we compare our results with those obtained from classical estimation discuss the relative merits of the bayesian framework and conclude that the flexibility of the bayesian approach offers some substantial advantages although selection of prior distributions is not straightforward

intra cluster correlation selection for cluster randomized trials in this paper we give focus to cluster randomized trials also known as group randomized trials which randomize clusters or groups of subjects to different trial arms such as intervention or control outcomes from subjects within the same cluster tend to exhibit an exchangeable correlation measured by the intra cluster correlation coefficient icc our primary interest is to test if the intervention has an impact on the marginal mean of an outcome using recently developed methods we propose how to select a working icc structure with the goal of choosing the structure that results in the smallest standard errors for regression parameter estimates and thus the greatest power for this test specifically we utilize small sample corrections for the estimation of the covariance matrix of regression parameter estimates this matrix is incorporated within correlation selection criteria proposed in the generalized estimating equations literature to choose one of multiple working icc structures under consideration we demonstrate the potential power and utility of this approach when used in cluster randomized trial settings via a simulation study and application example and we discuss practical considerations for its use in practice copyright 2016 john wiley sons ltd

improving small sample inference in group randomized trials with binary outcomes group randomized trials grts randomize groups of people to treatment or control arms instead of individually randomizing subjects when each subject has a binary outcome over dispersed binomial data may result quantified as an intra cluster correlation icc typically grts have a small number bin of independent clusters each of which can be quite large treating the icc as a nuisance parameter inference for a treatment effect can be done using quasi likelihood with a logistic link a wald statistic which under standard regularity conditions has an asymptotic standard normal distribution can be used to test for a marginal treatment effect however we have found in our setting that the wald statistic may have a variance less than 1 resulting in a test size smaller than its nominal value this problem is most apparent when marginal probabilities are close to 0 or 1 particularly when n is small and the icc is not negligible when the icc is known we develop a method for adjusting the estimated standard error appropriately such that the wald statistic will approximately have a standard normal distribution we also propose ways to handle non nominal test sizes when the icc is estimated we demonstrate the utility of our methods through simulation results covering a variety of realistic settings for grts

subspace information criterion for nonquadratic regularizers model selection for sparse regressors nonquadratic regularizers in particular the l sub 1 norm regularizer can yield sparse solutions that generalize well in this work we propose the generalized subspace information criterion gsic that allows to predict the generalization error for this useful family of regularizers we show that under some technical assumptions gsic is an asymptotically unbiased estimator of the generalization error gsic is demonstrated to have a good performance in experiments with the l sub 1 norm regularizer as we compare with the network information criterion nic and cross validation in relatively large sample cases however in the small sample case gsic tends to fail to capture the optimal model due to its large variance therefore also a biased version of gsic is introduced which achieves reliable model selection in the relevant and challenging scenario of high dimensional data and few samples

estimating intraclass correlation for binary data summary this paper reviews many different estimators of intraclass correlation that have been proposed for binary data and compares them in an extensive simulation study some of the estimators are very specific while others result from general methods such as pseudo likelihood and extended quasi likelihood estimation the simulation study identifies several useful estimators one of which does not seem to have been considered previously for binary data estimators based on extended quasi likelihood are found to have a substantial bias in some circumstances

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

model selection using wavelet decomposition and applications in this paper we discuss how to use wavelet decompositions to select a regression model the methodology relies on a minimum description length criterion which is used to determine the number of nonzero coefficients in the vector of wavelet coefficients consistency properties of the selection rule are established and simulation studies reveal information on the distribution of the minimum description length selector we then apply the selection rule to specific problems including testing for pure white noise the power of this test is investigated via simulation studies and the selection criterion is also applied to testing for no effect in nonparametric regression

a new approach to variable selection using the tls approach the problem of variable selection is one of the most important model selection problems in statistical applications it is also known as the subset selection problem and arises when one wants to explain the observations or data adequately by a subset of possible explanatory variables the objective is to identify factors of importance and to include only variables that contribute significantly to the reduction of the prediction error numerous selection procedures have been proposed in the classical multiple linear regression model we extend one of the most popular methods developed in this context the backward selection procedure to a more general class of models in the basic linear regression model errors are present on the observations only if errors are present on the regressors as well one gets the errors in variables model which for gaussian noise becomes the total least squares tls model this is the context considered here

