learning a decision boundary from stochastic examples incremental algorithms with and without queries even if it is not possible to reproduce a target input output relation a learning machine should be able to minimize the probability of making errors a practical learning algorithm should also be simple enough to go without memorizing example data if possible incremental algorithms such as error backpropagation satisfy this requirement we propose incremental algorithms that provide fast convergence of the machine parameter to its optimal choice o with respect to the number of examples t we will consider the binary choice model whose target relation has a blurred boundary and the machine whose parameter specifies a decision boundary to make the output prediction the question we wish to address here is how fast can approach o depending upon whether in the learning stage the machine can specify inputs as queries to the target relation or the inputs are drawn from a certain distribution if queries are permitted the machine can achieve the fastest convergence o 2 o t 1 if not o t 1 convergence is generally not attainable for learning without queries we showed in a previous paper that the error minimum algorithm exhibits a slow convergence o 2 o t 2 3 we propose here a practical algorithm that provides a rather fast convergence o t 4 5 it is possible to further accelerate the convergence by using more elaborate algorithms the fastest convergence turned out to be o lnt 2t 1 this scaling is considered optimal among possible algorithms and is not due to the incremental nature of our algorithm

maximum likelihood estimation of a multi dimensional log concave density fn is shown to have smaller mean integrated squared error compared with kernel based methods even when we allow the use of a theoret ical optimal fixed bandwidth for the kernel estimator that would not be available in practice we also present a real data clustering example which shows that our methodology can be used in conjunction with the expectation maximization algorithm to fit finite mixtures of log concave densities

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

acceleration of the shiftable mbi o 1 algorithm for bilateral filtering and nonlocal means a direct implementation of the bilateral filter requires o s2 operations per pixel where s is the effective width of the spatial kernel a fast implementation of the bilateral filter that required o 1 operations per pixel with respect to s was recently proposed this was done by using trigonometric functions for the range kernel of the bilateral filter and by exploiting their so called shiftability property in particular a fast implementation of the gaussian bilateral filter was realized by approximating the gaussian range kernel using raised cosines later it was demonstrated that this idea could be extended to a larger class of filters including the popular non local means filter as already observed a flip side of this approach was that the run time depended on the width r of the range kernel for an image with dynamic range 0 t the run time scaled as o t2 r2 with r this made it difficult to implement narrow range kernels particularly for images with large dynamic range in this paper we discuss this problem and propose some simple steps to accelerate the implementation in general and for small r in particular we provide some experimental results to demonstrate the acceleration that is achieved using these modifications

new donsker classes several classes of functions are shown to be donsker by an argument based on partitioning the sample space one example is the class of all nondecreasing functions f r r such that 0 f f for a given function f with f 2 dp p

complex linear phase filters for efficient image coding with the exception of the haar basis real valued orthogonal wavelet filter banks with compact support lack symmetry and therefore do not possess linear phase this has led to the use of biorthogonal filters for coding of images and other multidimensional data there are however complex solutions permitting the construction of compactly supported orthogonal linear phase qmf filter banks by explicitly seeking solutions in which the imaginary part of the filter coefficients is small enough to be approximated to zero real symmetric filters can be obtained that achieve excellent compression performance

the sorted effects method discovering heterogeneous effects beyond their averages the partial ceteris paribus effects of interest in nonlinear and interactive linear models are heterogeneous as they can vary dramatically with the underlying observed or unobserved covariates despite the apparent importance of heterogeneity a common practice in modern empirical work is to largely ignore it by reporting average partial effects or at best average effects for some groups while average effects provide very convenient scalar summaries of typical effects by definition they fail to reflect the entire variety of the heterogeneous effects in order to discover these effects much more fully we propose to estimate and report sorted effects a collection of estimated partial effects sorted in increasing order and indexed by percentiles by construction the sorted effect curves completely represent and help visualize the range of the heterogeneous effects in one plot they are as convenient and easy to report in practice as the conventional average partial effects they also serve as a basis for classification analysis where we divide the observational units into most or least affected groups and summarize their characteristics we provide a quantification of uncertainty standard errors and confidence bands for the estimated sorted effects and related classification analysis and provide confidence sets for the most and least affected groups the derived statistical results rely on establishing key new mathematical results on hadamard differentiability of a multivariate sorting operator and a related classification operator which are of independent interest we apply the sorted effects method and classification analysis to demonstrate several striking patterns in the gender wage gap we find that this gap is particularly strong for married women ranging from 60 to 0 between the 2 and 98 percentiles as a function of observed and unobserved characteristics while the gap for never married women ranges from 40 to 20 the most adversely affected women tend to be married do not have college degrees work in sales and have high levels of potential experience

