component selection and smoothing in multivariate nonparametric regression we propose a new method for model selection and model fitting in multivariate nonparametric regression models in the framework of smoothing spline anova the cosso is a method of regularization with the penalty functional being the sum of component norms instead of the squared norm employed in the traditional smoothing spline method the cosso provides a unified framework for several recent proposals for model selection in linear models and smoothing spline anova models theoretical properties such as the existence and the rate of convergence of the cosso estimator are studied in the special case of a tensor product design with periodic functions a detailed analysis reveals that the cosso does model selection by applying a novel soft thresholding type operation to the function components we give an equivalent formulation of the cosso estimator which leads naturally to an iterative algorithm we compare the cosso with mars a popular method that builds functional anova models in simulations and real examples the cosso method can be extended to classification problems and we compare its performance with those of a number of machine learning algorithms on real datasets the cosso gives very competitive performance in these studies

cross validation for correlated data abstract k fold cross validation cv with squared error loss is widely used for evaluating predictive models especially when strong distributional assumptions cannot be taken however cv with sq

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

on testing an unspecified function through a linear mixed effects model with multiple variance components summary we examine a generalized f test of a nonparametric function through penalized splines and a linear mixed effects model representation with a mixed effects model representation of penalized splines we imbed the test of an unspecified function into a test of some fixed effects and a variance component in a linear mixed effects model with nuisance variance components under the null the procedure can be used to test a nonparametric function or varying coefficient with clustered data compare two spline functions test the significance of an unspecified function in an additive model with multiple components and test a row or a column effect in a two way analysis of variance model through a spectral decomposition of the residual sum of squares we provide a fast algorithm for computing the null distribution of the test which significantly improves the computational efficiency over bootstrap the spectral representation reveals a connection between the likelihood ratio test lrt in a multiple variance components model and a single component model we examine our methods through simulations where we show that the power of the generalized f test may be higher than the lrt depending on the hypothesis of interest and the true model under the alternative we apply these methods to compute the genome wide critical value and p value of a genetic association test in a genome wide association study gwas where the usual bootstrap is computationally intensive up to 108 simulations and asymptotic approximation may be unreliable and conservative

a note on hausdorff separation in l top this note shows that a careful study of the history of hausdorff type separation axioms in many valued topology is necessary we make a few observations regarding a hausdorff type axiom for l topological spaces reinvented recently by shi in particular we put it into the adjoint situation associated with the functors w l and i l with l a continuous lattice

smoothing spline nonlinear nonparametric regression models almost all of the current nonparametric regression methods such as smoothing splines generalized additive models and varyingcoefficients models assume a linear relationship when nonparametric functions are regarded as parameters in this article we propose a general class of smoothing spline nonlinear nonparametric models that allow nonparametric functions to act nonlinearly they arise in many fields as either theoretical or empirical models our new estimation methods are based on an extension of the gauss newton method to infinite dimensional spaces and the backfitting procedure we extend the generalized cross validation and generalized maximum likelihood methods to estimate smoothing parameters we establish connections between some nonlinear nonparametric models and nonlinear mixed effects models we derive approximate bayesian confidence intervals for inference we illustrate the methods with an application to term structure of interest rates and conduct simulations to evaluate the finite sampl

determining the effective sample size of a parametric prior we present a definition for the effective sample size of a parametric prior distribution in a bayesian model and propose methods for computing the effective sample size in a variety of settings our approach first constructs a prior chosen to be vague in a suitable sense and updates this prior to obtain a sequence of posteriors corresponding to each of a range of sample sizes we then compute a distance between each posterior and the parametric prior defined in terms of the curvature of the logarithm of each distribution and the posterior minimizing the distance defines the effective sample size of the prior for cases where the distance cannot be computed analytically we provide a numerical approximation based on monte carlo simulation we provide general guidelines for application illustrate the method in several standard cases where the answer seems obvious and then apply it to some nonstandard settings

inference after model selection typical modeling strategies involve model selection which has a significant effect on inference of estimated parameters common practice is to use a selected model ignoring uncertainty introduced by the process of model selection this could yield overoptimistic inferences resulting in false discovery in this article we develop a general methodology via optimal approximation for estimating the mean and variance of complex statistics that involve the process of model selection this allows us to make approximately unbiased inferences taking into account the selection process we examine the operating characteristics of the proposed methodology via asymptotic analyses and simulations these results show that the proposed methodology yields correct inferences and outperforms common alternatives