an extended quasi likelihood function summary wedderburn s original definition of quasi likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions the relationship between generalized linear models and the use of transformations of the response variable is explored and the ideas are illustrated by three examples generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined in making this extension wedderburn widened the scope of generalized linear models in a way very similar to that of gauss when he replaced the assumption of normality in classical linear models by that of equal variance for generalized linear models with distributions in the exponential family likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions these methods are also applicable with wedderburn s form of quasi likelihood however neither of these methods is suitable for the comparison of different variance functions in this paper we introduce an extended quasi likelihood function which allows for the comparison of various forms of all the components of a generalized linear model i e the linear predictor the link function and the variance function we then apply the ideas to the analysis of several sets of 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

multi parameter regularization techniques for ill conditioned linear systems when a system of linear equations is ill conditioned regularization techniques provide a quite useful tool for trying to overcome the numerical inherent difficulties the ill conditioned system is replaced by another one whose solution depends on a regularization term formed by a scalar and a matrix which are to be chosen in this paper we consider the case of several regularizations terms added simultaneously thus overcoming the problem of the best choice of the regularization matrix the error of this procedure is analyzed and numerical results prove its efficiency

model selection in estimating equations summary model selection is a necessary step in many practical regression analyses but for methods based on estimating equations such as the quasi likelihood and generalized estimating equation gee approaches there seem to be few well studied model selection techniques in this article we propose a new model selection criterion that minimizes the expected predictive bias epb of estimating equations a bootstrap smoothed cross validation bcv estimate of epb is presented and its performance is assessed via simulation for overdispersed generalized linear models for illustration the method is applied to a real data set taken from a study of the development of ewe embryos

testing the fit of a parametric function abstract general methods for testing the fit of a parametric function are proposed the idea underlying each method is to accept the prescribed parametric model if and only if it is chosen by a model selection criterion several different selection criteria are considered including one based on a modified version of the akaike information criterion and others based on various score statistics the tests have a connection with nonparametric smoothing because they use orthogonal series estimators to detect departures from a parametric model an important aspect of the tests is that they can be applied in a wide variety of settings including generalized linear models spectral analysis the goodness of fit problem and longitudinal data analysis implementation using standard statistical software is straightforward asymptotic distribution theory for several test statistics is described and the tests are shown to be consistent against essentially any alternative hypothesis simulations and a data exampl

on the robust variance estimator in generalised estimating equations summary the variance matrix of the estimated regression coefficient in the liang zeger generalised estimating equation approach can be consistently estimated by the so called sandwich or robust estimator in this note we propose a modification to the liang zeger prescription for implementing the robust variance estimator analytical and numerical evidence shows the superior performance of our proposal

sample size calculations for clustered binary data in this paper we propose a sample size calculation method for testing on a binomial proportion when binary observations are dependent within clusters in estimating the binomial proportion in clustered binary data two weighting systems have been popular equal weights to clusters and equal weights to units within clusters when the number of units varies cluster by cluster performance of these two weighting systems depends on the extent of correlation among units within each cluster in addition to them we will also use an optimal weighting method that minimizes the variance of the estimator a sample size formula is derived for each of the estimators with different weighting schemes we apply these methods to the sample size calculation for the sensitivity of a periodontal diagnostic test simulation studies are conducted to evaluate a finite sample performance of the three estimators we also assess the influence of misspecified input parameter values on the calculated sample size the optimal estimator requires equal or smaller sample sizes and is more robust to the misspecification of an input parameter than those assigning equal weights to units or clusters

