information theoretic determination of minimax rates of convergence we present some general results determining minimax bounds on statistical risk for density estimation based on certain information theoretic considerations these bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence

local behavior of solutions of quasilinear elliptic equations with general structure this paper is motivated by the observation that solutions to certain variational inequalities involving partial differential operators of the form diva x u vu b x u vu where a and b are borel measurable are solutions to the equation diva x u vu b x u vu u for some nonnegative radon measure i among other things it is shown that if u is a holder continuous solution to this equation then the measure u satisfies the growth property j b x r 1 is given by the structure of the differential operator conversely if u is assumed to satisfy this growth condition then it is shown that u satisfies a harnack type inequality thus proving that u is locally bounded under the additional assumption that a is strongly monotonic it is shown that u is holder continuous

limits as p of p laplacian concave convex problems abstract we study the behavior as p of the sequel of positive weak solutions of the concave convex problem p div u p 2 u u q p u r p in u 0 in u 0 on where r n is a bounded domain 0 and the exponents q r satisfy lim p q p p 1 q lim p r p p 1 r with 0 q 1 r we characterize any positive uniform limit of a sequence of weak solutions of p as a viscosity solution of min u max u q u r u 0 in notice that the limit process decouples the nonlinearity we obtain existence non existence and global multiplicity of positive viscosity solutions of the limit problem in terms of the parameter

on estimating a density using hellinger distance and some other strange facts on s interesse ici aux possibles vitesses d estimation d une densite a support compact dans m sous des hypotheses de regularite lorsque la perte est mesuree par le carre de la distance de hellinger on regardera aussi le cas connu des normes mathbb l q pour 1 q 2 et le risque est le risque minimax sur la famille on donne une methode generale permettant de traiter les problemes dans le cadre de la theorie de l approximation sous des conditions concernant l entropie metrique et l e capacite des familles a estimer les rapports entre regularite et entropie metrique etant bien connus nous pourrons aussi traiter les cas classiques et d autres qui le sont moins sous des conditions de bornes inferieures les vitesses sont celles observees pour la norme mathbb l q mais elles different dans le cas general on montre aussi que les restrictions sur la compacite du support ou la regularite sont indispensables et que leur absence mene a l impossibilite d obtenir une estimation raisonnable en ce sens que n importe quelle suite d estimateurs sera arbitrairement mauvaise en un point au moins un resultat analogue est vrai sous des conditions de regularite

extremal solutions of p laplacian problems in d1 p rn via wolff potential estimates abstract we prove the existence of smallest positive and greatest negative solutions in d 1 p r n of the p laplacian problem of logistic type in r n with 1 p n u d 1 p r n p u a x u p 2 u g u and consider its extended version with a harvesting term h as well unlike papers on the topic here we only assume a superlinear growth for the nonlinearity g r r at zero and at infinity i e even supercritical growth is allowed and moreover no monotonicity for s g s s p 1 is required we prove a vazquez type maximum principle for nonnegative d 1 p r n solutions of some p laplacian problem and establish a sub supersolution method in d 1 p r n which along with wolff potential estimates for superharmonic functions will be used as our main tools

the dirichlet problem for the p fractional laplace equation abstract we deal with a class of equations driven by nonlocal possibly degenerate integro differential operators of differentiability order s 0 1 and summability growth p 1 whose model is the fractional p laplacian operator with measurable coefficients we review several recent results for the corresponding weak solutions supersolutions as comparison principles a priori bounds lower semicontinuity boundedness holder continuity up to the boundary and many others we then discuss the good definition of s p superharmonic functions and the nonlocal counterpart of the perron method in nonlinear potential theory together with various related results we briefly mention some basic results for the obstacle problem for nonlinear integro differential equations finally we present the connection amongst the fractional viscosity solutions the weak solutions and the aforementioned s p superharmonic functions together with other important results for this class of equations when involving general measure data and a surprising fractional version of the gehring lemma we sketch the corresponding proofs of some of the results presented here by especially underlining the development of new fractional localization techniques and other recent tools various open problems are listed throughout the paper