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

neural network modeling for small datasets neural network modeling for small datasets can be justified from a theoretical point of view according to some of bartlett s results showing that the generalization performance of a multilayer perceptron mlp depends more on the l1 norm c 1 of the weights between the hidden layer and the output layer rather than on the total number of weights in this article we investigate some geometrical properties of mlps and drawing on linear projection theory we propose an equivalent number of degrees of freedom to be used in neural model selection criteria like the akaike information criterion and the bayes information criterion and in the unbiased estimation of the error variance this measure proves to be much smaller than the total number of parameters of the network usually adopted and it does not depend on the number of input variables moreover this concept is compatible with bartlett s results and with similar ideas long associated with projection based models and kernel models some numerical studies i

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

markov chain estimation for test theory without an answer key this study develops markov chain monte carlo mcmc estimation theory for the general condorcet model gcm an item response model for dichotomous response data which does not presume the analyst knows the correct answers to the test a priori answer key in addition to the answer key respondent ability guessing bias and difficulty parameters are estimated with respect to data fit the study compares between the possible gcm formulations using mcmc based methods for model assessment and model selection real data applications and a simulation study show that the gcm can accurately reconstruct the answer key from a small number of respondents

blow up of h1 solution for the nonlinear schrodinger equation we consider the blow up of the solution in h1 for the following nonlinear schrodinger equation i tu u u p 1u x rn t 0 u 0 x u0 x x rn t 0 where n 2 and 1 4 n p min n 2 n 2 5 we prove that if the initial data u0 in h1 are radially symmetric and have negative energy then the solution of in h1 blows up in finite time we do not assume that xu0 l2 and therefore our result is the generalization of the results of glassey 4 and m tsutsumi 18 for the radially symmetric case

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

on the distributional properties of model selection criteria abstract it is commonly accepted that statistical modeling should follow the parsimony principle namely that simple models should be given priority whenever possible but little quantitative knowledge is known concerning the amount of penalty for complexity regarded as allowable we try to understand the parsimony principle in the context of model selection in particular the generalized final prediction error criterion is considered and we argue that the penalty term should be chosen between 1 5 and 5 for most practical situations applying our results to the cross validation criterion we obtain insights into how the partition of data should be done we also discuss the small sample performance of our methods

remarks on global existence for the supercritical nonlinear schrodinger equation with a harmonic potential abstract this paper is concerned with the nonlinear schrodinger equation with a harmonic potential which describes the attractive bose einstein condensate under the magnetic trap by combining the best constant of gagliardo nirenberg s inequality with the characteristic of this equation we derive out a global existence condition for the supercritical equation which coincides with the critical case

inverse limits in category ltop ii this paper is devoted to the study of the inverse limits in ltop the category of l fuzzy topological spaces and continuous l valued zadeh functions between them where l is a completely distributive complete lattice structures of such limits are described and a series of their properties are investigated furthermore the mappings between two arbitrary inverse systems is defined and some of their properties are discussed for the case of category ltop

measuring explained variation in linear mixed effects models we generalize the well known r2 measure for linear regression to linear mixed effects models our work was motivated by a cluster randomized study conducted by the eastern cooperative oncology group to compare two different versions of informed consent document we quantify the variation in the response that is explained by the covariates under the linear mixed model and study three types of measures to estimate such quantities the first type of measures make direct use of the estimated variances the second type of measures use residual sums of squares in analogy to the linear regression the third type of measures are based on the kullback leibler information gain all the measures can be easily obtained from software programs that fit linear mixed models we study the performance of the measures through monte carlo simulations and illustrate the usefulness of the measures on data sets copyright 2003 john wiley sons ltd

cauchy problem and ehrenfest s law of nonlinear schrodinger equations with potentials abstract in this paper we prove the long time existence of the flow in q h for the nonlinear schrodinger equation in r n i t 1 2 v p 1 1 p 4 n where h 1 2 v and q h is the form domain of h when the potential v is bounded below and satisfies the condition that d v is bounded for all 2 we also show that the nonlinear analogue of the classical ehrenfest s law for the linear schrodinger equation holds for the above equation