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

selecting the order of autoregressive models from small samples the weak parameter criterion wpc is introduced as a means for model order selection it is based on the same principles as mallows c p and the fpe and aic criteria of akaike according to the wpc parameters are weak and should be removed if the squares of their estimates are less than twice the expectation for a white noise signal roughly speaking the square of an estimate must exceed twice its variance due to the conceptual simplicity this criterion remains useful for small samples where the asymptotical properties are no longer valid if the maximum order considered is sqrt n or less the difference between akaike s fpe and aic criteria on one hand and the wpc on the other hand remains small and the use of aic or fpe may be justified however it is advised to use the wpc for higher maximum orders by using different variances in the wpc for yule walker and for burg estimates it is achieved that the average selected wpc order in small samples depends mainly on the given data and no longer on the method of parameter estimation

toward optimal model averaging in regression models with time series errors abstract consider a regression model with infinitely many parameters and time series errors we are interested in choosing weights for averaging across generalized least squares gls estimators obtained from a set of approximating models however gls estimators depending on the unknown inverse covariance matrix of the errors are usually infeasible we therefore construct feasible generalized least squares fgls estimators using a consistent estimator of the unknown inverse matrix based on this inverse covariance matrix estimator and fgls estimators we develop a feasible autocovariance corrected mallows model averaging criterion to select weights thereby providing an fgls model averaging estimator of the true regression function we show that the generalized squared error loss of our averaging estimator is asymptotically equivalent to the minimum one among those of gls model averaging estimators with the weight vectors belonging to a continuous set which includes the discrete weight set used in hansen 2007 as its proper subset

estimation and accuracy after model selection classical statistical theory ignores model selection in assessing estimation accuracy here we consider bootstrap methods for computing standard errors and confidence intervals that take model selection into account the methodology involves bagging also known as bootstrap smoothing to tame the erratic discontinuities of selection based estimators a useful new formula for the accuracy of bagging then provides standard errors for the smoothed estimators two examples nonparametric and parametric are carried through in detail a regression model where the choice of degree linear quadratic cubic is determined by the cp criterion and a lasso based estimation problem

smoothing spline anova models for large data sets with bernoulli observations and the randomized gacv we propose the randomized generalized approximate cross validation rangacv method for choosing multiple smoothing parameters in penalized likelihood estimates for bernoulli data the method is intended for application with penalized likelihood smoothing spline anova models in addition we propose a class of approximate numerical methods for solving the penalized likelihood variational problem which in conjunction with the rangacv method allows the application of smoothing spline anova models with bernoulli data to much larger data sets than previously possible these methods are based on choosing an approximating subset of the natural representer basis functions for the variational problem simulation studies with synthetic data including synthetic data mimicking demographic risk factor data sets is used to examine the properties of the method and to compare the approach with the grkpack code of wang 1997c bayesian confidence intervals are obtained for the fits and are shown in the simulation studies to have the across the function property usually claimed for these confidence intervals finally the method is applied to an observational data set from the beaver dam eye study with scientifically interesting results

binary regression using an extended beta binomial distribution with discussion of correlation induced by covariate measurement errors abstract the beta binomial distribution is extended to allow negative correlations among binary variates within an experimental unit regression models are proposed for both the binary variate response rate and for the pairwise correlation between binary variates and corresponding likelihood estimation procedures are described binary regression problems are also considered in which measurement errors in the regression variables represent the sole source of overdispersion some corresponding response rate estimation procedures are described and illustrated

discussion local rademacher complexities and oracle inequalities in risk minimization in this magnificent paper professor koltchinskii offers general and powerful performance bounds for empirical risk minimization a fundamental principle of statistical learning theory since the elegant pioneering work of vapnik and chervonenkis in the early 1970s various such bounds have been known that relate the performance of empirical risk minimizers to combinatorial and geometrical features of the class over which the minimization is performed this area of research has been a rich source of motivation and a major field of applications of empirical process theory the appearance of advanced concentration inequalities in the 1990s primarily thanks to talagrand s influential work provoked major advances in both empirical process theory and statistical learning theory and led to a much deeper understanding of some of the basic phenomena in the discussed paper professor koltchinskii develops a powerful new methodology iterative localization which with the help of concentration inequalities is able to explain most of the recent results and go significantly beyond them in many cases the main motivation behind professor koltchinskii s paper is based on classical problems of statistical learning theory such as binary classification and regression in which given a sample xi y i i 1 n of independent and identically distributed pairs of random variables where the xi take their values in some feature space x and the yi are say real valued the goal is to find a function f x r whose risk defined in terms of the expected value of an appropriately chosen loss function is as small as possible in the remaining part of this discussion we point out how the performance bounds of professor koltchinskii s paper can be used to study a seemingly different model motivated by nonparametric ranking problems which has received increasing attention both in the statistical and machine learning literature indeed in several applications such as the search engine problem or credit risk screening the goal is to learn how to rank or to score observations rather than just classify them in this case performance measures involve pairs of observations as can be seen for instance with the auc area under an roc curve criterion in this

