prediction error and its estimation for subset selected models strategies are compared for development of a linear regression model and the subsequent assessment of its predictive ability simulations were performed as a designed experiment over a range of data structures approaches using a forward selection of variables resulted in slightly smaller prediction errors and less biased estimators of predictive accuracy than all possible subsets selection but often did not improve on the full model random and balanced data splitting resulted in increased prediction errors and estimators with large mean squared error to maximize predictive accuracy while retaining a reliable estimate of that accuracy it is recommended that the entire sample usually he used for model development and assessment

bounds on the length of functional pir and batch codes a functional k private information retrieval k pir code of dimension s consists of n servers storing linear combinations of s linearly independent information symbols any linear combination of the s information symbols can be recovered by k disjoint subsets of servers the goal is to find the minimum number of servers for given k and s we provide lower bounds on the minimum number of servers and constructions which yield upper bounds on this number for k 4 exact bounds on this number are proved furthermore we provide some asymptotic bounds the problem coincides with the well known pir problem based on a coded database to reduce the storage overhead when each linear combination contains exactly one information symbol if any multiset of size k of linear combinations from the linearly independent information symbols can be recovered by k disjoint subset of servers then the servers form a functional k batch code a functional k batch code is a functional k pir code where all the k linear combinations in the multiset are equal we provide some bounds on the minimum number of servers for functional k batch codes in particular we present a random construction and a construction based on simplex codes write once memory wom codes and random i o rio codes

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

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

binary batch codes with improved redundancy a primitive k batch code encodes a string boldsymbol x of length n into a string boldsymbol y of length n such that each multiset of k symbols from boldsymbol x has k mutually disjoint recovering sets from boldsymbol y in this paper we discuss new constructions of binary primitive batch codes first we develop novel explicit and random coding constructions of linear primitive batch codes based on finite geometries second a new explicit coding construction of binary primitive batch codes based on bivariate lifted multiplicity codes is provided for any k n varepsilon with varepsilon in 0 0 47 setminus 1 5 1 4 our proposed codes have a better trade off between the redundancy and the parameters k n than previously known batch codes

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

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

sequential selection procedures and false discovery rate control summary we consider a multiple hypothesis testing setting where the hypotheses are ordered and one is only permitted to reject an initial contiguous block of hypotheses a rejection rule in this setting amounts to a procedure for choosing the stopping point k this setting is inspired by the sequential nature of many model selection problems where choosing a stopping point or a model is equivalent to rejecting all hypotheses up to that point and none thereafter we propose two new testing procedures and prove that they control the false discovery rate in the ordered testing setting we also show how the methods can be applied to model selection by using recent results on p values in sequential model selection settings

a bottom up approach to testing hypotheses that have a branching tree dependence structure with error rate control modern statistical analyses often involve testing large numbers of hypotheses in many situations these hypotheses may have an underlying tree structure that both helps determine the order that te

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

bayesian variable selection in linear regression abstract this article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable it is based on a bayesian approach intended to be as objective as possible a probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model the method is not fully bayesian however because the ultimate choice of prior distribution from this family is affected by the data it is assumed that the predictors represent distinct observables the corresponding regression coefficients are assigned independent prior distributions for each regression coefficient subject to deletion from the model the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere that is a spike and slab distribution the random error component is assigned a normal distribution with mean 0 and standard deviation

controlling variable selection by the addition of pseudovariables we propose a new approach to variable selection designed to control the false selection rate fsr defined as the proportion of uninformative variables included in selected models the method works by adding a known number of pseudovariables to the real dataset running a variable selection procedure and monitoring the proportion of pseudovariables falsely selected information obtained from bootstrap like replications of this process is used to estimate the proportion of falsely selected real variables and to tune the selection procedure to control the fsr

a robust version of mallows s c p abstract we present a robust version of mallows s c p for regression models it is defined by rc p w p 2 u p v p where w p i 2 i r 2 i is a weighted residual sum of squares computed from a robust fit of model p 2 is a robust and consistent estimator of 2 in the full model and u p and v p are constants depending on the weight function and the number of parameters in model p good subset models are those with rc p close to v p or smaller than v p when the weights are identically 1 w p becomes the residual sum of squares of a least squares fit and rc p reduces to mallows s c p the robust model selection procedure based on rc p allows us to choose the models that fit the majority of the data by taking into account the presence of outliers and possible departures from the normality assumption on the error distribution together with the classical c p the robust version suggests several models from which we can choose

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

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

predictive model selection we consider the problem of selecting one model from a large class of plausible models a predictive bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model using criteria derived from a certain predictive density and a prior specification that emphasizes the observables we implement the proposed methodology for three common problems arising in normal linear models variable subset selection selection of a transformation of predictor variables and estimation of a parametric variance function interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria relationships between the proposed criteria and other well known criteria are examined

on ky fan s section theorem abstract in this paper we first give a new section theorem of ky fan type and its equivalent theorem in an h space and next as their applications we obtain some minimax theorems and a coincidence theorem

the covariance inflation criterion for adaptive model selection we propose a new criterion for model selection in prediction problems the covariance inflation criterion adjusts the training error by the average covariance of the predictions and responses when the prediction rule is applied to permuted versions of the data set this criterion can be applied to general prediction problems e g regression or classification and to general prediction rules e g stepwise regression tree based models and neural nets as a by product we obtain a measure of the effective number of parameters used by an adaptive procedure we relate the covariance inflation criterion to other model selection procedures and illustrate its use in some regression and classification problems we also revisit the conditional bootstrap approach to model selection

variable selection in data mining building a predictive model for bankruptcy we predict the onset of personal bankruptcy using least squares regression although well publicized only 2 244 bankruptcies occur in our dataset of 2 9 million months of credit card activity we use stepwise selection to find predictors of these from a mix of payment history debt load demographics and their interactions this combination of rare responses and over 67 000 possible predictors leads to a challenging modeling question how does one separate coincidental from useful predictors we show that three modifications turn stepwise regression into an effective methodology for predicting bankruptcy our version of stepwise regression 1 organizes calculations to accommodate interactions 2 exploits modern decision theoretic criteria to choose predictors and 3 conservatively estimates p values to handle sparse data and a binary response omitting any one of these leads to poor performance a final step in our procedure calibrates regression predictions with these modifications stepwise regr

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

two stage sensitivity based group screening in computer experiments sophisticated computer codes that implement mathematical models of physical processes can involve large numbers of inputs and screening to determine the most active inputs is critical for understanding the input output relationship this article presents a new two stage group screening methodology for identifying active inputs in stage 1 groups of inputs showing low activity are screened out in stage 2 individual inputs from the active groups are identified inputs are evaluated through their estimated total effect sensitivity indices tsis which are compared with a benchmark null tsi distribution created from added low noise inputs examples show that compared with other procedures the proposed method provides more consistent and accurate results for high dimensional screening additional details and computer code are provided in supplementary materials available online

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

rejoinder gene hunting with hidden markov model knockoffs in this paper we deepen and enlarge the reflection on the possible advantages of a knockoff approach to genome wide association studies sesia et al 2018 starting from the discussions in bottolo richardson 2019 jewell witten 2019 rosenblatt et al 2019 and marchini 2019 the discussants bring up a number of important points either related to the knockoffs methodology in general or to its specific application to genetic studies in the following we offer some clarifications mention relevant recent developments and highlight some of the still open problems

on the choice of a model to fit data from an exponential family let x 1 cdots x n be iid observations coming from an exponential family the problem of interest is this given a finite number of models m j smoothly curved manifolds in mathbb r k choose the best model to fit the observations with some penalty for choosing models with dimensions which are too large a result of schwarz is made more specific and is extended to the case where the models are curved manifolds if s y n j is up to a constant c n independent of the model the log of the posterior probability of the j th model where the sample mean y n 1 n sum n i 1 x i has been replaced by y schwarz suggested an asymptotic expansion of s y n j whose leading terms are gamma y n j n sup psi in m j cap theta y psi b psi frac 1 2 k j log n in the case where the models are affine subspaces of mathbb r k we establish a similar asymptotic expansion including the next term with uniform bounds for y in a compact neighborhood of nabla b theta where theta is the true value of the parameter we suggest a criterion for the choice of the best model that consists of maximizing the three leading terms in the expansion s y n j we show that the criterion gives the correct model with probabilities p n theta rightarrow 1 as n rightarrow infty

tuning variable selection procedures by adding noise many variable selection methods for linear regression depend critically on tuning parameters that control the performance of the method for example entry and stay significance levels in forward and backward selection however most methods do not adapt the tuning parameters to particular datasets we propose a general strategy for adapting variable selection tuning parameters that effectively estimates the tuning parameters so that the selection method avoids overfitting and underfitting the strategy is based on the principle that overfitting and underfitting can be directly observed in estimates of the error variance after adding controlled amounts of additional independent noise to the response variable then running a variable selection method it is related to the simulation technique simex found in the measurement error literature we focus on forward selection because of its simplicity and ability to handle large numbers of explanatory variables monte carlo studies show that the new method c

multiple shrinkage and subset selection in wavelets this paper discusses bayesian methods for multiple shrinkage estimation in wavelets wavelets are used in applications for data denoising via shrinkage of the coefficients towards zero and for data compression by shrinkage and setting small coefficients to zero we approach wavelet shrinkage by using bayesian hierarchical models assigning a positive prior probability to the wavelet coefficients being zero the resulting estimator for the wavelet coefficients is a multiple shrinkage estimator that exhibits a wide variety of nonlinear patterns we discuss fast computational implementations with a focus on easy to compute analytic approximations as well as importance sampling and markov chain monte carlo methods multiple shrinkage estimators prove to have excellent mean squared error performance in reconstructing standard test functions we demonstrate this in simulated test examples comparing various implementations of multiple shrinkage to commonly used shrinkage rules finally we illustrate our approach with an application to the so called glint data