the effect of improper priors on gibbs sampling in hierarchical linear mixed models abstract often either from a lack of prior information or simply for convenience variance components are modeled with improper priors in hierarchical linear mixed models although the posterior distributions for these models are rarely available in closed form the usual conjugate structure of the prior specification allows for painless calculation of the gibbs conditionals thus the gibbs sampler may be used to explore the posterior distribution without ever having established propriety of the posterior an example is given showing that the output from a gibbs chain corresponding to an improper posterior may appear perfectly reasonable thus one cannot expect the gibbs output to provide a red flag informing the user that the posterior is improper the user must demonstrate propriety before a markov chain monte carlo technique is used a theorem is given that classifies improper priors according to the propriety of the resulting posteriors applications concerning bayesian analysis of animal breeding

a stone type duality for st0 stratified alexandrov l topological spaces abstract we show that for every complete lattice a both the set of completely prime elements and the set of completely coprime elements are one to one corresponding to ch a the set of complete lattice homomorphisms from a to the two element lattice 2 a complete lattice is called completely generated a cg lattice for short if it is generated by the set of completely prime elements or equivalently by the set of completely coprime elements then we restudy the duality between the category of t 0 alexandrov topological spaces and the category of cg lattices by means of ch a with these preparations for a frame l as the truth value table we introduce s t 0 separation axiom for stratified alexandrov l topological spaces and finally establish a duality between the category of s t 0 stratified alexandrov l topological spaces and the category of completely generated complete l ordered sets we also investigate some properties of the s t 0 axiom for example it is hereditary by closed subspaces and productive

when does more regularization imply fewer degrees of freedom sufficient conditions and counterexamples regularization aims to improve prediction performance by trading an increase in training error for better agreement between training and prediction errors which is often captured through decreased degrees of freedom in this paper we give examples which show that regularization can increase the degrees of freedom in common models including the lasso and ridge regression in such situations both training error and degrees of freedom increase making the regularization inherently without merit two important scenarios are described where the expected reduction in degrees of freedom is guaranteed all symmetric linear smoothers and convex constrained linear regression models like ridge regression and the lasso when compared to unconstrained linear regression

combining multiple biomarker models in logistic regression in medical research there is great interest in developing methods for combining biomarkers we argue that selection of markers should also be considered in the process traditional model variable selection procedures ignore the underlying uncertainty after model selection in this work we propose a novel model combining algorithm for classification in biomarker studies it works by considering weighted combinations of various logistic regression models five different weighting schemes are considered in the article the weights and algorithm are justified using decision theory and risk bound results simulation studies are performed to assess the finite sample properties of the proposed model combining method it is illustrated with an application to data from an immunohistochemical study in prostate cancer

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

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

separation axioms in l fuzzy topological spaces i t0 and t1 abstract using the notion of remote neighborhood we define the separation axioms t 0 and t 1 in l fuzzy topological spaces l fts the relations between our definitions hutton and reilly s and wang s are discussed and the separations of hutton s fuzzy unit interval and gantner s fuzzy real line are examined characterizations of these l fts are proved and a series of properties of them are investigated moreover some results on minimal t 0 l fts and t 1 l fts are established which will be used immediately in our subsequent discussion

on the behaviour of marginal and conditional aic in linear mixed models in linear mixed models model selection frequently includes the selection of random effects two versions of the akaike information criterion aic have been used based either on the marginal or on the conditional distribution we show that the marginal aic is not an asymptotically unbiased estimator of the akaike information and favours smaller models without random effects for the conditional aic we show that ignoring estimation uncertainty in the random effects covariance matrix as is common practice induces a bias that can lead to the selection of any random effect not predicted to be exactly zero we derive an analytic representation of a corrected version of the conditional aic which avoids the high computational cost and imprecision of available numerical approximations an implementation in an r package r development core team 2010 is provided all theoretical results are illustrated in simulation studies and their impact in practice is investigated in an analysis of childhood malnutrition in zambia