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

a regularization approach to the many instruments problem abstract this paper focuses on the estimation of a finite dimensional parameter in a linear model where the number of instruments is very large or infinite in order to improve the small sample properties of standard instrumental variable iv estimators we propose three modified iv estimators based on three different ways of inverting the covariance matrix of the instruments these inverses involve a regularization or smoothing parameter it should be stressed that no restriction on the number of instruments is needed and that all the instruments are used in the estimation we show that the three estimators are asymptotically normal and attain the semiparametric efficiency bound higher order analysis of the mse reveals that the bias of the modified estimators does not depend on the number of instruments finally we suggest a data driven method for selecting the regularization parameter interestingly our regularization techniques lead to a consistent nonparametric estimation of the optimal instrument

a covariance estimator for gee with improved small sample properties summary in this paper we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations gee hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small resampling methods such as the jackknife and bootstrap have been suggested for covariance estimation when the number of clusters is small a drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near zero cell counts caused by resampling we propose a bias corrected covariance estimator that avoids this problem in a small simulation study we compare the bias corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10 40 subjects with equal and unequal cluster sizes of 16 64 observations the bias corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2 3 times the nominal level the methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease

data adaptive rank shaping methods for solving least squares problems there are two types of problems in the theory of least squares signal processing parameter estimation and signal extraction parameter estimation is called inversion and signal extraction is called filtering in this paper we present a unified theory of rank shaping for solving overdetermined and underdetermined versions of these problems we develop several data dependent rank shaping methods and evaluate their performance our key result is a data adaptive wiener filter that automatically adjusts its gains to accommodate realizations that are a priori unlikely the adaptive filter dramatically outperforms the wiener filter on a typical realizations and just slightly under performs it on typical realizations this is the most one can hope for in a data adaptive filter

adaptive estimation in autoregression or mixing regression via model selection we study the problem of estimatingsome unknown regression function in a beta mixing dependent framework to this end we consider some collection of models which are finite dimensional spaces a penalized least squares estimator plse is built on a data driven selected model among this collection we state non asymptotic risk bounds for this plse and give several examples where the procedure can be applied autoregression regression with arithmetically beta mixing design points regression with mixing errors estimation in additive frameworks estimation of the order of the autoregression in addition we show that under a weak moment condition on the errors our estimator is adaptive in the minimax sense simultaneously over some family of besov balls

a note on the efficiency of sandwich covariance matrix estimation the sandwich estimator also known as robust covariance matrix estimator heteroscedasticity consistent covariance matrix estimate or empirical covariance matrix estimator has achieved increasing use in the econometric literature as well as with the growing popularity of generalized estimating equations its virtue is that it provides consistent estimates of the covariance matrix for parameter estimates even when the fitted parametric model fails to hold or is not even specified surprisingly though there has been little discussion of properties of the sandwich method other than consistency we investigate the sandwich estimator in quasi likelihood models asymptotically and in the linear case analytically we show that under certain circumstances when the quasi likelihood model is correct the sandwich estimate is often far more variable than the usual parametric variance estimate the increased variance is a fixed feature of the method and the price that one pays to obtain consistency even when the p