inference after variable selection in linear regression models summary we explore the impact of variable selection on statistical inferences in linear regression models in particular the generalized final prediction error criterion of shibata 1984 is considered and it is found among other things that inferences on the regression coefficients are impaired by the variable selection procedure most notably the size of the nominal confidence sets tend to be inflated if they are derived based on the selected model on the other hand variable selection does not seem to have much impact on the inferences for the error variance our results complement those obtained by potscher 1991 in which testing procedures are used for variable selection

nearly optimal constructions of pir and batch codes in this paper we study two families of codes with availability namely private information retrieval pir codes and batch codes while the former requires that every information symbol has k mutually disjoint recovering sets the latter imposes this property for each multiset request of k information symbols the main problem under this paradigm is to minimize the number of redundancy symbols we denote this value by r p n k and r b n k for pir codes and batch codes respectively where n is the number of information symbols previous results showed that for any constant k r p n k theta sqrt n and r b n k mathcal o sqrt n log n in this paper we study the asymptotic behavior of these codes for non constant k and specifically for k theta n epsilon we also study the largest value of k such that the rate of the codes approaches 1 and show that for all epsilon r p n n epsilon o n and r b n n epsilon o n furthermore several more results are proved for the case of fixed k

prior elicitation variable selection and bayesian computation for logistic regression models summary bayesian selection of variables is often difficult to carry out because of the challenge in specifying prior distributions for the regression parameters for all possible models specifying a prior distribution on the model space and computations we address these three issues for the logistic regression model for the first we propose an informative prior distribution for variable selection several theoretical and computational properties of the prior are derived and illustrated with several examples for the second we propose a method for specifying an informative prior on the model space and for the third we propose novel methods for computing the marginal distribution of the data the new computational algorithms only require gibbs samples from the full model to facilitate the computation of the prior and posterior model probabilities for all possible models several properties of the algorithms are also derived the prior specification for the first challenge focuses on the observables in that the elicitation is based on a prior prediction yo for the response vector and a quantity ao quantifying the uncertainty in yo then yo and ao are used to specify a prior for the regression coefficients semi automatically examples using real data are given to demonstrate the methodology

fixed point and maximal element theorems with applications to abstract economies and minimax inequalities in this paper we establish some fixed point theorems for a family of multivalued maps under mild conditions by using our fixed point theorems we derive some maximal element theorems for a particular family of multivalued maps namely the condensing multivalued maps as applications of our results we prove some general equilibrium existence theorems in the generalized abstract economies with preference correspondences further applications of our results are also given to minimax inequalities for a family of functions

simultaneous factor selection and collapsing levels in anova when performing an analysis of variance the investigator often has two main goals to determine which of the factors have a significant effect on the response and to detect differences among the levels of the significant factors level comparisons are done via a post hoc analysis based on pairwise differences this article proposes a novel constrained regression approach to simultaneously accomplish both goals via shrinkage within a single automated procedure the form of this shrinkage has the ability to collapse levels within a factor by setting their effects to be equal while also achieving factor selection by zeroing out entire factors using this approach also leads to the identification of a structure within each factor as levels can be automatically collapsed to form groups in contrast to the traditional pairwise comparison methods these groups are necessarily nonoverlapping so that the results are interpretable in terms of distinct subsets of levels the proposed procedure is shown to have the oracle property in that asymptotically it performs as well as if the exact structure were known beforehand a simulation and real data examples show the strong performance of the method

fast robust model selection in large datasets large datasets are increasingly common in many research fields in particular in the linear regression context it is often the case that a huge number of potential covariates are available to explain a response variable and the first step of a reasonable statistical analysis is to reduce the number of covariates this can be done in a forward selection procedure that includes selecting the variable to enter deciding to retain it or stop the selection and estimating the augmented model least squares plus t tests can be fast but the outcome of a forward selection might be suboptimal when there are outliers in this article we propose a complete algorithm for fast robust model selection including considerations for huge sample sizes because simply replacing the classical statistical criteria with robust ones is not computationally possible we develop simplified robust estimators selection criteria and testing procedures for linear regression the robust estimator is a one step weighted m estimato

variable selection via gibbs sampling abstract a crucial problem in building a multiple regression model is the selection of predictors to include the main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets this procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices in this framework the promising subsets of predictors can be identified as those with higher posterior probability the computational burden is then alleviated by using the gibbs sampler to indirectly sample from this multinomial posterior distribution on the set of possible subset choices those subsets with higher probability the promising ones can then be identified by their more frequent appearance in the gibbs sample

an optimal selection of regression variables summary an asymptotically optimal selection of regression variables is proposed the key assumption is that the number of control variables is infinite or increases with the sample size it is also shown that mallows s qp akaike s fpe and aic methods are all asymptotically equivalent to this method

an intersection theorem in l convex spaces with applications a new intersection theorem is obtained in l convex spaces without linear structure as its applications a fixed point theorem a maximal element theorem a coincidence theorem some new minimax inequalities and a saddle point theorem are given in l convex spaces our results generalize many known theorems in the literature

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