symmetrization approach to concentration inequalities for empirical processes we introduce a symmetrization technique that allows us to translate a problem of controlling the deviation of some functionals on a product space from their mean into a problem of controlling the deviation between two independent copies of the functional as an application we give a new easy proof of talagrand s concentration inequality for empirical processes where besides symmetrization we use only talagrand s concentration inequality on the discrete cube 0 1 n as another application of this technique we prove new vapnik chervonenkis type inequalities for example for vc classes of functions we prove a classical inequality of vapnik and chervonenkis only with normalization by the sum of variance and sample variance

adaptive nonparametric bayesian inference using location scale mixture priors we study location scale mixture priors for nonparametric statistical problems including multivariate regression density estimation and classification we show that a rate adaptive procedure can be obtained if the prior is properly constructed in particular we show that adaptation is achieved if a kernel mixture prior on a regression function is constructed using a gaussian kernel an inverse gamma bandwidth and gaussian mixing weights

posterior consistency of gaussian process prior for nonparametric binary regression consider binary observations whose response probability is an unknown smooth function of a set of covariates suppose that a prior on the response probability function is induced by a gaussian process mapped to the unit interval through a link function in this paper we study consistency of the resulting posterior distribution if the covariance kernel has derivatives up to a desired order and the bandwidth parameter of the kernel is allowed to take arbitrarily small values we show that the posterior distribution is consistent in the l 1 distance as an auxiliary result to our proofs we show that under certain conditions a gaussian process assigns positive probabilities to the uniform neighborhoods of a continuous function this result may be of independent interest in the literature for small ball probabilities of gaussian processes

fast rates for support vector machines using gaussian kernels for binary classification we establish learning rates up to the order of n 1 for support vector machines svms with hinge loss and gaussian rbf kernels these rates are in terms of two assumptions on the considered distributions tsybakov s noise assumption to establish a small estimation error and a new geometric noise condition which is used to bound the approximation error unlike previously proposed concepts for bounding the approximation error the geometric noise assumption does not employ any smoothness assumption 1 introduction in recent years support vector machines svms have been the subject of many theoretical considerations despite this effort their learning performance on restricted classes of distributions is still widely unknown in particular it is unknown under which nontrivial circumstances svms can guarantee fast learning rates the aim of this work is to use concepts like tsybakov s noise assumption and local rademacher averages to establish learning rates up to the order of n 1 for nontrivial distributions in addition to these concepts that are used to deal with the stochastic part of the analysis we also introduce a geometric assumption for distributions that allows us to estimate the approximation properties of gaussian rbf kernels unlike many other concepts introduced for bounding the approximation error our geometric assumption is not in terms of smoothness but describes the concentration and the noisiness of the data generating distribution near the decision boundary let us formally introduce the statistical classification problem to this end let us fix a subset x r d we write y 1 1 given a finite training set

testing the order of a model using locally conic parametrization population mixtures and stationary arma processes in this paper we address the problem of testing hypotheses using the likelihood ratio test statistic in nonidentifiable models with application to model selection in situations where the parametrization for the larger model leads to nonidentifiability in the smaller model we give two major applications the case where the number of populations has to be tested in a mixture and the case of stationary arma p q processes where the order p q has to be tested we give the asymptotic distribution for the likelihood ratio test statistic when testing the order of the model in the case of order selection for armas the asymptotic distribution is invariant with respect to the parameters generating the process a locally conic parametrization is a key tool in deriving the limiting distributions it allows one to discover the deep similarity between the two problems

renyi divergence and kullback leibler divergence renyi divergence is related to renyi entropy much like kullback leibler divergence is related to shannon s entropy and comes up in many settings it was introduced by renyi as a measure of information that satisfies almost the same axioms as kullback leibler divergence and depends on a parameter that is called its order in particular the renyi divergence of order 1 equals the kullback leibler divergence we review and extend the most important properties of renyi divergence and kullback leibler divergence including convexity continuity limits of sigma algebras and the relation of the special order 0 to the gaussian dichotomy and contiguity we also show how to generalize the pythagorean inequality to orders different from 1 and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs for all orders and present several other minimax results

concentration inequalities large and moderate deviations for self normalized empirical processes we consider the supremum w n of self normalized empirical processes indexed by unbounded classes of functions f such variables are of interest in various statistical applications for example the likelihood ratio tests of contamination using the herbst method we prove an exponential concentration inequality for w n under a second moment assumption on the envelope function of f this inequality is applied to obtain moderate deviations for w n we also provide large deviations results for some unbounded parametric classes f

self normalized large deviations let x x n n geq 1 be a sequence of independent and identically distributed random variables the classical cramer chernoff large deviation states that lim n to infty n 1 ln p sum i 1 n x i n geq x ln rho x if and only if the moment generating function of x is finite in a right neighborhood of zero this paper uses n p 1 p v n p n p 1 p sum i 1 n x i p 1 p p 1 as the normalizing constant to establish a self normalized large deviation without any moment conditions a self normalized moderate deviation that is the asymptotic probability of p s n v n p geq x n for x n o n p 1 p is also found for any x in the domain of attraction of a normal or stable law as a consequence a precise constant in the self normalized law of the iterated logarithm of griffin and kuelbs is obtained applications to the limit distribution of self normalized sums the asymptotic probability of the t statistic as well as to the erdos renyi shepp law of large numbers are also discussed

global existence and nonexistence results for a generalized davey stewartson system we consider a system of three equations which will be called generalized davey stewartson equations involving three coupled equations two for the long waves and one for the short wave propagating in an infinite elastic medium we classify the system according to the signs of the parameters conserved quantities related to mass momentum and energy are derived as well as a specific instance of the so called virial theorem using these conservation laws and the virial theorem both global existence and nonexistence results are established under different constraints on the parameters in the elliptic elliptic elliptic case

a test of non identifying restrictions and confidence regions for partially identified parameters abstract we propose an easily implementable test of the validity of a set of theoretical restrictions on the relationship between economic variables which do not necessarily identify the data generating process the restrictions can be derived from any model of interactions allowing censoring and multiple equilibria when the restrictions are parameterized the test can be inverted to yield confidence regions for partially identified parameters thereby complementing other proposals primarily chernozhukov et al chernozhukov v hong h tamer e 2007 estimation and confidence regions for parameter sets in econometric models econometrica 75 1243 1285

on uniform deviations of general empirical risks with unboundedness dependence and high dimensionality the statistical learning theory of risk minimization depends heavily on probability bounds for uniform deviations of the empirical risks classical probability bounds using hoeffding s inequality cannot accommodate more general situations with unbounded loss and dependent data the current paper introduces an inequality that extends hoeffding s inequality to handle these more general situations we will apply this inequality to provide probability bounds for uniform deviations in a very general framework which can involve discrete decision rules unbounded loss and a dependence structure that can be more general than either martingale or strong mixing we will consider two examples with high dimensional predictors autoregression ar with l1 loss and arx model with variable selection for sign classification which uses both lagged responses and exogenous predictors

model averaging in semiparametric estimation of treatment effects choosing the covariates and functional form of the propensity score is an important choice for estimating treatment effects this paper proposes a data driven way of averaging the estimators over candidate specifications to resolve the specification uncertainty in the propensity score weighting estimation of the att the proposed procedures minimize the estimated mse of the att estimator in a local asymptotic framework we formulate model averaging as a statistical decision problem in a limit experiment and derive an averaging scheme that is bayes optimal with respect to a given prior the averaging estimator outperforms selection estimators and the estimators in any of the candidate models in terms of bayes asymptotic mse our monte carlo studies illustrate the size of the mse gains we apply the averaging procedure to evaluate the effect of a labor market program

optimal aggregation of classifiers in statistical learning classification can be considered as nonparametric estimation of sets where the risk is defined by means of a specific distance between sets associated with misclassification error it is shown that the rates of convergence of classifiers depend on two parameters the complexity of the class of candidate sets and the margin parameter the dependence is explicitly given indicating that optimal fast rates approaching o n 1 can be attained where n is the sample size and that the proposed classifiers have the property of robustness to the margin the main result of the paper concerns optimal aggregation of classifiers we suggest a classifier that automatically adapts both to the complexity and to the margin and attains the optimal fast rates up to a logarithmic factor

boundedly pivotal structural change tests in continuous updating gmm with strong weak identification and completely unidentified cases abstract this paper develops structural change tests in the continuous updating gmm framework that are robust to weak identification we propose likelihood ratio like anderson rubin 1949 estimation of the parameters of a single equation in a complete system of stochastic equations annals of mathematical statistics 20 46 63 and kleinbergen 2005 testing parameters in gmm without assuming that they are identified manuscript brown university types of tests since the limits of the test statistics are not nuisance parameter free bounds for the limit of the test statistics are derived the bounds are nuisance parameter free and robust to identification problems simulations show that the anderson rubin 1949 and kleinbergen 2005 testing parameters in gmm without assuming that they are identified manuscript brown university type of tests have very good small sample properties in the case of weak instruments the sup lm test of andrews 1993a tests for parameter instability and structural change with unknown change point econometrica 61 821 856 rejects the true null of parameter stability more than the nominal level and also has low power in the weak instrument setup

the focusing singularity of the davey stewartson equations for gravity capillary surface waves abstract we analyze the self focusing of gravity capillary surface waves modeled by the davey stewartson equations this system governs the coupling of the wave amplitude to the induced mean flow and is an anisotropic two dimensional schrodinger equation with a non local cubic nonlinearity with accurate numerical simulations based on dynamic rescaling and asymptotic analysis we show that the dynamics of singularity formation is critical as in the usual two dimensional cubic schrodinger equation which is the deep water limit of the davey stewartson equations we also derive a sharp upper bound for the initial amplitude of the wave that prevents singularity formation

complexity an interpretability criterion for multiple comparisons abstract this article deals with multiple comparison methods for a set of populations which can be described in terms of testing hypotheses of homogeneity of subsets criteria for evaluating procedures are defined in terms of the set of outcomes of all tests termed the pattern of decisions in particular complexity is introduced as a new criterion based on interpretability of that pattern the criteria are defined and discussed in a general framework and then applied to the comparison of normal means for the cases considered results favor an f over a range procedure and are mixed in the comparison of the f and the range procedure with a new gap procedure

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

new concentration inequalities in product spaces we introduce three new ways to measure the distance from a point to a subset of a product space and we prove corresponding concentration inequalities each of them allows to control the fluctuation of a new class of random variables

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

smoothed quantile regression for panel data abstract this paper studies fixed effects estimation of quantile regression models for panel data under an asymptotic framework where both the numbers of individuals and time periods grow at the same rate we show that the fixed effects estimator for the smoothed objective function has a limiting normal distribution with a bias in the mean and provide the analytic form of the asymptotic bias we propose a one step bias correction estimator based on the analytic bias formula obtained from the asymptotic analysis importantly our results cover the case that observations are dependent over time we illustrate the effects of the bias correction through simulations

on the bootstrap of the maximum score estimator this paper shows that the bootstrap does not consistently estimate the asymptotic distribution of the maximum score estimator the theory developed also applies to other estimators within a cube root convergence class for some single parameter estimators in this class the results suggest a simple method for inference based upon the bootstrap

closing the gap in the purely elliptic generalized davey stewartson system abstract in this note we improve the results presented previously on global existence and global nonexistence for the solutions of the purely elliptic generalized davey stewartson system these results left a gap in the parameter range where neither a global existence result nor a global nonexistence result could be established here we are able to show that when the coupling parameter is negative there is no gap moreover in the case where the coupling parameter is positive we reduce the size of the gap

standing waves for a generalized davey stewartson system in this paper we establish the existence of non trivial solutions for a semi linear elliptic partial differential equation with a non local term this result allows us to prove the existence of standing wave ground state solutions for a generalized davey stewartson system a sharp upper bound is also obtained on the size of the initial values for which solutions exist globally

convex clustering via l1 fusion penalization summary we study the large sample behaviour of a convex clustering framework which minimizes the sample within cluster sum of squares under an l1 fusion constraint on the cluster centroids this recently proposed approach has been gaining in popularity however its asymptotic properties have remained mostly unknown our analysis is based on a novel representation of the sample clustering procedure as a sequence of cluster splits determined by a sequence of maximization problems we use this representation to provide a simple and intuitive formulation for the population clustering procedure we then demonstrate that the sample procedure consistently estimates its population analogue and we derive the corresponding rates of convergence the proof conducts a careful simultaneous analysis of a collection of m estimation problems whose cardinality grows together with the sample size on the basis of the new perspectives gained from the asymptotic investigation we propose a key post processing modification of the original clustering framework we show both theoretically and empirically that the resulting approach can be successfully used to estimate the number of clusters in the population using simulated data we compare the proposed method with existing number of clusters and modality assessment approaches and obtain encouraging results we also demonstrate the applicability of our clustering method to the detection of cellular subpopulations in a single cell virology study

universal prediction of individual binary sequences in the presence of noise the problem of predicting the next outcome of an individual binary sequence based on noisy observations of the past is considered the goal of the predictor is to perform for each individual sequence almost as well as the best in a set of experts where performance is evaluated using a general loss function a comprehensive approach to prediction in this noisy setting is presented and proven generally efficient under appropriate conditions as an illustration of the applicability of the approach suggested for concrete situations two important special cases are explicitly treated the first is the case where the data corrupting noise process is binary valued where the observed bit is the bitwise xor of the clean bit and the noise bit the second case is that of real valued additive noise it is shown that even in this more challenging situation where the information available to the predictor regarding the past sequence is incomplete a predictor can be guaranteed to successfully compete with a whole set of experts in considerably strong senses

convexity classification and risk bounds many of the classification algorithms developed in the machine learning literature including the support vector machine and boosting can be viewed as minimum contrast methods that minimize a convex surrogate of the 0 1 loss function the convexity makes these algorithms computationally efficient the use of a surrogate however has statistical consequences that must be balanced against the computational virtues of convexity to study these issues we provide a general quantitative relationship between the risk as assessed using the 0 1 loss and the risk as assessed using any nonnegative surrogate loss function we show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function that it satisfies a pointwise form of fisher consistency for classification the relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications we also present a refined version of this result in the

two stage plans for estimating the inverse of a monotone function this study investigates two stage plans based on nonparametric procedures for estimating an inverse regression function at a given point specifically isotonic regression is used at stage one to obtain an initial estimate followed by another round of isotonic regression in the vicinity of this estimate at stage two it is shown that such two stage plans accelerate the convergence rate of one stage procedures and are superior to existing two stage procedures that use local parametric approximations at stage two when the available budget is moderate and or the regression function is ill behaved both wald and likelihood ratio type confidence intervals for the threshold value of interest are investigated and the latter are recommended in applications due to their simplicity and robustness the developed plans are illustrated through a comprehensive simulation study and an application to car fuel efficiency data

the sharp threshold and limiting profile of blow up solutions for a davey stewartson system abstract the blow up solutions of the cauchy problem for the davey stewartson system which is a model equation in the theory of shallow water waves are investigated firstly the existence of the ground state for the system derives the best constant of a gagliardo nirenberg type inequality and the variational character of the ground state secondly the blow up threshold of the davey stewartson system is developed in r 3 thirdly the mass concentration is established for all the blow up solutions of the system in r 2 finally the existence of the minimal blow up solutions in r 2 is constructed by using the pseudo conformal invariance the profile of the minimal blow up solutions as t t blow up time is in detail investigated in terms of the ground state

moment inequalities for functions of independent random variables a general method for obtaining moment inequalities for functions of independent random variables is presented it is a generalization of the entropy method which has been used to derive concentration inequalities for such functions boucheron lugosi and massart ann probab 31 2003 1583 1614 and is based on a generalized tensorization inequality due to latala and oleszkiewicz lecture notes in math 1745 2000 147 168 the new inequalities prove to be a versatile tool in a wide range of applications we illustrate the power of the method by showing how it can be used to effortlessly re derive classical inequalities including rosenthal and kahane khinchine type inequalities for sums of independent random variables moment inequalities for suprema of empirical processes and moment inequalities for rademacher chaos and u statistics some of these corollaries are apparently new in particular we generalize talagrand s exponential inequality for rademacher chaos of order 2 to any order we also discuss applications for other complex functions of independent random variables such as suprema of boolean polynomials which include as special cases subgraph counting problems in random graphs

on the power of tests for multiple comparison of three normal means abstract this article gives theoretical reasons for the findings of a simulation study by shaffer 1981 two well established procedures for multiple comparison of three normal means are compared an f procedure and an r procedure it turns out that for a large set of parameters the average number of rejected pairwise hypotheses is larger when the f procedure is applied to achieve our result we have to prove a special instance of a conjecture by david lachenbruch and brandis 1972

using a monotone single index model to stabilize the propensity score in missing data problems and causal inference the augmented inverse weighting method is one of the most popular methods for estimating the mean of the response in causal inference and missing data problems an important component of this method is the propensity score popular parametric models for the propensity score include the logistic probit and complementary log log models a common feature of these models is that the propensity score is a monotonic function of a linear combination of the explanatory variables to avoid the need to choose a model we model the propensity score via a semiparametric single index model in which the score is an unknown monotonic nondecreasing function of the given single index under this new model the augmented inverse weighting estimator aiwe of the mean of the response is asymptotically linear semiparametrically efficient and more robust than existing estimators moreover we have made a surprising observation the inverse probability weighting and aiwes based on a correctly specified parametric model may have worse performance than their counterparts based on a nonparametric model a heuristic explanation of this phenomenon is provided a real data example is used to illustrate the proposed methods

a comparison of one sided methods to identify significant individual outcomes in a multiple outcome setting stepwise tests or global tests with closed testing we compare two approaches to the identification of individual significant outcomes when a comparison of two groups involves multiple outcome variables the approaches are all designed to control the familywise error rate fwe with any subset of the null hypotheses being true in the strong sense the first approach is initially to use a global test of the overall hypothesis that the groups are equivalent for all variables followed by an application of the closed testing algorithm of marcus peritz and gabriel the global tests considered here are ordinary least squares ols generalized least squares gls an approximation to a likelihood ratio test alr and a new test based on an approximation to the most powerful similar test for simple alternatives the second approach is that of stepwise testing which tests the univariate hypotheses in a specific order with appropriate adjustment to the univariate p values for multiplicity the stepwise tests considered include both step down and step up tests of a general type and likewise permutation tests that incorporate the dependence structure of the data we illustrate the tests with two examples of birth outcomes a comparison of cocaine exposed new borns to control new borns on neurobehavioural and physical growth variables and in a separate study a comparison of babies born to diabetic mothers and babies born to non diabetic mothers on minor malformations after describing the methods and analysing the birth outcome data we use simulations on gaussian data to provide guidelines for the use of these procedures in terms of power and computation

parametric rate inference for one sided differentiable parameters suppose one has a collection of parameters indexed by a possibly infinite dimensional set given data generated from some distribution the objective is to estimate the maximal parameter in this

new dependence coefficients examples and applications to statistics to measure the dependence between a real valued random variable x and a algebra we consider four distances between the conditional distribution function of x given and the distribution function of x the coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations in particular we show that they are well adapted to functions of mixing sequences iterated random functions and dynamical systems starting from a new covariance inequality we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence we obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis under a minimal condition on the coefficients using recent results we show that our coefficients may be also used to obtain various exponential inequalities a concentration inequality for lipschitz functions and a berry esseen type inequality

a note on two sided distribution free treatment versus control multiple comparisons abstract distribution free treatment versus control multiple comparison procedures using joint ranking and pairwise ranking are compared the pairwise ranking method is extended to allow for all contrast comparisons among the treatments and the control the author shows by counterexample that the joint ranking method does not control the maximum type 1 error rate

the central limit theorem and the law of iterated logarithm for empirical processes under local conditions a clt and a lil are proved under weak l2 gaussian bracketing conditions weaker than the usual ones these results have wide applicability and in particular provide an improvement of the jain marcus central limit theorem for c s valued random variables

concentration inequalities using the entropy method we investigate a new methodology worked out by ledoux and massart to prove concentration of measure inequalities the method is based on certain modified logarithmic sobolev inequalities we provide some very simple and general ready to use inequalities one of these inequalities may be considered as an exponential version of the efron stein inequality the main purpose of this paper is to point out the simplicity and the generality of the approach we show how the new method can recover many of talagrand s revolutionary inequalities and provide new applications in a variety of problems including rademacher averages rademacher chaos the number of certain small subgraphs in a random graph and the minimum of the empirical risk in some statistical estimation problems

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

efficient shrinkage in parametric models this paper introduces shrinkage for general parametric models we show how to shrink maximum likelihood estimators towards parameter subspaces de ned by general nonlinear restrictions we derive the asymptotic distribution and risk of the generalized shrinkage estimator using a local asymptotic framework we show that if the shrinkage dimension exceeds two the asymptotic risk of the shrinkage estimator is strictly less than that of the mle this reduction holds globally in the parameter space we show that the reduction in asymptotic risk is substantial even for moderately large values of the parameters the formula simplify in a very convenient way in the context of high dimensional models we derive a simple bound for the asymptotic risk we also provide a new large sample minimax e ciency bound we use the concept of local asymptotic minimax bounds a generalization of the conventional asymptotic minimax bounds the

delta method in large deviations and moderate deviations for estimators the delta method is a popular and elementary tool for deriving limiting distributions of transformed statistics while applications of asymptotic distributions do not allow one to obtain desirable accuracy of approximation for tail probabilities the large and moderate deviation theory can achieve this goal motivated by the delta method in weak convergence a general delta method in large deviations is proposed the new method can be widely applied to driving the moderate deviations of estimators and is illustrated by examples including the wilcoxon statistic the kaplan meier estimator the empirical quantile processes and the empirical copula function we also improve the existing moderate deviations results for m estimators and l statistics by the new method some applications of moderate deviations to statistical hypothesis testing are provided

inegalites de concentration pour les processus empiriques de classes de parties we propose new concentration inequalities for maxima of set indexed empirical processes our approach is based either on entropy inequalities or on martingale methods the improvements we get concern the rate function which is exactly the large deviations rate function of a binomial law in most of the cases

symmetry full symmetry groups and some exact solutions to a generalized davey stewartson system the lie symmetry algebra of a generalized davey stewartson gds system is obtained the general element of this algebra depends on eight arbitrary functions of time which has a kac moody virasoro loop algebra structure and is isomorphic to that of the standard integrable davey stewartson equations under certain conditions imposed on parameters and arbitrary functions then based on the symmetry group direct method proposed by lou and ma j phys a 38 l129 2005 the full symmetry groups of the gds system are obtained from the full symmetry groups both the lie symmetry group and a group of discrete transformations can be obtained finally some exact solutions involving sech sech2 sech2 and tanh tanh2 tanh2 type solitary wave solutions are presented by a generalized subequation expansion method

complexity regularization via localized random penalties in this article model selection via penalized empirical loss minimization in nonparametric classification problems is studied data dependent penalties are constructed which are based on estimates of the complexity of a small subclass of each model class containing only those functions with small empirical loss the penalties are novel since those considered in the literature are typically based on the entire model class oracle inequalities using these penalties are established and the advantage of the new penalties over those based on the complexity of the whole model class is demonstrated

likelihood ratio tests for monotone functions we study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function the likelihood ratio test for this hypothesis is derived in the particular case of interval censoring or current status data and its limiting distribution is obtained the limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving brownian motion t 2 and greatest convex minorants thereof inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function we also study the behavior of the statistic under local and fixed alternatives 1 introduction we shall consider likelihood ratio tests and the corresponding confidence intervals in a class of problems involving nonparametric estimation of a monotone function the problem in each case involves testing the null hypothesis h 0 that the monotone function has a particular value at afixed point t 0 in the domain of the function of course with each testing problem there is a related problem of finding confidence intervals here are some examples of the problems we have in mind

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

rates of convergence for the gaussian mixture sieve gaussian mixtures provide a convenient method of densityestimation that lies somewhere between parametric models and kernel densityestimators when the number of components of the mixture is allowed to increase as sample size increases the model is called a mixture sieve we establish a bound on the rate of convergence in hellinger distance for densityestimation using the gaussian mixture sieve assuming that the true density is itself a mixture of gaussians the underlying mixing measure of the true densityis not necessarilyassumed to have finite support computing the rate involves some delicate calculations since the size of the sieve as measured bybracketing entropy and the saturation rate cannot be found using standard methods when the mixing measure has compact support using kn n 2 3 log n 1 3 components in the mixture yields a rate of order log n 1 6 n1 6 for every 0 the rates depend heavilyon the tail behavior of the true density the sensitivity to the tail behavior is diminished byusing a robust sieve which includes a long tailed component in the