disaggregation procedures for generating serially correlated flow vectors the structure of disaggregation models places severe constraints on the feasible values of the lagged covariance of generated flow vectors a new and simple class of disaggregation models is presented which employ the valencia schaake disaggregation model structure but allow the models innovations to be serially correlated these models can reproduce 1 the covariance matrix of the disaggregated flows 2 their covariance with the upper level flows and 3 reasonable approximations to the lag one covariance of the disaggregated flow vectors given the constraints imposed by a disaggregation approach the mejia rousselle disaggregation model is shown in general to fail to reproduce the anticipated variances and covariances of the disaggregated flows because the model and its parameter estimators are not self consistent the paper closes with a discussion of practical modeling considerations and of staged disaggregation procedures which reduce the size of multisite multiseason models

monte carlo sure a black box optimization of regularization parameters for general denoising algorithms we consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white gaussian noise to achieve this we propose to minimize stein s unbiased risk estimate sure which provides a means of assessing the true mean squared error mse purely from the measured data without need for any knowledge about the noise free signal specifically we present a novel monte carlo technique which enables the user to calculate sure for an arbitrary denoising algorithm characterized by some specific parameter setting our method is a black box approach which solely uses the response of the denoising operator to additional input noise and does not ask for any information about its functional form this therefore permits the use of sure for optimization of a wide variety of denoising algorithms we justify our claims by presenting experimental results for sure based optimization of a series of popular image denoising algorithms such as total variation denoising wavelet soft thresholding and wiener filtering smoothing splines in the process we also compare the performance of these methods we demonstrate numerically that sure computed using the new approach accurately predicts the true mse for all the considered algorithms we also show that sure uncovers the optimal values of the parameters in all cases

a group bridge approach for variable selection in multiple regression problems when covariates can be naturally grouped it is important to carry out feature selection at the group and within group individual variable levels simultaneously the existing methods including the lasso and group lasso are designed for either variable selection or group selection but not for both we propose a group bridge approach that is capable of simultaneous selection at both the group and within group individual variable levels the proposed approach is a penalized regularization method that uses a specially designed group bridge penalty it has the oracle group selection property in that it can correctly select important groups with probability converging to one in contrast the group lasso and group least angle regression methods in general do not possess such an oracle property in group selection simulation studies indicate that the group bridge has superior performance in group and individual variable selection relative to several existing methods

generalized finite element analysis with t complete boundary solution functions the use of a complete and nonsingular set of trefftz functions in the solution of quasi harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary type approximation procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated

accumulated prediction errors information criteria and optimal forecasting for autoregressive time series the predictive capability of a modification of rissanen s accumulated prediction error ape criterion ape n is investigated in infinite order autoregressive ar oo models instead of accumulating squares of sequential prediction errors from the beginning ape n is obtained by summing these squared errors from stage n n where n is the sample size and 1 n n 1 1 n may depend on n under certain regularity conditions an asymptotic expression is derived for the mean squared prediction error mspe of an ar predictor with order determined by ape n this expression shows that the prediction performance of ape n can vary dramatically depending on the choice of n another interesting finding is that when n approaches 1 at a certain rate ape n can achieve asymptotic efficiency in most practical situations an asymptotic equivalence between ape n and an information criterion with a suitable penalty term is also established from the mspe point of view this offers new perspectives for understanding the information and prediction based model selection criteria finally we provide the first asymptotic efficiency result for the case when the underlying ar oo model is allowed to degenerate to a finite autoregression

small sample adjustments for wald type tests using sandwich estimators the sandwich estimator of variance may be used to create robust wald type tests from estimating equations that are sums of k independent or approximately independent terms for example for repeated measures data on k individuals each term relates to a different individual these tests applied to a parameter may have greater than nominal size if k is small or more generally if the parameter to be tested is essentially estimated from a small number of terms in the estimating equation we offer some practical modifications to these robust wald type tests which asymptotically approach the usual robust wald type tests we show that one of these modifications provides exact coverage for a simple case and examine by simulation the modifications applied to the generalized estimating equations of liang and zeger 1986 conditional logistic regression and the cox proportional hazard model

asymptotic analysis of the squared estimation error in misspecified factor models abstract in this paper we obtain asymptotic approximations to the squared error of the least squares estimator of the common component in large approximate factor models with possibly misspecified number of factors the approximations are derived under both strong and weak factors asymptotics assuming that the cross sectional and temporal dimensions of the data are comparable we develop consistent estimators of these approximations and propose to use them for model comparison and for selection of the number of factors we show that the estimators of the number of factors that minimize these loss estimators are asymptotically loss efficient in the sense of shibata 1980 li 1987 and shao 1997