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

goodness of fit statistics for age specific reference intervals the age specific reference interval is a commonly used screening tool in medicine it involves estimation of extreme quantile curves such as the 5th and 95th centiles of a reference distribution of clinically normal individuals it is crucial that models used to estimate such intervals fit the data extremely well however few procedures to assess goodness of fit have been proposed in the literature and even fewer have been evaluated systematically here we consider procedures based on the distribution of the z scores standardized residuals from a model and on pearson chi 2 statistics for observed and expected counts in groups defined by age and the estimated reference centile curves two of the procedures q and grid tests are mainly inferential whereas the third permutation bands and b tests is essentially graphical we obtain approximations to the null distributions of several relevant test statistics and examine their size and power for a range of models based on real data sets we recommend q tests in all situations where z scores are available since they are general simple to calculate and usually have the highest power among the three classes of test considered for the cases considered the grid tests are always inferior to the q and b tests

statistical mechanics of the nonlinear schrodinger equation we investigate the statistical mechanics of a complex fieldo whose dynamics is governed by the nonlinear schrodinger equation such fields describe in suitable idealizations langmuir waves in a plasma a propagating laser field in a nonlinear medium and other phenomena their hamiltonian h phi int omega frac 1 2 nabla phi 2 1 p phi p dx is unbounded below and the system will under certain conditions develop self focusing singularities in a finite time we show that when is the circle and thel 2 norm of the field which is conserved by the dynamics is bounded byn the gibbs measure obtained is absolutely continuous with respect to wiener measure and normalizable if and only ifp andn are such that classical solutions exist for all time no collapse of the solitons this measure is essentially the same as that of a one dimensional version of the more realisitc zakharov model of coupled langmuir and ion acoustic waves in a plasma we also obtain some properties of the gibbs state by both analytic and numerical methods asn and the temperature are varied

optimal geostatistical model selection in many fields of science predicting variables of interest over a study region based on noisy data observed at some locations is an important problem two popular methods for the problem are kriging and smoothing splines the former assumes that the underlying process is stochastic whereas the latter assumes it is purely deterministic kriging performs better than smoothing splines in some situations but is outperformed by smoothing splines in others however little is known regarding selecting between kriging and smoothing splines in addition how to perform variable selection in a geostatistical model has not been well studied in this article we propose a general methodology for selecting among arbitrary spatial prediction methods based on approximately unbiased estimation of mean squared prediction errors using a data perturbation technique the proposed method accounts for estimation uncertainty in both kriging and smoothing spline predictors and is shown to be optimal in terms of two mean sq

quantile regression in reproducing kernel hilbert spaces in this article we consider quantile regression in reproducing kernel hilbert spaces which we call kernel quantile regression kqr we make three contributions 1 we propose an efficient algorithm that computes the entire solution path of the kqr with essentially the same computational cost as fitting one kqr model 2 we derive a simple formula for the effective dimension of the kqr model which allows convenient selection of the regularization parameter and 3 we develop an asymptotic theory for the kqr model

calibrating the degrees of freedom for automatic data smoothing and effective curve checking curve fitting and curve checking based on the local polynomial regression technique are commonly used data analytic methods in statistics this article examines in nonparametric settings both the asymptotic expressions and empirical formulas for degrees of freedom df a notion introduced by hastie and tibshirani of linear smoothers the asymptotic results give useful insights into the nonparametric modeling complexity meanwhile by substituting the exact dfs by the empirical formula an empirical version of the generalized cross validation egcv is obtained an automatic bandwidth selection method based on minimizing egcv is proposed for conducting local smoothing this procedure preserves full benefits of the ordinary and generalized cross validation but offers a substantial reduction in computational burden furthermore the egcv minimizing bandwidth can be extended in a very simple manner to fit multivariate models such as the varying coefficient models applications of calibrating dfs to impo

diagnostics for assessing regression models abstract we concern ourselves with diagnostics for checking the overall and local goodness of fit of a model s x used in the regression of y on x u 0 1 d the model for s x is a functional form that depends on a finite number of unknown parameters two statistics and w are proposed that measure the level of agreement between the model fit to the data and the nonparametric kernel estimator on m preselected points in u conditions are given under which and w are asymptotically equivalent both of these statistics measure overall lack of fit and are related to the deviance their asymptotic distribution under the null model and under local alternatives is derived this work is motivated by the local mean deviance plot of landwehr pregibon and shoemaker for assessing overall lack of fit in logistic regression their plot is summarized by our test statistics and is extended to other likelihood based regressions of y on x