heuristics of instability and stabilization in model selection in model selection usually a best predictor is chosen from a collection hat mu cdot s of predictors where hat mu cdot s is the minimum least squares predictor in a collection mathsf u s of predictors here s is a complexity parameter that is the smaller s the lower dimensional smoother the models in mathsf u s if mathsf l is the data used to derive the sequence hat mu cdot s the procedure is called unstable if a small change in mathsf l can cause large changes in hat mu cdot s with a crystal ball one could pick the predictor in hat mu cdot s having minimum prediction error without prescience one uses test sets cross validation and so forth the difference in prediction error between the crystal ball selection and the statistician s choice we call predictive loss for an unstable procedure the predictive loss is large this is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods unstable procedures can be stabilized by perturbing the data getting a new predictor sequence hat mu cdot s and then averaging over many such predictor sequences

gradient estimates for problems with orlicz growth abstract we study a general nonlinear elliptic equation in the orlicz setting with data not belonging to the dual of the energy space we provide several lorentz type and morrey type estimates for the gradients of solutions under various conditions on the data

density estimation by wavelet thresholding density estimation is a commonly used test case for nonparametric estimation methods we explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients minimax rates of convergence are studied over a large range of besov function classes b pq and for a range of global l p error measures 1 p p some form of nonlinearity is essential since the minimax linear estimators are suboptimal by polynomial powers of n a second approach using an approximation of a gaussian white noise model in a mallows metric is used to attain exactly optimal rates of convergence for quadratic error p 2

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

quasilinear equations with source terms on carnot groups in this paper we give necessary and sufficient conditions for the existence of solutions to quasilinear equations of lane emden type with measure data on a carnot group g of arbitrary step the quasilinear part involves operators of the p laplacian type g p 1 p these results are based on new a priori estimates of solutions in terms of nonlinear potentials of th wolff s type as a consequence we characterize completely removable singularities and prove a liouville type theorem for supersolutions of quasilinear equations with source terms which has been known only for equations involving the sub laplacian p 2 on the heisenberg group

isolated singularities of solutions to quasi linear elliptic equations with absorption we study the problem of removability of isolated singularities for a general second order quasi linear equation in divergence form diva x u u a0 x u g x u 0 in a punctured domain 0 where is a domain in rn n 3 the model example is the equation pu gu u p 2 u u q 1 0 q p 1 0 p n assuming that the lower order terms satisfy certain non linear kato type conditions we prove that for q n p 1 n p all point singularities of the above equation are removable thus extending the seminal result of brezis and veron

on the weak continuity of elliptic operators and applications to potential theory in this paper we establish weak continuity results for quasilinear elliptic and subellip tic operators of divergence form acting on corresponding classes of subharmonic functions these results are analogous to our earlier results for fully nonlinear fc hessian operators from the weak continuity we derive various potential theoretic results including capacity estimates potential esti mates and the wiener criterion for regular boundary points our methods make substantial use of harnack inequalities for solutions 1 introduction there are numerous papers on the local behavior of solu tions subsolutions and supersolutions of second order quasilinear elliptic diver gence structure partial differential equations as well as extensions to degenerate elliptic and subelliptic equations in this paper we show that many properties including the quasi continuity of subharmonic functions asymptotic behavior of fundamental solutions removability of singularities wiener criteria for regular boundary points can be derived via harnack inequalities for solutions the two main ingredients in our approach which follow from our treatment of the fully nonlinear hessian operators in 45 46 47 together with that of labutin 30 are a weak convergence result and a pointwise potential estimate for subharmonic functions the weak convergence result is analogous to that for hessian operators 45 46 while the potential estimate extends that found for quasilinear elliptic operators by kilpelainen and maly 27 weak convergence results in the theory of nonlinear partial differential equa tions have been addressed in the survey 13 in this paper we treat nonlinear elliptic divergence structure operators for which harnack type inequalities are available to state our result we need an appropriate notion of subharmonic func tion in the fully nonlinear case 46 we employed the viscosity notion based on differential inequalities for smooth test functions for quasilinear divergence structure equations the traditional definition using the homogeneous equation is more convenient accordingly we shall say that a continuous function w in a domain q in euclidean space ir is harmonic with respect to an operator or c harmonic in short if it is a solution of the homogeneous equation c u 0

existence of minimal solutions to quasilinear elliptic equations with several sub natural growth terms abstract we study the existence of positive solutions to quasilinear elliptic equations of the type p u u q in r n in the sub natural growth case 0 q p 1 where p u u p 2 u is the p laplacian with 1 p n and and are nonnegative radon measures on r n we construct minimal generalized solutions under certain generalized energy conditions on and to prove this we give new estimates for interaction between measures we also construct solutions to equations with several sub natural growth terms using the same methods

consistency of cross validation for comparing regression procedures theoretical developments on cross validation cv have mainly focused on selecting one among a list of finite dimensional models e g subset or order selection in linear regression or selecting a smoothing parameter e g bandwidth for kernel smoothing however little is known about consistency of cross validation when applied to compare between parametric and nonparametric methods or within nonparametric methods we show that under some conditions with an appropriate choice of data splitting ratio cross validation is consistent in the sense of selecting the better procedure with probability approaching 1 our results reveal interesting behavior of cross validation when comparing two models procedures converging at the same nonparametric rate in contrast to the parametric case it turns out that the proportion of data used for evaluation in cv does not need to be dominating in size furthermore it can even be of a smaller order than the proportion for estimation while not affecting the consistency property

existence of weak solutions of doubly nonlinear parabolic equations abstract we deal with a cauchy dirichlet problem with homogeneous boundary conditions on the parabolic boundary of a space time cylinder for doubly nonlinear parabolic equations whose prototype is t u div u m 1 d u p 2 d u f with a non negative lebesgue function f on the right hand side where p 2 n n 2 and m 0 the central objective is to establish the existence of weak solutions under the optimal integrability assumption on the inhomogeneity f the constructed solution is obtained by a limit of approximations i e we use solutions of regularized cauchy dirichlet problems and pass to the limit to receive a solution for the original cauchy dirichlet problem

choice of v for v fold cross validation in least squares density estimation this paper studies v fold cross validation for model selection in least squares density estimation the goal is to provide theoretical grounds for choosing v in order to minimize the least squares loss of the selected estimator we first prove a non asymptotic oracle inequality for v fold cross validation and its bias corrected version v fold penalization in particular this result implies that v fold penalization is asymptotically optimal in the nonparametric case then we compute the variance of v fold cross validation and related criteria as well as the variance of key quantities for model selection performance we show that these variances depend on v like 1 4 v 1 at least in some particular cases suggesting that the performance increases much from v 2 to v 5 or 10 and then is almost constant overall this can explain the common advice to take v 5 at least in our setting and when the computational power is limited as supported by some simulation experiments an oracle inequality and exact formulas for the variance are also proved for monte carlo cross validation also known as repeated cross validation where the parameter v is replaced by the number b of random splits of the data

flexible parsimonious smoothing and additive modeling a simple method is presented for fitting regression models that are nonlinear in the explanatory variables despite its simplicity or perhaps because of it the method has some powerful characteristics that cause it to be competitive with and often superior to more sophisticated techniques especially for small data sets in the presence of high noise

model selection for estimating treatment effects summary researchers often believe that a treatment s effect on a response may be heterogeneous with respect to certain baseline covariates this is an important premise of personalized medicine several methods for estimating heterogeneous treatment effects have been proposed however little attention has been given to the problem of choosing between estimators of treatment effects models that best estimate the regression function may not be best for estimating the effect of a treatment therefore there is a need for model selection methods that are targeted to treatment effect estimation we demonstrate an application of the focused information criterion in this setting and develop a treatment effect cross validation aimed at minimizing treatment effect estimation errors theoretically treatment effect cross validation has a model selection consistency property when the data splitting ratio is properly chosen practically treatment effect cross validation has the flexibility to compare different types of models we illustrate the methods by using simulation studies and data from a clinical trial comparing treatments of patients with human immunodeficiency virus

riesz potentials and pointwise estimates of solutions to anisotropic porous medium equation abstract for an anisotropic porous medium equation with l 1 regular lower order terms we prove the local boundedness and continuity of weak solutions in terms of the linear riesz potential of the right hand side of equation

approximate efficiency of a selection procedure for the number of regression variables we first show a simplification and a refinement of the result obtained by shibata 1976 it is an evaluation of the mean squared error of estimates of regression parameters when the number of regression variables is selected by a certain procedure the evaluation yields a good approximation to the efficiency of a selection procedure when the number of regression variables is small the approximate efficiency of fpe akaike 1970 aic akaike 1973 or cp mallows 1973 turns out to be not satisfactorily high although they are all asymptotically efficient in the sense of shibata 1981 based on such an approximation a procedure for choosing a out of the family fpe bhansali downham 1977 is suggested a further generalization is also developed consider a regression model y xji k l e where y yl yj is a vector of observations x is an n x k design matrix and f k fl1 fk 0 0 is a vector of regression parameters for simplicity we assume e el ej is a vector of independent normally distributed random variables with mean zero and variance a we call the above model model k in this paper we are concerned only with the problem of selecting the number k from a given range 1 k k under the model k we have a least squares estimate of the regression parameters 43 k h f1k which is a solution of x k x k k x k y here x k is an n x k submatrix consisting of the first k column vectors of the design matrix x hereafter f k is occasionally considered as the corresponding k dimensional vector with the undefined entries being zero suppose that observations yl yn come from a model ko 1 ko k with the regression parameter b such that f3k0 0 our main concern is the mean squared error

regularity symmetry and uniqueness of some integral type quasilinear equations abstract we study integral equations corresponding to some quasilinear equations with nonlinearities of lane emden and hartree type regularity symmetry and uniqueness of these equations are considered we obtain the uniqueness of the ground state of h 1 critical hartree equation and extend the moving plane method of integral equation in w chen c li b ou classification of solutions for an integral equation comm pure appl math lix 2006 0330 0343 w chen c li b ou classification of solutions for a system of integral equations comm partial differential equations 30 1 3 2005 59 65 to some integral equations corresponding to the p laplace equation we use ideas from the potential theories for the p laplace equations and hessian equations

on the harm that ignoring pretesting can cause in econometrics the same data set is typically used to select the model and to estimate the parameters in the selected model in applied econometrics practice however one typically acts as if the model had been given a priori thus ignoring the fact that the estimators are in fact pretest estimators hence one assumes incorrectly that the estimator is unbiased and that the reported variance conditional on the selected model is equal to its unconditional variance in this paper we nd the unconditional rst and second moments of the pretest estimator in fact of a more general estimator the wals estimator taking full account of the fact that model selection and estimation are an integrated procedure and show that the error in not reporting the correct moments can be large we also show that this error can vary substantially between di2erent model selection procedures finally we ask how the error increases when the number of auxiliary regressors increases

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

potential estimates of superquadratic elliptic systems with vmo coefficients in reifenberg domains abstract the purpose of this paper is to study an inhomogeneous dirichlet problem of a superquadratic nonlinear elliptic system with vmo coefficients in reifenberg flat domains we derive pointwise potential estimates for weak solution to the dirichlet problem as a consequence we derive global integrability estimates for solutions with respect to lorentz and morrey norms

reverse khas minskii condition the aim of this paper is to present and discuss some equivalent characterizations of p parabolicity for complete riemannian manifolds in terms of existence of special exhaustion functions in particular khas minskii in ergodic properties of recurrent diffusion prossesses and stabilization of solution to the cauchy problem for parabolic equations theor prob appl 5 2 1960 proved that if there exists a 2 superharmonic function mathcal k defined outside a compact set on a complete riemannian manifold r such that lim x to infty mathcal k x infty then r is 2 parabolic and sario and nakai in classification theory of riemann surfaces springer berlin 1970 were able to improve this result by showing that r is 2 parabolic if and only if there exists an evans potential i e a 2 harmonic function e r setminus k to mathbb r with lim x to infty mathcal e x infty in this paper we will prove a reverse khas minskii condition valid for any p 1 and discuss the existence of evans potentials in the nonlinear case

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

calibration and empirical bayes variable selection for the problem of variable selection for the normal linear model selection criteria such as aic c p bic and ric have fixed dimensionality penalties such criteria are shown to correspond to selection of maximum posterior models under implicit hyperparameter choices for a particular hierarchical bayes formulation based on this calibration we propose empirical bayes selection criteria that use hyperparameter estimates instead of fixed choices for obtaining these estimates both marginal and conditional maximum likelihood methods are considered as opposed to traditional fixed penalty criteria these empirical bayes criteria have dimensionality penalties that depend on the data their performance is seen to approximate adaptively the performance of the best fixed penalty criterion across a variety of orthogonal and nonorthogonal set ups including wavelet regression empirical bayes shrinkage estimators of the selected coefficients are also proposed

holder estimates of p harmonic extension operators it is now a well known fact that for 1 p the p harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a 1 p poincare inequality are locally holder continuous in this note we provide a characterization of domains in such metric spaces for which p harmonic extensions of holder continuous boundary data are globally holder continuous we also provide a link between this regularity property of the domain and the uniform p fatness of the complement of the domain

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

keller osserman a priori estimates and the harnack inequality for the evolution p laplace equation with singular absorption term abstract in this paper we study quasilinear equations of type u t d i v u p 2 u v x f u 0 p 2 u 0 despite the lack of comparison principle we prove a priori estimates of keller osserman type using these estimates we give a proof of the intrinsic harnack inequality

on the large sample minimal coverage probability of confidence intervals after model selection we give a large sample analysis of the minimal coverage probability of the usual confidence intervals for regression parameters when the underlying model is chosen by a conservative or overconsistent model selection procedure we derive an upper bound for the large sample limit minimal coverage probability of such intervals that applies to a large class of model selection procedures including the akaike information criterion as well as various pretesting procedures this upper bound can be used as a safeguard to identify situations where the actual coverage probability can be far below the nominal level we illustrate that the asymptotic upper bound can be statistically meaningful even in rather small samples

two theorems of n wiener for solutions of quasilinear elliptic equations relatively little is known about boundary behavior of solutions of quasilinear elliptic partial differential equations as compared to that of harmonic functions in this paper two results which in the harmonic case are due to n wiener are generalized to a nonlinear situation suppose that g is a bounded domain in r n we consider functions u g r which are free extremals of the variational integral

sign changing solutions for coupled nonlinear schrodinger equations with critical growth abstract we consider the following nonlinear schrodinger system with critical growth u j j u j i 1 k i j u i 2 2 u j 2 2 2 u j in u j 0 on j 1 k where is a bounded smooth domain in r n 2 2 n n 2 0 j 1 j 1 k 1 is the first eigenvalue of with zero dirichlet boundary condition we consider the repulsive case namely j j 0 j 1 k i j j i 0 i j i j 1 k the existence of infinitely many sign changing solutions as bound states is proved provided n 7 by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions

catching up faster by switching sooner a predictive approach to adaptive estimation with an application to the aic bic dilemma summary prediction and estimation based on bayesian model selection and model averaging and derived methods such as the bayesian information criterion bic do not always converge at the fastest possible rate we identify the catch up phenomenon as a novel explanation for the slow convergence of bayesian methods which inspires a modification of the bayesian predictive distribution called the switch distribution when used as an adaptive estimator the switch distribution does achieve optimal cumulative risk convergence rates in non parametric density estimation and gaussian regression problems we show that the minimax cumulative risk is obtained under very weak conditions and without knowledge of the underlying degree of smoothness unlike other adaptive model selection procedures such as the akaike information criterion aic and leave one out cross validation bic and bayes factor model selection are typically statistically consistent we show that this property is retained by the switch distribution which thus solves the aic bic dilemma for cumulative risk the switch distribution has an efficient implementation we compare its performance with aic bic and bayesian model selection and averaging on a regression problem with simulated data

can the strengths of aic and bic be shared a conflict between model indentification and regression estimation a traditional approach to statistical inference is to identify the true or best model first with little or no consideration of the specific goal of inference in the model identification stage can the pursuit of the true model also lead to optimal regression estimation in model selection it is well known that bic is consistent in selecting the true model and aic is minimax rate optimal for estimating the regression function a recent promising direction is adaptive model selection in which in contrast to aic and bic the penalty term is data dependent some theoretical and empirical results have been obtained in support of adaptive model selection but it is still not clear if it can really share the strengths of aic and bic model combining or averaging has attracted increasing attention as a means to overcome the model selection uncertainty can bayesian model averaging be optimal for estimating the regression function in a minimax sense we show that the answers to these questions are basically in the negative for any model selection criterion to be consistent it must behave suboptimally for estimating the regression function in terms of minimax rate of covergence and bayesian model averaging cannot be minimax rate optimal for regression estimation

the p laplace system with right hand side in divergence form inner and up to the boundary pointwise estimates abstract in this note we collect some very recent pointwise bounds for the gradient of solutions and for the solutions themselves to the p laplace system with right hand side in divergence form both estimates inside the domain for local solutions and global estimates for solutions to boundary value problems are discussed their formulation involves sharp maximal operators whose properties enable us to translate some aspects of the elliptic regularity theory into a merely harmonic analytic framework as a consequence a flexible comprehensive approach to estimates for solutions to the p laplace system for a broad class of norms is derived in particular global estimates under minimal boundary regularity are presented

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

a strongly consistent procedure for model selection in a regression problem summary we consider the multiple regression model yn xn en where yn and en are n vector random variables xn is an n x m matrix and 83 is an m vector of unknown regression parameters each component of 3 may be zero or nonzero which gives rise to 2 possible models for multiple regression we provide a decision rule for the choice of a model which is strongly consistent for the true model as n oo the result is proved under certain mild conditions for instance without assuming normality of the distribution of the components of en

harnack inequality and continuity of solutions to quasi linear degenerate parabolic equations with coefficients from kato type classes abstract for a general class of divergence type quasi linear degenerate parabolic equations with measurable coefficients and lower order terms from nonlinear kato type classes we prove local boundedness and continuity of solutions and the intrinsic harnack inequality for positive solutions

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

order selection for same realization predictions in autoregressive processes assume that observations are generated from an infinite order autoregressive ar process shibata ann statist 8 1980 147 164 considered the problem of choosing a finite order ar model allowing the order to become infinite as the number of observations does in order to obtain a better approximation he showed that for the purpose of predicting the future of an independent replicate akaike s information criterion aic and its variants are asymptotically efficient although shibata s concept of asymptotic efficiency has been widely accepted in the literature it is not a natural property for time series analysis this is because when new observations of a time series become available they are not independent of the previous data to overcome this difficulty in this paper we focus on order selection for forecasting the future of an observed time series referred to as same realization prediction we present the first theoretical verification that aic and its variants are still asymptotically elficient in the sense defined in section 4 for same realization predictions to obtain this result a technical condition easily met in common practice is introduced to simplify the complicated dependent structures among the selected orders estimated parameters and future observations in addition a simulation study is conducted to illustrate the practical implications of aic this study shows that aic also yields a satisfactory saute realization prediction in finite samples on the other hand a limitation of aic in same realization settings is pointed out it is interesting to note that this limitation of aic does not exist for corresponding independent cases

minimum complexity density estimation the authors introduce an index of resolvability that is proved to bound the rate of convergence of minimum complexity density estimators as well as the information theoretic redundancy of the corresponding total description length the results on the index of resolvability demonstrate the statistical effectiveness of the minimum description length principle as a method of inference the minimum complexity estimator converges to true density nearly as fast as an estimator based on prior knowledge of the true subclass of densities interpretations and basic properties of minimum complexity estimators are discussed some regression and classification problems that can be examined from the minimum description length framework are considered

on the wiener criterion and quasilinear obstacle problems we study the wiener criterion and variational inequalities with irregular obstacles for quasilinear elliptic operators a a x vu vu 3 vu p in r local solutions are continuous at wiener points of the obstacle func tion if p n 1 the converse is also shown to be true if p n 1 then a characterization of the thinness of a set at a point is given in terms of a superharmonic functions 1 introduction in this paper we study the wiener criterion and variational

local rademacher complexities and oracle inequalities in risk minimization let be a class of measurable functions f s 0 1 defined on a probability space s mathcal a p given a sample x1 xn of i i d random variables taking values in s with common distribution p let pn denote the empirical measure based on x1 xn we study an empirical risk minimization problem pnf min f given a solution fn of this problem the goal is to obtain very general upper bounds on its excess risk mathcal e p hat f n p hat f n inf f in mathcal f pf expressed in terms of relevant geometric parameters of the class using concentration inequalities and other empirical processes tools we obtain both distribution dependent and data dependent upper bounds on the excess risk that are of asymptotically correct order in many examples the bounds involve localized sup norms of empirical and rademacher processes indexed by functions from the class we use these bounds to develop model selection techniques in abstract risk minimization problems that can be applied to more specialized frameworks of regression and classification

regularity for quasilinear equations on degenerate singular sets we prove a new universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at singular set of degeneracy fancyscript s u x d u x 0 our main theorem reveals that along fancyscript s u u is asymptotically as regular as solutions to constant coefficient equations in particular along the critical set fancyscript s u du enjoys a modulus of continuity much superior than the possibly low continuity feature of the coefficients the results are new even in the context of linear elliptic equations where it is herein shown that h 1 weak solutions to text div left a ij x du right 0 with a ij elliptic and dini continuous are actually c 1 1 along fancyscript s u the results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations beyond typical elliptic regularity estimates precisely where the diffusion attributes of the equation collapse

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

on the generalized um pf classes of de giorgi ladyzhenskaya ural tseva and pointwise estimates of solutions to high order elliptic equations via wolff potentials abstract we consider quasilinear elliptic 2m order m 2 partial differential equations which prototype is m 1 d d m u p 2 d u f x x where is a bounded open set in r n n m p and f l 1 using an analogue of the kilpelainen mal method we obtain the local boundedness and continuity of arbitrary weak solution u w m p via the wolff potential w m p f

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

estimates at infinity for positive solutions to a class of p laplacian problems in rn abstract in this paper we consider a class of logistic type problems for the p laplacian in the whole space using minimization we prove existence of a positive solution and its behavior at infinity we also consider questions of uniqueness and sharp estimates at infinity

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

can one estimate the conditional distribution of post model selection estimators we consider the problem of estimating the conditional distribution of a post model selection estimator where the conditioning is on the selected model the notion of a post model selection estimator here refers to the combined procedure resulting from first selecting a model e g by a model selection criterion such as aic or by a hypothesis testing procedure and then estimating the parameters in the selected model e g by least squares or maximum likelihood all based on the same data set we show that it is impossible to estimate this distribution with reasonable accuracy even asymptotically in particular we show that no estimator for this distribution can be uniformly consistent not even locally this follows as a corollary to local minimax lower bounds on the performance of estimators for this distribution similar impossibility results are also obtained for the conditional distribution of linear functions e g predictors of the post model selection estimator

tuning parameter selectors for the smoothly clipped absolute deviation method summary the penalized least squares approach with smoothly clipped absolute deviation penalty has been consistently demonstrated to be an attractive regression shrinkage and selection method it not only automatically and consistently selects the important variables but also produces estimators which are as efficient as the oracle estimator however these attractive features depend on appropriate choice of the tuning parameter we show that the commonly used generalized crossvalidation cannot select the tuning parameter satisfactorily with a nonignorable overfitting effect in the resulting model in addition we propose a bic tuning parameter selector which is shown to be able to identify the true model consistently simulation studies are presented to support theoretical findings and an empirical example is given to illustrate its use in the female labor supply data

an asymptotic property of model selection criteria probability models are estimated by use of penalized log likelihood criteria related to akaike 1973 information criterion aic and minimum description length mdl the accuracies of the density estimators are shown to be related to the tradeoff between three terms the accuracy of approximation the model dimension and the descriptive complexity of the model classes the asymptotic risk is determined under conditions on the penalty term and is shown to be minimax optimal for some cases as an application we show that the optimal rate of convergence is simultaneously achieved for log densities in sobolev spaces w sub 2 sup s u without knowing the smoothness parameter s and norm parameter u in advance applications to neural network models and sparse density function estimation are also provided

riesz potential estimates for a class of double phase problems abstract a double phase problem with a bounded borel measure on the right hand side is studied we prove an optimal pointwise gradient estimate for such a measure data problem via riesz potentials of the measure under log dini continuity assumption on the modulating coefficients as a consequence we find an optimized c 1 regularity criterion

on the boundary behavior of kahler einstein metrics on log canonical pairs in this paper we study the boundary behavior of the negatively curved kahler einstein metric attached to a log canonical pair x d such that k x d is ample in the case where x is smooth and d has simple normal crossings support but possibly negative coefficients we provide a very precise estimate on the potential of the ke metric near the boundary d in the more general singular case d being assumed effective though we show that the ke metric has mixed cone and cusp singularities near d on the snc locus of the pair as a corollary we derive the behavior in codimension one of the ke metric of a stable variety

gradient estimates via linear and nonlinear potentials we prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems involving possibly degenerate quasilinear operators whose prototype is given by pu in particular no matter the nonlinearity of the equations considered we show that in the case p 2 a pointwise gradient estimate is possible using standard linear riesz potentials the proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem and on the other allows to encode the weaker coercivity properties of the operators considered in the case p 2 in the case p 2 we prove a new gradient estimate employing nonlinear potentials of wolff type

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

asymptotic properties of criteria for selection of variables in multiple regression in normal linear regression analysis many model selection rules proposed from various viewpoints are available for the information criteria aic fpe c p pss and bic the asymptotic distribution of the selected model and the asymptotic quadratic risk based on each criterion are explicitly obtained

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

decay estimates for wolff potentials in rn and gradient dependent quasilinear elliptic equations abstract we prove decay estimates of positive weak solutions u d 1 p r n c 1 r n and their gradients u of the p laplacian problem 1 p n u d 1 p r n p u a x where the measurable function a satisfies the decay estimate 0 0 a x 1 1 x n x r n the positive borel measure on r n generated by the right hand side function a and its associated wolff potential is known to be the main tool for pointwise estimates of u x 0 and u x therefore one of the main goal of this paper is to prove decay estimates for corresponding wolff potentials in r n the obtained decay estimates in conjunction with a sub supersolution method which is developed here for quasilinear elliptic equations are then applied to prove the existence of positive solutions for classes of gradient dependent quasilinear equations in the form u d 1 p r n p u a x f u u

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

asymptotic optimality for c_p c_l cross validation and generalized cross validation discrete index set c p c l cross validation and generalized cross validation are useful data driven techniques for selecting a good estimate from a proposed class of linear estimates the asymptotic behaviors of these procedures are studied some easily interpretable conditions are derived to demonstrate the asymptotic optimality it is argued that cross validation and generalized cross validation can be viewed as some special ways of applying c l applications in nearest neighbor nonparametric regression and in model selection are discussed in detail

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