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

modeling of two dimensional random fields by parametric cepstrum a method is presented for parametric modeling of stationary random fields the class of parametric models considered allows the most general elliptic field and by linear constraints can include such special cases as isotropic quarter plane and separable fields the technique based on the cepstrum has the great advantage of requiring only the use of fast fourier transforms in the fitting process thus unlike the fitting of two dimensional autoregressions no iteration is necessary other advantages are that any wiener filters constructed from the fitted spectrum are guaranteed to be stable and that the spectrum is guaranteed to be positive statistical tests for determining various special types of field from data are developed the choice of model order is discussed as well

kernel smoothing of data with correlated errors abstract kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y f x e where f x is a smooth deterministic mean function and e is an error process with mean zero in this article the mean squared error of kernel estimators is computed for processes with correlated errors and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence the standard techniques for bandwidth selection such as cross validation and generalized cross validation are shown to perform very badly when the errors are correlated standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative the selection criteria can however be adjusted to correct for the effect of correlation in simulations the standard selection criteria are shown to behave as predicted the corrected criteria are shown to be very effective when the correlation functi

convergence of the red tower method for removing noise from data by coupling the wavelet transform with a particular nonlinear shrinking function the red telescopic optimal wavelet estimation of the risk tower method is introduced for removing noise from signals it is shown that the method yields convergence of the l sub 2 risk to the actual solution with optimal rate moreover the method is proved to be asymptotically efficient when the regularization parameter is selected by the generalized cross validation criterion gcv or the mallows criterion numerical experiments based on synthetic data are provided to compare the performance of the red tower method with hard thresholding soft thresholding and neigh coeff thresholding furthermore the numerical tests are also performed when the tower method is applied to hard thresholding soft thresholding and neigh coeff thresholding for which the full convergence results are still open

improved hypothesis testing for coefficients in generalized estimating equations with small samples of clusters the sandwich standard error estimator is commonly used for making inferences about parameter estimates found as solutions to generalized estimating equations gee for clustered data the sandwich tends to underestimate the variability in the parameter estimates when the number of clusters is small and reference distributions commonly used for hypothesis testing poorly approximate the distribution of wald test statistics consequently tests have greater than nominal type i error rates we propose tests that use bias reduced linearization brl to adjust the sandwich estimator and satterthwaite or saddlepoint approximations for the reference distribution of resulting wald t tests we conducted a large simulation study of tests using a variety of estimators traditional sandwich brl mancl and derouen s bc estimator and a modification of an estimator proposed by kott and approximations to reference distributions under diverse settings that varied the distribution of the explanatory variables the values of coefficients and the degree of intra cluster correlation icc our new method generally worked well providing accurate estimates of the variability of fitted coefficients and tests with near nominal type i error rates when the icc is small our method works less well when the icc is large but it continues to out perform the traditional sandwich and other alternatives

optimal model assessment selection and combination central to statistical theory and application is statistical modeling which typically involves choosing a single model or combining a number of models of different sizes and from different sources whereas model selection seeks a single best modeling procedure model combination combines the strength of different modeling procedures in this article we look at several key issues and argue that model assessment is the key to model selection and combination most important we introduce a general technique of optimal model assessment based on data perturbation thus yielding optimal selection in particular model selection and combination from a frequentist perspective we advocate model combination over a selected subset of modeling procedures because it controls bias while reducing variability hence yielding better performance in terms of the accuracy of estimation and prediction to realize the potential of model combination we develop methodologies for determining the optimal tuning parameter such