the behavior of the npmle of a decreasing density near the boundaries of the support we investigate the behavior of the nonparametric maximum likelihood estimator f n for a decreasing density f near the boundaries of the support of f we establish the limiting distribution of f n n where we need to distinguish between different values of 0 a 1 similar results are obtained for the upper endpoint of the support in the case it is finite this yields consistent estimators for the values of f at the boundaries of the support the limit distribution of these estimators is established and their performance is compared with the penalized maximum likelihood estimator

semiparametric estimation of the random utility model with rank ordered choice data abstract we propose semiparametric methods for estimating random utility models using rank ordered choice data our primary method is the generalized maximum score gms estimator with partially rank ordered data the gms estimator allows for arbitrary forms of interpersonal heteroskedasticity with fully rank ordered data the gms estimator becomes considerably more flexible allowing for random coefficients and alternative specific heteroskedasticity and correlations the gms estimator has a non standard asymptotic distribution and a convergence rate of n 1 3 we proceed to construct its smoothed version which is asymptotically normal with a faster convergence rate of n d 2 d 1 where d 2 increases in the strength of smoothness assumptions

testing for threshold effects in regression models in this article we develop a general method for testing threshold effects in regression models using sup likelihood ratio lr type statistics although the sup lr type test statistic has been considered in the literature our method for establishing the asymptotic null distribution is new and nonstandard the standard approach in the literature for obtaining the asymptotic null distribution requires that there exist a certain quadratic approximation to the objective function the article provides an alternative novel method that can be used to establish the asymptotic null distribution even when the usual quadratic approximation is intractable we illustrate the usefulness of our approach in the examples of the maximum score estimation maximum likelihood estimation quantile regression and maximum rank correlation estimation we establish consistency and local power properties of the test we provide some simulation results and also an empirical application to tipping in racial segregation this ar

asymptotic normality of the l_1 error of the grenander estimator groeneboom introduced a jump process that can be used among other things to study the asymptotic properties of the grenander estimator of a monotone density in this paper we derive the asymptotic normality of a suitably rescaled version of the l 1 error of the grenander estimator using properties of this jump process

maximum likelihood estimation for multivariate observations of markov sources parameter estimation for multivariate functions of markov chains a class of versatile statistical models for vector random processes is discussed the model regards an ordered sequence of vectors as noisy multivariate observations of a markov chain mixture distributions are a special case the foundations of the theory presented here were established by baum petrie soules and weiss a powerful representation theorem by fan is employed to generalize the analysis of baum em et al to a larger class of distributions

robust out of sample inference this paper presents analytical empirical and simulation results concerning inference about the moments of nondifferentiable functions of out of sample forecasts and forecast errors special attention is given to the measurement of a model s predictive ability using the test of equal mean absolute error tests for equal mean absolute error and mean square error are used to evaluate predictions of excess returns to the s p 500 composite simulations indicate that appropriately constructed tests for equal mean absolute error can provide more accurately sized and more powerful tests than inappropriately constructed tests for equal mean absolute error and mean square error

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

nonparametric tests for constant tail dependence with an application to energy and finance the present paper proposes new tests for detecting structural breaks in the tail dependence of multivariate time series using the concept of tail copulas to obtain asymptotic properties we derive a new limit result for the sequential empirical tail copula process moreover consistency of both the tests and a change point estimator are proven we analyze the finite sample behavior of the tests by monte carlo simulations finally and crucial from a risk management perspective we apply the new findings to datasets from energy and financial markets