partitioning degrees of freedom in hierarchical and other richly parameterized models hodges and sargent 2001 have developed a measure of a hierarchical model s complexity degrees of freedom df that is consistent with definitions for scatterplot smoothers is interpretable in terms of simple models and enables control of a fit s complexity by means of a prior distribution on complexity but although df describes the complexity of the whole fitted model in general it remains unclear how to allocate df to individual effects here we present a new definition of df for arbitrary normal error linear hierarchical models consistent with that of hodges and sargent that naturally partitions the n observations into df for individual effects and for error the new conception of an effect s df is the ratio of the effect s modeled variance matrix to the total variance matrix this provides a way to describe the sizes of different parts of a model e g spatial clustering vs heterogeneity to place df based priors on smoothing parameters and to describe how a smoothed effect competes with

variable selection in nonparametric regression with continuous covariates in a nonparametric regression setup where the covariates are continuous the problem of estimating the number of covariates will be discussed in this paper the kernel method is used to estimate the regression function and the selection criterion is based on minimizing the cross validation estimate of the mean squared prediction error we consider choosing both the bandwidth and the number of covariates based on the data unlike the case of linear regression it turns out that the selection is consistent and efficient even when the true model has only a finite number of covariates in addition we also observe the curse of dimensionality at work

model selection for multi component frailty models various frailty models have been developed and are now widely used for analysing multivariate survival data it is therefore important to develop an information criterion for model selection however in frailty models there are several alternative ways of forming a criterion and the particular criterion chosen may not be uniformly best in this paper we study an akaike information criterion aic on selecting a frailty structure from a set of possibly non nested frailty models we propose two new aic criteria based on a conditional likelihood and an extended restricted likelihood erl given by lee and nelder j r statist soc b 1996 58 619 678 we compare their performance using well known practical examples and demonstrate that the two criteria may yield rather different results a simulation study shows that the aic based on the erl is recommended when attention is focussed on selecting the frailty structure rather than the fixed effects

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

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

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

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

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

on what is noise j a scales and r snieder geophysics 63 1122 1124 with reference to scales and snieder s 1998 discussion of what is noise i would like to offer a statistical interpretation of this question any experiment involving measurement will have variation this is what allows us to make inference upon the results and it is the partition of this variation which is key to the noise question from an experimenter s point of view noise may be that part of the data variation which we choose not to explain although scales and snieder are correct to point out that we must then have a model to explain the rest since a purely statistical one i e assuming that the remainder is in some way random may not be adequate from a statistical viewpoint however noise may be better described as that part of the variation which we cannot explain with a parsimonious deterministic model and the signal is then the interesting part of the overall model

testing the goodness of fit of a linear model via nonparametric regression techniques abstract this article investigates the use of nonparametric regression methodology to test the adequacy of a parametric linear model the large sample properties of parametric goodness of fit tests for linearity are considered the inadequacies of such tests lead to the proposal of new tests that are constructed from nonparametric regression fits to the residuals from linear regression large sample theory is derived for two variants of this type of statistic the results demonstrate that such tests are consistent against all fixed smooth alternatives to linearity but are incapable of detecting local alternatives converging to a linear model at the parametric rate n 1 2 simulation experiments involving a test based on fitting cubic smoothing splines to residuals reveals that this test has good power properties against several reasonable alternatives

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

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

the risk inflation criterion for multiple regression a new criterion is proposed for the evaluation of variable selection procedures in multiple regression this criterion which we call the risk inflation is based on an adjustment to the risk essentially the risk inflation is the maximum increase in risk due to selecting rather than knowing the correct predictors a new variable selection procedure is obtained which in the case of orthogonal predictors substantially improves on aic c p and bic and is close to optimal in contrast to aic c p and bic which use dimensionality penalties of 2 2 and log n respectively this new procedure uses a penalty 2 log p where p is the number of available predictors for the case of nonorthogonal predictors bounds for the optimal penalty are obtained

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

model selection in the presence of incidental parameters this paper considers model selection in panels where incidental parameters are present primary interest centers on selecting a model that best approximates the underlying structure involving parameters that are common within the panel it is well known that conventional model selection procedures are often inconsistent in panel models and this can be so even without nuisance parameters modifications are then needed to achieve consistency new model selection information criteria are developed here that use either the kullback leibler information criterion based on the profile likelihood or the bayes factor based on the integrated likelihood with a bias reducing prior these model selection criteria impose heavier penalties than those associated with standard information criteria such as aic and bic the additional penalty which is data dependent properly reflects the model complexity arising from the presence of incidental parameters a particular example is studied in detail involving lag order selection in dynamic panel models with fixed effects the new criteria are shown to control for over under selection probabilities in these models and lead to consistent order selection criteria

some results on i l and r l abstract this paper gives a partial solution to problem 1 in wang and lu 1992 and a suggestion about how to construct a better fuzzy unit interval it defines a new kind of fuzzy real line and investigates a series of its properties

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

pointwise influence matrices for functional response regression we extend the notion of an influence or hat matrix to regression with functional responses and scalar predictors for responses depending linearly on a set of predictors our definition is shown to reduce to the conventional influence matrix for linear models the pointwise degrees of freedom the trace of the pointwise influence matrix are shown to have an adaptivity property that motivates a two step bivariate smoother for modeling nonlinear dependence on a single predictor this procedure adapts to varying complexity of the nonlinear model at different locations along the function and thereby achieves better performance than competing tensor product smoothers in an analysis of the development of white matter microstructure in the brain

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

conditional akaike information for mixed effects models summary this paper focuses on the akaike information criterion aic for linear mixed effects models in the analysis of clustered data we make the distinction between questions regarding the population and questions regarding the particular clusters in the data we show that the aic in current use is not appropriate for the focus on clusters and we propose instead the conditional akaike information and its corresponding criterion the conditional aic caic the penalty term in caic is related to the effective degrees of freedom p for a linear mixed model proposed by hodges sargent 2001 p reflects an intermediate level of complexity between a fixed effects model with no cluster effect and a corresponding model with fixed cluster effects the caic is defined for both maximum likelihood and residual maximum likelihood estimation a pharmacokinetics data appli cation is used to illuminate the distinction between the two inference settings and to illustrate the use of the conditional aic in model selection

variable selection in nonparametric regression with categorical covariates abstract this article extends the problem of variable selection to a nonparametric regression model with categorical covariates two selection criteria are considered the cross validation cv criterion and the accumulated prediction error ape criterion we find that asymptotically the cv criterion performs well only when the true model is infinite dimensional while the ape criterion is appropriate when the true model is finite dimensional this is very similar to the case of linear regression model a simulation study reveals some interesting small sample properties of these criteria to be more specific suppose that we have observations x 1 y 1 xn yn that are iid random vectors and x x 1 x 2 where the x i s are categorical we allow y to be of any type now a new observation x has arrived and we want to predict the corresponding y such a framework is more appropriate than regressions with fixed covariates in situations where the covariates are observational rather than bei

quantitative domains via fuzzy sets part ii fuzzy scott topology on fuzzy directed complete posets this paper pursues an investigation on quantitative domains via fuzzy sets initiated by the first author this time we define an l topology called the fuzzy scott topology on fuzzy dcpos and investigate its properties scott convergence of stratified l filters is also defined and studied we show that a fuzzy dcpo x e is continuous if and only if for any stratified l filter on x scott convergence coincides with the convergence with respect to the fuzzy scott topology at last we show that the category of fuzzy dcpos with fuzzy scott continuous maps is cartesian closed

blow up threshold for the gross pitaevskii system with combined nonlocal nonlinearities abstract in this paper we study the gross pitaevskii system with combined nonlocal nonlinearities first we establish both the stable regime and the unstable regime then via a construction of cross minimization problem the blow up threshold is established

conditional akaike information under generalized linear and proportional hazards mixed models we study model selection for clustered data when the focus is on cluster specific inference such data are often modelled using random effects and conditional akaike information was proposed in vaida blanchard 2005 and used to derive an information criterion under linear mixed models here we extend the approach to generalized linear and proportional hazards mixed models outside the normal linear mixed models exact calculations are not available and we resort to asymptotic approximations in the presence of nuisance parameters a profile conditional akaike information is proposed bootstrap methods are considered for their potential advantage in finite samples simulations show that the performance of the bootstrap and the analytic criteria are comparable with bootstrap demonstrating some advantages for larger cluster sizes the proposed criteria are applied to two cancer datasets to select models when the cluster specific inference is of interest

counting degrees of freedom in hierarchical and other richly parameterised models summary drawing on linear model theory we rigorously extend the notion of degrees of freedom to richly parameterised models including linear hierarchical and random effect models some smoothers and spatial models and combinations of these the number of degrees of freedom is often much smaller than the number of parameters our notion of degrees of freedom is compatible with similar ideas long associated with smoothers but is applicable to new classes of models and can be interpreted using the projection theory of linear models we use an example to illustrate the two applications of setting prior distributions for variances and fixing model complexity by fixing degrees of freedom

differential geometric least angle regression a differential geometric approach to sparse generalized linear models sparsity is an essential feature of many contemporary data problems remote sensing various forms of automated screening and other high throughput measurement devices collect a large amount of information typically about few independent statistical subjects or units in certain cases it is reasonable to assume that the underlying process generating the data is itself sparse in the sense that only a few of the measured variables are involved in the process we propose an explicit method of monotonically decreasing sparsity for outcomes that can be modelled by an exponential family in our approach we generalize the equiangular condition in a generalized linear model although the geometry involves the fisher information in a way that is not obvious in the simple regression setting the equiangular condition turns out to be equivalent with an intuitive condition imposed on the rao score test statistics in certain special cases the method can be tweaked to obtain l1 penalized generalized linear model solution paths but the method itself defines sparsity more directly although the computation of the solution paths is not trivial the method compares favourably with other path following algorithms

bayesian measures of explained variance and pooling in multilevel hierarchical models explained variance r2 is a familiar summary of the fit of a linear regression and has been generalized in various ways to multilevel hierarchical models the multilevel models that we consider in this article are characterized by hierarchical data structures in which individuals are grouped into units which themselves might be further grouped into larger units and variables are measured on individuals and each grouping unit the models are based on regression relationships at different levels with the first level corresponding to the individual data and subsequent levels corresponding to between group regressions of individual predictor effects on grouping unit variables we present an approach to defining r2 at each level of the multilevel model rather than attempting to create a single summary measure of fit our method is based on comparing variances in a single fitted model rather than with a null model in simple regression our measure generalizes the classical adjusted r2 we also discuss

smoothing balanced single error term analysis of variance we present an approach to smoothing balanced single error term analysis of variance anova descended from smith that also allows spatial temporal or spatiotemporal smoothing the approach addresses unreplicated designs masked contrasts in effects with many degrees of freedom and subgroup analysis demonstrated using a study of denture lining materials our approach is bayesian but can be viewed as a way to generate frequentist procedures a simulation experiment compares four priors unsmoothed anova and dropping nonsignificant interactions three priors have advantages when some interactions are absent dropping nonsignificant interactions has serious flaws we contrast our approach with the approaches of nobile green and gelman

unified low rank matrix estimate via penalized matrix least squares approximation low rank matrix estimation arises in a number of statistical and machine learning tasks in particular the coefficient matrix is considered to have a low rank structure in multivariate linear regression and multivariate quantile regression in this paper we propose a method called penalized matrix least squares approximation pmlsa toward a unified yet simple low rank matrix estimate specifically pmlsa can transform many different types of low rank matrix estimation problems into their asymptotically equivalent least squares forms which can be efficiently solved by a popular matrix fast iterative shrinkage thresholding algorithm furthermore we derive analytic degrees of freedom for pmlsa with which a bayesian information criterion bic type criterion is developed to select the tuning parameters the estimated rank based on the bic type criterion is verified to be asymptotically consistent with the true rank under mild conditions extensive experimental studies are performed to confirm our assertion

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

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

remarks on products of l fuzzy topological spaces two essential theorems about fuzzy projection are improved and a negative answer to wang s question in 1987 on products of l fuzzy topological spaces is given

discussion on model confidence bounds for variable selection by yang li yuetian luo davide ferrari xiaonan hu and yichen qin

a proximal iteration for deconvolving poisson noisy images using sparse representations we propose an image deconvolution algorithm when the data is contaminated by poisson noise the image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms our key contributions are as follows first we handle the poisson noise properly by using the anscombe variance stabilizing transform leading to a nonlinear degradation equation with additive gaussian noise second the deconvolution problem is formulated as the minimization of a convex functional with a data fidelity term reflecting the noise properties and a nonsmooth sparsity promoting penalty over the image representation coefficients e g lscr1 norm an additional term is also included in the functional to ensure positivity of the restored image third a fast iterative forward backward splitting algorithm is proposed to solve the minimization problem we derive existence and uniqueness conditions of the solution and establish convergence of the iterative algorithm finally a gcv based model selection procedure is proposed to objectively select the regularization parameter experimental results are carried out to show the striking benefits gained from taking into account the poisson statistics of the noise these results also suggest that using sparse domain regularization may be tractable in many deconvolution applications with poisson noise such as astronomy and microscopy

model selection for generalized estimating equations accommodating dropout missingness the generalized estimating equation gee has been a popular tool for marginal regression analysis with longitudinal data and its extension the weighted gee approach can further accommodate data that are missing at random mar model selection methodologies for gee however have not been systematically developed to allow for missing data we propose the missing longitudinal information criterion mlic for selection of the mean model and the mlic for correlation mlicc for selection of the correlation structure in gee when the outcome data are subject to dropout monotone missingness and are mar our simulation results reveal that the mlic and mlicc are effective for variable selection in the mean model and selecting the correlation structure respectively we also demonstrate the remarkable drawbacks of naively treating incomplete data as if they were complete and applying the existing gee model selection method the utility of proposed method is further illustrated by two real applications involving missing longitudinal outcome data

order homomorphisms on fuzzes abstract in this paper basic properties of the concept of order homomorphisms on fuzzes 4 which is a proper generalization of the concept of fuzz functions 1 are studied conditions for fuzz functions to be functions of zadeh s type are obtained moreover the concepts of continuous closed and open order homomorphisms are introduced and studied

fence methods for mixed model selection many model search strategies involve trading off model fit with model complexity in a penalized goodness of fit measure asymptotic properties for these types of procedures in settings like linear regression and arma time series have been studied but these do not naturally extend to nonstandard situations such as mixed effects models where simple definition of the sample size is not meaningful this paper introduces a new class of strategies known as fence methods for mixed model selection which includes linear and generalized linear mixed models the idea involves a procedure to isolate a subgroup of what are known as correct models of which the optimal model is a member this is accomplished by constructing a statistical fence or barrier to carefully eliminate incorrect models once the fence is constructed the optimal model is selected from among those within the fence according to a criterion which can be made flexible in addition we propose two variations of the fence the first is a stepwise procedure to handle situations of many predictors the second is an adaptive approach for choosing a tuning constant we give sufficient conditions for consistency of fence and its variations a desirable property for a good model selection procedure the methods are illustrated through simulation studies and real data analysis

adaptive model selection most model selection procedures use a fixed penalty penalizing an increase in the size of a model these nonadaptive selection procedures perform well only in one type of situation for instance bayesian information criterion bic with a large penalty performs well for small models and poorly for large models and akaike s information criterion aic does just the opposite this article proposes an adaptive model selection procedure that uses a data adaptive complexity penalty based on a concept of generalized degrees of freedom the proposed procedure combining the benefit of a class of nonadaptive procedures approximates the best performance of this class of procedures across a variety of different situations this class includes many well known procedures such as aic bic mallows s cp and risk inflation criterion ric the proposed procedure is applied to wavelet thresholding in nonparametric regression and variable selection in least squares regression simulation results and an asymptotic

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 new approach to l t2 l urysohn and l completely hausdorff axioms in this paper new l t 2 l urysohn and l completely hausdorff axioms are introduced in l topological spaces relations between these axioms are established the l extended real line and its subspaces as well as pointwise metric spaces satisfy these axioms

on the predictive risk in misspecified quantile regression abstract in the present paper we investigate the predictive risk of possibly misspecified quantile regression functions the in sample risk is well known to be an overly optimistic estimate of the predictive risk and we provide two relatively simple asymptotic characterizations of the associated bias also called expected optimism we propose estimates for the expected optimism and the predictive risk and establish their uniform consistency under mild conditions our results hold for models of moderately growing size and allow the quantile function to be incorrectly specified empirical evidence from our estimates is encouraging as it compares favorably with cross validation