parametric or nonparametric a parametricness index for model selection in model selection literature two classes of criteria perform well asymptotically in different situations bayesian information criterion bic as a representative is consistent in selection when the true model is finite dimensional parametric scenario akaike s information criterion aic performs well in an asymptotic efficiency when the true model is infinite dimensional nonparametric scenario but there is little work that addresses if it is possible and how to detect the situation that a specific model selection problem is in in this work we differentiate the two scenarios theoretically under some conditions we develop a measure parametricness index pi to assess whether a model selected by a potentially consistent procedure can be practically treated as the true model which also hints on aic or bic is better suited for the data for the goal of estimating the regression function a consequence is that by switching between aic and bic based on the pi the resulting regression estimator is simultaneously asymptotically efficient for both parametric and nonparametric scenarios in addition we systematically investigate the behaviors of pi in simulation and real data and show its usefulness

some notes on the nonnegative garrote some notes on the use of the nonnegative garrote ng are given in this article the main result is that compared with other penalized least squares methods the ng has a natural selection of penalty function according to an estimator of prediction risk furthermore two natural estimators of the tuning parameter which only involve very simple computations are proposed corresponding to the akaike information criterion aic and bayesian information criterion bic respectively this indicates that to select tuning parameters it may be unnecessary to optimize a model selection criterion repeatedly several reasonable ng estimators with natural tuning parameters are proposed for multicollinearity problems and other problems simulation results and a real data analysis are reported for testing the results obtained previously

predictive value of statistical models a review is given of different ways of estimating the error rate of a prediction rule based on a statistical model a distinction is drawn between apparent optimum and actual error rates moreover it is shown how cross validation can be used to obtain an adjusted predictor with smaller error rate a detailed discussion is given for ordinary least squares logistic regression and cox regression in survival analysis finally the splitsample approach is discussed and demonstrated on two data sets

kernel based partially linear models and nonlinear identification in this note we propose partially linear models with least squares support vector machines ls svms for nonlinear arx models we illustrate how full black box models can be improved when prior information about model structure is available a real life example based on the silverbox benchmark data shows significant improvements in the generalization ability of the structured model with respect to the full black box model reflected also by a reduction in the effective number of parameters

robust estimation in heteroscedastic linear models we consider a heteroscedastic linear model in which the variances are given by a parametric function of the mean responses and a parameter theta we propose robust estimates for the regression parameter beta and show that as long as a reasonable starting estimate of theta is available our estimates of beta are asymptotically equivalent to the natural estimate obtained with known variances a particular method for estimating theta is proposed and shown by monte carlo to work quite well especially in power and exponential models for the variances we also briefly discuss a feedback estimate of beta

on the use of the quasi likelihood method in teratological experiments this paper concerns the impact of litter effects on the inference of dose response relationships in teratological experiments with binary response kupper et al 1986 biometrics 42 85 98 concluded that when intra litter correlations are dose dependent the use of a common intra litter correlation likelihood based on the beta binomial distribution could lead to severe bias in estimation of the parameter beta which characterizes the dose response relationship we show here that the problems of bias and coverage probability for beta could still be substantial when one uses the likelihood with heterogeneous intra litter correlations we then examine through a simulation study the performance of the quasi likelihood method wedderburn 1974 biometrika 61 439 447 and recommend that this method with a common intra litter correlation parameter be used when the number of litters is small or modest

the focused information criterion a variety of model selection criteria have been developed of general and specific types most of these aim at selecting a single model with good overall properties for example formulated via average prediction quality or shortest estimated overall distance to the true model the akaike the bayesian and the deviance information criteria along with many suitable variations are examples of such methods these methods are not concerned however with the actual use of the selected model which varies with context and application the present article takes the view that the model selector should instead focus on the parameter singled out for interest in particular a model that gives good precision for one estimand may be worse when used for inference for another estimand we develop a method that for a given focus parameter estimates the precision of any submodel based estimator the framework is that of large sample likelihood inference using an unbiased estimate of limiting risk we propose a foc