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

maximum score estimation of disequilibrium models and the role of anticipatory price setting abstract in this paper a general model of disequilibrium quantity determination under median regression assumptions is combined with weak assumptions on the price adjustment to yield a consistent maximum score estimator in contrast to sapra 1986 the supply and demand parameters are separately identifiable and the estimator employs all of the observed data the price adjustment assumptions can be motivated by the anticipatory pricing model of green and laffont 1981 in contrast to the green laffont model however our assumptions only require that long run walrasian prices have positive probability as an illustrative application maximum score estimates of a model of the us commercial loan market are computed and compared to maximum likelihood estimates monte carlo results are also reported

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

persistence of fractional brownian motion with moving boundaries and applications we consider various problems related to the persistence probability of fractional brownian motion fbm which is the probability that the fbm x stays below a certain level until time t recently oshanin et al 2012 arxiv 1209 3313v2 have studied a physical model where persistence properties of fbm are shown to be related to scaling properties of a quantity jn called the steady state current it turns out that for this analysis it is important to determine persistence probabilities of fbm with a moving boundary we show that one can add a boundary of logarithmic order to an fbm without changing the polynomial rate of decay of the corresponding persistence probability which proves a result needed in oshanin et al 2013 phys rev lett at press arxiv 1209 3313v2 moreover we complement their findings by considering the continuous time version of jt finally we use the results for moving boundaries in order to improve estimates by molchan 1999 commun math phys 205 97 111 concerning the persistence properties of other quantities of interest such as the time when an fbm reaches its maximum on the time interval 0 1 or the last zero in the interval 0 1

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

fast and provably accurate bilateral filtering the bilateral filter is a non linear filter that uses a range filter along with a spatial filter to perform edge preserving smoothing of images a direct computation of the bilateral filter requires o s operations per pixel where s is the size of the support of the spatial filter in this paper we present a fast and provably accurate algorithm for approximating the bilateral filter when the range kernel is gaussian in particular for box and gaussian spatial filters the proposed algorithm can cut down the complexity to o 1 per pixel for any arbitrary s the algorithm has a simple implementation involving n 1 spatial filterings where n is the approximation order we give a detailed analysis of the filtering accuracy that can be achieved by the proposed approximation in relation to the target bilateral filter this allows us to estimate the order n required to obtain a given accuracy we also present comprehensive numerical results to demonstrate that the proposed algorithm is competitive with the state of the art methods in terms of speed and accuracy

a massive data framework for m estimators with cubic rate the divide and conquer method is a common strategy for handling massive data in this article we study the divide and conquer method for cubic rate estimators under the massive data framework we develop a general theory for establishing the asymptotic distribution of the aggregated m estimators using a weighted average with weights depending on the subgroup sample sizes under certain condition on the growing rate of the number of subgroups the resulting aggregated estimators are shown to have faster convergence rate and asymptotic normal distribution which are more tractable in both computation and inference than the original m estimators based on pooled data our theory applies to a wide class of m estimators with cube root convergence rate including the location estimator maximum score estimator and value search estimator empirical performance via simulations and a real data application also validate our theoretical findings

an efficient calculation of the moments of matched and mismatched hidden markov models streit 1990 analysed the classification of an unknown hidden markov model hmm using a set of prescribed hmms he proposed a suboptimal test statistic that can be approximated by certain moments for this classification in the present paper the algorithm given by streit to derive these moments is reformulated in a matrix algebra setting that gives a better insight into the algorithm also an asymptotic analysis of the algorithm is derived using the reformulation

asymptotic stochastic programs consider a stochastic program with unique solution by the notion of epi convergence in distribution in local coordinates we define the asymptotic stochastic program associated to it stochastic programs may be classified according to the type of the pertaining asymptotic program in particular three such types are studied in detail the normal shift program the wiener type program and the poisson hyperplane program conditions for the convergence to each of the three types are given

on integrated l 1 convergence rate of an isotonic regression estimator for multivariate observations we consider a general monotone regression estimation where we allow for independent and dependent regressors we propose a modification of the classical isotonic least squares estimator and establish its rate of convergence for the integrated l 1 loss function the methodology captures the shape of the data without assuming additivity or a parametric form for the regression function furthermore the degree of smoothing is chosen automatically and no auxiliary tuning is required for the theoretical analysis some simulations and two real data illustrations complement the study of the proposed estimator

the moments of matched and mismatched hidden markov models an algorithm for computing the moments of matched and mismatched hidden markov models hmms from their defining parameters is presented the algorithm is an extension of the usual forward backward linear recursion the algorithm computes the joint moments of the posterior likelihood functions i e the scores by a multilinear recursion involving the joint moments of the random variables associated with the hidden states of the markov chain examples comparing the first two theoretical moments to simulation results are presented they are of independent interest because they indicate that the distribution of the posterior likelihood function scores for mismatched models are asymptotically log normal in important special cases and therefore are characterized asymptotically by the first two moments alone one example shows that classification using a suboptimal statistic reliably distinguishes between sufficiently long quasi stationary signals with a reasonable amount of computational effort

semiparametric identification of binary decision games of incomplete information with correlated private signals abstract this paper studies the identification and estimation of a static binary decision game of incomplete information we make no parametric assumptions on the joint distribution of private signals and allow them to be correlated we show that the parameters of interest can be point identified subject to a scale normalization under mild support requirements for the regressors publicly observed state variables and errors private signals following manski and tamer 2002 we propose a maximum score type estimator for the structural parameters and establish the asymptotic properties of the estimator

estimation of a function under shape restrictions applications to reliability this paper deals with a nonparametric shape respecting estimation method for u shaped or unimodal functions a general upper bound for the nonasymptotic l 1 risk of the estimator is given the method is applied to the shape respecting estimation of several classical functions among them typical intensity functions encountered in the reliability field in each case we derive from our upper bound the spatially adaptive property of our estimator with respect to the l 1 metric it approximately behaves as the best variable binwidth histogram of the function under estimation

robust inference for threshold regression models abstract this paper considers robust inference in threshold regression models when the practitioners do not know whether at the threshold point the true specification has a kink or a jump nesting previous works that assume either continuity or discontinuity at the threshold we find that the parameter values under the kink restriction are irregular points of the hessian matrix destroying the asymptotic normality and inducing the cube root convergence rate for the threshold estimate however we are able to obtain the same asymptotic distribution as hansen 2000 for the quasi likelihood ratio statistic for the unknown threshold we propose to construct confidence intervals for the threshold by bootstrap test inversion finite sample performances of the proposed procedures are examined through monte carlo simulations and an economic empirical application is given

universal donsker classes and metric entropy when mathfrak f is a universal donsker class then for independent indetically distributed i i d observation mathbf x 1 ldots mathbf x n with an unknown law p for any mathfrak f i in mathfrak f i 1 ldots m quad n 1 2 left mathfrak f 1 left mathbf x 1 right ldots mathfrak f i left mathbf x n right right 1 leq i leq m is asymptotically normal with mean vector n 1 2 left int mathfrak f i left mathbf x n right d mathbf p left x right right 1 leq i leq m and covariance matrix int mathfrak f i mathfrak f j d mathbf p int mathfrak f id mathbf p int mathfrak f jd mathbf p uniformly for mathfrak f i in mathfrak f then for certain statistics formed frome the mathfrak f i left mathbf x k right even where mathfrak f i may be chosen depending on the mathbf x k there will be asymptotic distribution as n rightarrow infty for example for mathbf x 2 statistics where f i are indicators of disjoint intervals depending suitably on mathbf x 1 ldots mathbf x n whose union is the real line mathbf x 2 quadratic forms have limiting distributions roy 1956 and watson 1958 which may however not be mathbf x 2 distributions and may depend on p chernoff and lehmann 1954 universal donsker classes of sets are up to mild measurability conditions just classes satisfying the vapnik cervonenkis comdinatorial conditions defined later in this section donsker the vapnik cervonenkis combinatorial conditions defined later in this section durst and dudley 1981 and dudley 1984 chapter 11 the use of such classes allows a variety of extensions of the roy watson results to general multidimensional sample spaces pollard 1979 and moore and subblebine 1981 vapnik and cervonenkis 1974 indicated application of their families of sets to classification pattern recognition problems more recently the classes have been applied to tree structured classifiacation breiman friedman olshen and stone 1984 chapter 12

rates of convergence for estimating regression coefficients in heteroskedastic discrete response models abstract in this paper we consider estimation of discrete response models exhibiting conditional heteroskedasticity of a multiplicative form where the latent error term is assumed to be the product of an unknown scale function and a homoskedastic error term it is first shown that for estimation of the slope coefficients in a binary choice model under this type of restriction the semiparametric information bound is zero even when the homoskedastic error term is parametrically specified hence it is impossible to attain the parametric convergence rate for the parameters of interest however for ordered response models where the response variable can take at least three different values the parameters of interest can be estimated at the parametric rate under the multiplicative heteroskedasticity assumption two estimation procedures are proposed the first estimator based on a parametric restriction on the homoskedastic component of the error term is a two step maximum likelihood estimators where the unknown scale function is estimated nonparametrically in the first stage the second procedure which does not require the parametric restriction estimates the parameters by a kernel weighted least squares procedure under regularity conditions which are standard in the literature both estimators are shown to be n consistent and asymptotically normal

a likelihood ratio test against stochastic ordering in several populations abstract the likelihood ratio test is often used to test hypotheses involving a stochastic ordering distribution theory for the likelihood ratio test has been developed only for two stochastically ordered distributions for testing equality of distributions against a stochastic ordering in several populations this paper derives the null asymptotic distribution of the likelihood ratio test statistic which is characterized by minimization problems and has no closed form a monte carlo simulation is conducted to study the limiting distribution because the limiting distribution depends on the specific values of the unknown distributions under the null hypothesis asymptotic and bootstrap approaches are proposed to overcome practical difficulties and implement tests based on the likelihood principle asymptotic validities for these tests are established and simulations are carried out to check their performances for finite sample sizes the tests are applied to an example involving data for survival time f

optimal third root asymptotic bounds in the statistical estimation of thresholds this paper is concerned with estimating the intersection point of two densities given a sample of both of the densities this problem arises in classification theory the main results provide lower bounds for the probability of the estimation errors to be large on a scale determined by the inverse cube root of the sample size as corollaries we obtain probabilistic bounds for the prediction error in a classification problem the key to the proof is an entropy estimate the lower bounds are based on bounds for general estimators which are applicable in other contexts as well furthermore we introduce a class of optimal estimators whose errors asymptotically meet the border permitted by the lower bounds

sharp asymptotics for isotonic regression the asymptotic behavior of the isotonic estimator of a monotone regression function that is the least squares estimator under monotonicity restriction is investigated in particular it is proved that the 1 distance between the isotonic estimator and the true function is of magnitude n 1 3 moreover it is proved that a centered version of this 1 distance converges at the n1 2 rate to a gaussian variable with fixed variance

the length of the shorth soit n 0 1 la longueur de l fraction la plus courte de l echantillon ordonne x 1 n x 2 n x n n on utilise le concept de derivees compactes des fonctionnelles statistiques pour etudier le comportement asymptotique de n lorsque n

fast o 1 bilateral filtering using trigonometric range kernels it is well known that spatial averaging can be realized in space or frequency domain using algorithms whose complexity does not scale with the size or shape of the filter these fast algorithms are generally referred to as constant time or o 1 algorithms in the image processing literature along with the spatial filter the edge preserving bilateral filter involves an additional range kernel this is used to restrict the averaging to those neighborhood pixels whose intensity are similar or close to that of the pixel of interest the range kernel operates by acting on the pixel intensities this makes the averaging process nonlinear and computationally intensive particularly when the spatial filter is large in this paper we show how the o 1 averaging algorithms can be leveraged for realizing the bilateral filter in constant time by using trigonometric range kernels this is done by generalizing the idea presented by porikli i e using polynomial kernels the class of trigonometric kernels turns out to be sufficiently rich allowing for the approximation of the standard gaussian bilateral filter the attractive feature of our approach is that for a fixed number of terms the quality of approximation achieved using trigonometric kernels is much superior to that obtained by porikli using polynomials

progressive image denoising image denoising continues to be an active research topic although state of the art denoising methods are numerically impressive and approch theoretical limits they suffer from visible artifacts while they produce acceptable results for natural images human eyes are less forgiving when viewing synthetic images at the same time current methods are becoming more complex making analysis and implementation difficult we propose image denoising as a simple physical process which progressively reduces noise by deterministic annealing the results of our implementation are numerically and visually excellent we further demonstrate that our method is particularly suited for synthetic images finally we offer a new perspective on image denoising using robust estimators

on the mathbb l _ p error of monotonicity constrained estimators we aim at estimating a function 0 1 r subject to the constraint that it is decreasing or increasing we provide a unified approach for studying the lp loss of an estimator defined as the slope of a concave or convex approximation of an estimator of a primitive of based on n observations our main task is to prove that the l p loss is asymptotically gaussian with explicit though unknown asymptotic mean and variance we also prove that the local lp risk at a fixed point and the global lp risk are of order n p 3 applying the results to the density and regression models we recover and generalize known results about grenander and brunk estimators also we obtain new results for the huang wellner estimator of a monotone failure rate in the random censorship model and for an estimator of the monotone intensity function of an inhomogeneous poisson process

semiparametric analysis of discrete response asymptotic properties of the maximum score estimator abstract this paper is concerned with the estimation of the model med y x x from a random sample of observations on sgn y x manski 1975 introduced the maximum score estimator of the normalized parameter vector in the present paper strong consistency is proved it is also proved that the maximum score estimate lies outside any fixed neighborhood of with probability that goes to zero at exponential rate

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

the asymptotics of s estimators in the linear regression model we consider the consistency and weak convergence of s estimators in the linear regression model sufficient conditions for consistency with varying dimension are given which are sufficiently weak to cover for example polynomial trends and i i d carriers a weak convergence theorem for the hampel rousseeuw least median of squares estimator is obtained and it is shown under rather general conditions that the correct norming factor is n 1 3 finally the asymptotic normality of s estimators with a smooth rho function is obtained again under weak conditions on the carriers

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

compressive bilateral filtering this paper presents an efficient constant time bilateral filter that produces a near optimal performance tradeoff between approximate accuracy and computational complexity without any complicated parameter adjustment called a compressive bilateral filter cblf the constant time means that the computational complexity is independent of its filter window size although many existing constant time bilateral filters have been proposed step by step to pursue a more efficient performance tradeoff they have less focused on the optimal tradeoff for their own frameworks it is important to discuss this question because it can reveal whether or not a constant time algorithm still has plenty room for improvements of performance tradeoff this paper tackles the question from a viewpoint of compressibility and highlights the fact that state of the art algorithms have not yet touched the optimal tradeoff the cblf achieves a near optimal performance tradeoff by two key ideas 1 an approximate gaussian range kernel through fourier analysis and 2 a period length optimization experiments demonstrate that the cblf significantly outperforms state of the art algorithms in terms of approximate accuracy computational complexity and usability

two step estimation of semiparametric censored regression models root n consistent estimators of the regression coecients in the linear censored regression model under conditional quantile restrictions on the error terms were pro posed by powell journal of econometrics 25 1984 303 325 32 1986a 143 155 while those estimators have desirable asymptotic properties under weak regularity conditions simulation studies have shown these estimators to exhibit a small sam ple bias in the opposite direction of the least squares bias for censored data this paper introduces two step estimators for these models which minimize convex objec tive functions and are designed to overcome thisnite sample bias the paper gives regularity conditions under which the proposed two step estimators are consistent and asymptotically normal a monte carlo study compares thenite sample behavior of the proposed methods with their one step counterparts 2001 elsevier science s a all rights reserved

estimation of a panel data sample selection model we consider the problem of estimation in a panel data sample selection model where both the selection and the regression equation of interest contain unobservable individual specific effects we propose a two step estimation procedure which differences out both the sample selection effect and the unobservable individual effect from the equation of interest in the first step the unknown coefficients of the selection equation are consistently estimated the estimates are then used to estimate the regression equation of interest the estimator proposed in this paper is consistent and asymptotically normal with a rate of convergence that can be made arbitrarily close to n 1 2 depending on the strength of certain smoothness assumptions the finite sample properties of the estimator are investigated in a small monte carlo simulation

brownian motion with a parabolic drift and airy functions let w t t be two sided brownian motion originating from zero and let v a be defined by v a sup t w t t a 2 is maximal then v a a is a markovian jump process running through the locations of maxima of two sided brownian motion with respect to the parabolas fa t t a 2 we give an analytic expression for the infinitesimal generators of the processes a in terms of airy functions in theorem 4 1 this makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators i e estimators derived under order restrictions an example of this is given in groeneboom 1985 where the asymptotic distribution of the standardized l1 distance between a decreasing density and the grenander maximum likelihood estimator of this density is determined on our way to theorem 4 1 we derive some other results for example we give an analytic expression for the joint density of the maximum and the location of the maximum of the process w t ct2 t where c is an aribrary positive constant we also determine the laplace transform of the integral over a brownian excursion these last results also have recently been derived by several other authors using a variety of methods

a two stage hybrid procedure for estimating an inverse regression function we consider a two stage procedure tsp for estimating an inverse regression function at a given point where isotonic regression is used at stage one to obtain an initial estimate and a local linear approximation in the vicinity of this estimate is used at stage two we establish that the convergence rate of the second stage estimate can attain the parametric n 1 2 rate furthermore a bootstrapped variant of tsp btsp is introduced and its consistency properties studied this variant manages to overcome the slow speed of the convergence in distribution and the estimation of the derivative of the regression function at the unknown target quantity finally the finite sample performance of btsp is studied through simulations and the method is illustrated on a data set

a consistent bootstrap procedure for the maximum score estimator abstract in this paper we propose a new model based smoothed bootstrap procedure for making inference on the maximum score estimator of manski 1975 1985 and prove its consistency we provide a set of sufficient conditions for the consistency of any bootstrap procedure in this problem we compare the finite sample performance of different bootstrap procedures through simulation studies the results indicate that our proposed smoothed bootstrap outperforms other bootstrap schemes including the m out of n bootstrap additionally we prove a convergence theorem for triangular arrays of random variables arising from binary choice models which may be of independent interest

the limit distribution of the l error of grenander type estimators let f be a nonincreasing function defined on 0 1 under standard regularity conditions we derive the asymptotic distribution of the supremum norm of the difference between f and its grenander type estimator on sub intervals of 0 1 the rate of convergence is found to be of order n log n 1 3 and the limiting distribution to be gumbel

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

cube root asymptotics we establish a new functional central limit theorem for empirical processes indexed by classes of functions in a neighborhood of a fixed parameter point an n 1 3 rescaling of the parameter is compensated for by an n 2 3 rescaling of the empirical measure resulting in a limiting gaussian process by means of a modified continuous mapping theorem for the location of the maximizing value we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure these statistics include the short rousseeuw s least median of squares estimator manski s maximum score estimator and the maximum likelihood estimator for a monotone density the limit theory depends on a simple new sufficient condition for a gaussian process to achieve its maximum almost surely at a unique point