a high dimensional wilks phenomenon a theorem by wilks asserts that in smooth parametric density estimation the difference between the maximum likelihood and the likelihood of the sampling distribution converges toward a chi square distribution where the number of degrees of freedom coincides with the model dimension this observation is at the core of some goodness of fit testing procedures and of some classical model selection methods this paper describes a non asymptotic version of the wilks phenomenon in bounded contrast optimization procedures using concentration inequalities for general functions of independent random variables it proves that in bounded contrast minimization as for example in statistical learning theory the difference between the empirical risk of the minimizer of the true risk in the model and the minimum of the empirical risk the excess empirical risk satisfies a bernstein like inequality where the variance term reflects the dimension of the model and the scale term reflects the noise conditions from a mathematical statistics viewpoint the significance of this result comes from the recent observation that when using model selection via penalization the excess empirical risk represents a minimum penalty if non asymptotic guarantees concerning prediction error are to be provided from the perspective of empirical process theory this paper describes a concentration inequality for the supremum of a bounded non centered actually non positive empirical process combining the now classical analysis of m estimation building on talagrand s inequality for suprema of empirical processes and versatile moment inequalities for functions of independent random variables this paper develops a genuine bernstein like inequality that seems beyond the reach of traditional tools

hybrid shrinkage estimators using penalty bases for the ordinal one way layout this paper constructs improved estimators of the means in the gaussian saturated one way layout with an ordinal factor the least squares estimator for the mean vector in this saturated model is usually inadmissible the hybrid shrinkage estimators of this paper exploit the possibility of slow variation in the dependence of the means on the ordered factor levels but do not assume it and respond well to faster variation if present to motivate the development candidate penalized least squares pls estimators for the mean vector of a one way layout are represented as shrinkage estimators relative to the penalty basis for the regression space this canonical representation suggests further classes of candidate estimators for the unknown means monotone shrinkage ms estimators or soft thresholding st estimators or most generally hybrid shrinkage hs estimators that combine the preceding two strategies adaptation selects the estimator within a candidate class that minimizes estimated risk under the gaussian saturated one way layout model such adaptive estimators minimize risk asymptotically over the class of candidate estimators as the number of factor levels tends to infinity thereby adaptive hs estimators asymptotically dominate adaptive ms and adaptive st estimators as well as the least squares estimator local annihilators of polynomials among them difference operators generate penalty bases suitable for a range of numerical examples in case studies adaptive hs estimators recover high frequency details in the mean vector more reliably than pls or ms estimators and low frequency details more reliably than st estimators

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

bias correction in generalized linear models summary in this paper we derive general formulae for first order biases of maximum likelihood estimates of the linear parameters linear predictors the dispersion parameter and fitted values in generalized linear models these formulae may be implemented in the glim program to compute bias corrected maximum likelihood estimates to order n 1 where n is the sample size with minimal effort by means of a supplementary weighted regression for linear logistic models it is shown that the asymptotic bias vector of j8 is almost collinear with f3 the approximate formula flp m for the bias of s in logistic models where p dim and m e mi is the sum of the binomial indices is derived and checked numerically

variable selection in semiparametric regression modeling in this paper we are concerned with how to select significant variables in semiparametric modeling variable selection for semiparametric regression models consists of two components model selection for nonparametric components and select significant variables for parametric portion thus it is much more challenging than that for parametric models such as linear models and generalized linear models because traditional variable selection procedures including stepwise regression and the best subset selection require model selection to nonparametric components for each submodel this leads to very heavy computational burden in this paper we propose a class of variable selection procedures for semiparametric regression models using nonconcave penalized likelihood the newly proposed procedures are distinguished from the traditional ones in that they delete insignificant variables and estimate the coefficients of significant variables simultaneously this allows us to establish the sampling properties of the resulting estimate we first establish the rate of convergence of the resulting estimate with proper choices of penalty functions and regularization parameters we then establish the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures perform as well as an oracle procedure semiparametric generalized likelihood ratio test is proposed to select significant variables in the nonparametric component we investigate the asymptotic behavior of the proposed test and demonstrate its limiting null distribution follows a chi squared distribution which is independent of the nuisance parameters extensive monte carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedures