optimal estimation of a general regression function let e be a convex subset of rd and for each 0 e e let f 0 dt be a probability on the line for any vector an an an2 ann where anl e e let xn1 xnn be independent observations the distribution of xn being f an dt the main results give a method of estimating the unknown regression function an based on a minimum distance recipe under regularity assumptions the proposed estimators are shown in an appropriate framework to be asymptotically normal locally asymptotically minimax and robust the abstract results are illustrated by application to the linear model and to exponential response models in general nothing at all is assumed about the form of the regression function accordingly this forces the limiting normal distributions of the proposed estimators to be located on infinite dimensional linear spaces

hellinger deviance tests efficiency breakdown points and examples abstract hellinger distance analogs of likelihood ratio tests are proposed for parametric inference the proposed tests are based on minimized hellinger distances between nonparametric density estimates and densities corresponding to null and unconstrained parametric models if the parametric model is correct the hellinger deviance test is asymptotically equivalent to the likelihood ratio test a simulation study examines the relative performance of these tests in finite samples breakdown properties of the hellinger deviance tests and likelihood ratio tests are compared the hellinger tests are generally high breakdown point tests whereas in many instances the likelihood ratio tests have breakdown points of 0 two numerical examples illustrate the implementation and performance of the proposed tests likelihood based methods are widely used in applications they provide a routine method for generating efficient estimates and inference statements in many instances however these methods are known to

nonparametric estimates of regression quantiles and their local bahadur representation let x y be a random vector such that x is d dimensional y is real valued and y theta x varepsilon where x and varepsilon are independent and the alpha th quantile of varepsilon is 0 alpha is fixed such that 0 0 and set r p m 2p d where m is a nonnegative integer smaller than p let t theta denote a derivative of theta of order m it is proved that there exists a pointwise estimate hat t n of t theta based on a set of i i d observations x 1 y 1 cdots s n y n that achieves the optimal nonparametric rate of convergence n r under appropriate regularity conditions further a local bahadur type representation is shown to hold for the estimate hat t n and this is used to obtain some useful asymptotic results

constructions of algebraic geometry codes based on curves over finite fields with many rational points we present two constructions of linear codes from local expansions of functions at a fixed rational point it turns out that codes from our constructions have the same bound on their parameters as goppa s 1981 geometry codes furthermore we prove that our second construction is equivalent to goppa s construction finally an additional construction of linear codes from maximal curves shows that these codes have better parameters than goppa s geometry codes from maximal curves for a certain interval of parameters

remarks on codes from hermitian curves corresp parameters and generator matrices are given for the codes obtained by applying goppa s algebraic geometric construction method to hermitian curves in pg 2 q where q 2 2s for some s in bf n automorphisms of these codes are also discussed and some results on self duality and weak self duality are given

on inconsistent bayes estimates of location on montre que dans certaines situations relativement naturelles les estimateurs de position bayesiens sont inconsistants

the resistant line and related regression methods abstract in a bivariate x y scatterplot the three group resistant line is that line for which the median residual in each outer third of the data ordered on x is zero it was proposed by tukey as an exploratory method resistant to outliers in x and y and suited to hand calculation a dual plot representation of the procedure yields a fast convergent algorithm nonparametric confidence intervals for the slope consistency influence function and asymptotic normality results monte carlo results show that small sample efficiencies exceed their asymptotic values in important cases both breakdown and efficiency are adequate for exploratory work replacing medians by other m estimators of location increases efficiency substantially without affecting breakdown or computational complexity the method thus combines bounded influence and acceptable efficiency with conceptual and computational simplicity

structural adaptation via mathbb l _p norm oracle inequalities in this paper we study the problem of adaptive estimation of a multivariate function satisfying some structural assumption we propose a novel estimation procedure that adapts simultaneously to unknown structure and smoothness of the underlying function the problem of structural adaptation is stated as the problem of selection from a given collection of estimators we develop a general selection rule and establish for it global oracle inequalities under arbitrary mathbb l p losses these results are applied for adaptive estimation in the additive multi index model

robust estimation via minimum distance methods given a fixed parametric family p it is desired to estimate however due to contaminations of various sorts the data actually collected by the statistician follow a distribution that is close to but possibly distinct from thep s it is proved that under these conditions and in an appropriate asymptotic framework the minimum distance estimators of the cramer von mises type are robust specification of a cramer von mises weight functionh defines a notion of distance each such choice ofh then delineates the kind of contamination possible and leads to an estimator which defends optimally against it when the theory is specialized to location models various choices ofh lead to estimators asymptotically equivalent to such familiar ones as trimmed mean median hodges lehmann estimator and so forth the framework developed herein provides some guidelines for choosing among the possible estimators and suggests that the standard cramer von mises estimator of location is probably as good a robust estimator as any

on the consistency of bayes estimates on etudie les proprietes de frequence des regles de bayes avec une attention particuliere a la consistence

non parametric applications of an infinite dimensional convolution theorem this paper gives a general approach to the statistical problem of efficient non parametric estimation the main tool is an abstract infinite dimensional convolution theorem of the hajek le cam type this result is applied to the problems of efficiently estimating measures quantile functions spectral functions of stationary processes minimum distance functionals m functional and so forth

efficiency constrained bias robust estimation of location abstract it has been correctly pointed out by an anonymous referee that the results of this paper amount to a rigorous analysis of the correctness of hampel s heuristic approach in the location setup still one might conjecture by analogy that in the case of more complicated models maximizing the efficiency under a constraint on the gross error sensitivity is almost the same as putting a constraint on the bias however the extension of our technique to more complicated models is by no means straightforward if al all possible and this matter deserves further study another interesting issue brought up by an anonymous referee is that if one leaves the realm of m estimates of location there is not necessarily a pay off between bias robustness and efficiency at the central gaussian model

on the automorphism group of various goppa codes we determine the automorphism group of various goppa codes spl cscr sub l d g associated with certain function fields f f sub q of genus g 0 it is well known that for deg d n 2g 2 the automorphism group aut sub d g f f sub q can be embedded into aut spl cscr sub l d g as a subgroup we show that under certain conditions on the divisors d and g aut sub d g f f sub q is actually isomorphic to aut spl cscr sub l d g

a regression type problem let x 1 x n be random vectors that take values in a compact set in r d d 1 2 let y 1 y n be random variables the responses which conditionally on x 1 x 1 x n x n are independent with densities f y x i b x i i l n assuming that lies in a sup norm compact space of real valued functions an l 1 consistent estimator of is constructed via empirical measures the rate of convergence of the estimator of the estimator to the true parameter depends on kolmogorov s entropy of

breakdown points of affine equivariant estimators of multivariate location and covariance matrices finite sample replacement breakdown points are derived for different types of estimators of multivariate location and covariance matrices the role of various equivariance properties is illustrated the breakdown point is related to a measure of performance based on large deviations probabilities finally we show that one step reweighting preserves the breakdown point

a generalization of the product limit estimator with an application to censored regression the product limit estimator ple and weighted empirical processes are two important ingredients of almost any censored regression analysis a link between them is provided by the generalized ples introduced in this paper these generalized ples are the product limit integrals of the empirical cumulative hazard function estimators in which weighted empirical processes are used to replace the standard empirical processes the weak convergence and some large sample approximations of the generalized ples are established as an application these generalized ples are used to define some minimum distance estimators which are shown to be asymptotically normal these estimators are qualitatively robust in some submodels an optimal choice of the weight matrix is the covariate matrix and some of these estimators are quite efficient at a few common survival distributions to implement these estimators some computational aspects are discussed and an algorithm is given from a real data example and some preliminary simulation results these estimators seem to be very competitive to and more robust than some more traditional estimators such as the buckley james estimator

indecomposable higher chow cycles on jacobians we construct some natural indecomposable elements of ch g j c 1 with trivial regulator and in particular prove that ch g ind j c 1 mb q is uncountable for c a generic curve or a generic hyperelliptic curve of genus g geq 3

probability inequalities for empirical processes and a law of the iterated logarithm given a class f of functions on x we can view vn as a stochastic process indexed by f and consider limit theorems for this process to prove such theorems it is often helpful to have bounds on the tail of the r v sup w i n f i see for example dudley 1978 kuelbs and dudley 1980 dudley and phillip 1983 or gine and zinn 1983 our main question of interest is for what classes f can best possible bounds be obtained so first we must ask what are these best possible bounds of course

robust estimation in models for independent non identically distributed data this paper concerns robust estimation of the parameter 0 which indexes a parametric model for independent non identically distributed data for reasonable choices of contamination neighborhood and of what is to be estimated when the parametric model does not hold we characterize asymptotically minimax robust estimates of 0 when applied to the normal regression model the theory yields recipes for the influence curves of optimal robust regression and scale estimates the contamination neighborhood does not assume regression plus error structure the regression and scale parameters are estimated simultaneously and the theory establishes r oles for estimates with redescending influence curves as well as for those with monotone influence curves when applied to the logit and probit models the theory recommends influence curves which differ markedly from those of the maximum likelihood estimates except in the i i d case

breakdown robustness of tests abstract power and level breakdown functions of a test statistic are introduced to obtain a unified analysis of local stability as typified by the influence function and global stability as illustrated by the breakdown point the power breakdown function gives the amount of contamination of each alternative distribution that can carry the test statistic to a null value the level breakdown function gives the amount of contamination of a null distribution that can carry the test statistic to each value in the alternative space in many cases the breakdown function bears an inverse relationship to the maximum bias as a function of the contamination as the maximum bias is not invariant to one to one transformations however equivalent test statistics can have different scales for bias a distinct advantage of the inverse scale is the invariance of the breakdown functions to such transformations the resulting stability curves depend on the parameters in a way analogous to the power curve the slopes o

information theoretic bounds for compressed sensing in this paper we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections we consider two settings output noise models where the noise enters after the projection and input noise models where the noise enters before the projection we consider two types of distortion for reconstruction support errors and mean squared errors our goal is to relate the number of measurements m and snr to signal sparsity k distortion level d and signal dimension n we consider support errors in a worst case setting we employ different variations of fano s inequality to derive necessary conditions on the number of measurements and snr required for exact reconstruction to derive sufficient conditions we develop new insights on max likelihood analysis based on a novel superposition property in particular this property implies that small support errors are the dominant error events consequently our ml analysis does not suffer the conservatism of the union bound and leads to a tighter analysis of max likelihood these results provide order wise tight bounds for output noise models we show that asymptotically an snr of n together with k n k measurements is necessary and sufficient for exact support recovery furthermore if a small fraction of support errors can be tolerated a constant snr turns out to be sufficient in the linear sparsity regime in contrast for input noise models we show that support recovery fails if the number of measurements scales as o n n snr implying poor compression performance for such cases motivated by the fact that the worst case setup requires significantly high snr and substantial number of measurements for input and output noise models we consider a bayesian setup to derive necessary conditions we develop novel extensions to fano s inequality to handle continuous domains and arbitrary distortions we then develop a new max likelihood analysis over the set of rate distortion quantization points to characterize tradeoffs between mean squared distortion and the number of measurements using rate distortion theory we show that with constant snr the number of measurements scales linearly with the rate distortion function of the sparse phenomena

information and asymptotic efficiency in parametric nonparametric models asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems for regular estimates and asymptotic minimax bounds the methods used involve i the notion of a hellinger differentiable root density where part of the differentiation is with respect to the nonparametric part of the model to obtain appropriate scores and ii calculation of the effective score for the real or vector finite dimensional parameter of interest as that component of the score function orthogonal to all nuisance parameter scores perhaps infinite dimensional the resulting asymptotic information for estimation of the parametric component of the model is just 4 times the squared l 2 norm of the effective score a corollary of these results is a simple necessary condition for adaptive estimation adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model examples considered include the one sample location model with and without symmetry mixture models the two sample shift model and cox s proportional hazards model

inference for empirical wasserstein distances on finite spaces summary the wasserstein distance is an attractive tool for data analysis but statistical inference is hindered by the lack of distributional limits to overcome this obstacle for probability measures supported on finitely many points we derive the asymptotic distribution of empirical wasserstein distances as the optimal value of a linear programme with random objective function this facilitates statistical inference e g confidence intervals for sample based wasserstein distances in large generality our proof is based on directional hadamard differentiability failure of the classical bootstrap and alternatives are discussed the utility of the distributional results is illustrated on two data sets

high rate locally correctable codes via lifting we present a general framework for constructing high rate error correcting codes that are locally correctable and hence locally decodable if linear with a sublinear number of queries based on lifting codes with respect to functions on the coordinates our approach generalizes the lifting of affine invariant codes of guo kopparty and sudan and its generalization automorphic lifting alluded to in the work of ben sasson et al but distinct from their degree lifting which lifts algebraic geometry codes with respect to a group of automorphisms of the code our notion of lifting is a natural alternative to the degree lifting of ben sasson et al and it carries two advantages first it overcomes the rate barrier inherent in degree lifting second it requires no special properties e g linearity and invariance of the base code and requires a very little structure on the set of functions on the coordinates of the code as an application we construct new explicit families of locally correctable codes by lifting algebraic geometry codes like the multiplicity codes of kopparty saraf yekhanin and the affine lifted codes of guo kopparty and sudan our codes of block length n can achieve n epsilon query complexity and 1 alpha rate for any given epsilon alpha 0 while correcting a constant fraction of errors in contrast to the reed muller codes and the degree lifted ag codes of ben sasson et al which face a rate barrier of epsilon o 1 epsilon however like the degree lifted ag codes our codes are over an alphabet significantly smaller than that obtained by reed muller codes affine lifted codes and multiplicity codes

pathologies of some minimum distance estimators on etudie des estimateurs de distance minimale dans le modele n 0 1 on montre que la consistance automatique n est pas necessaire lorsque le modele est seulement approximativement vrai

the automatic robustness of minimum distance functionals on montre que les estimateurs de distance minimale forment une classe d estimateurs automatiquement robustes dans le meme sens que l estimateur du maximum de vraisemblance

automorphism groups of one point codes from the curves y sup q y x sup qr 1 the automorphism groups are determined for the one point codes c sub m on the curve over f sub q 2r defined by y sup q y x sup qr 1 where r is an odd number this generalizes xing s theorem see ibid vol 41 p 1629 35 nov 1995 and extends a result of wesemeyer see ibid vol 44 p 630 43 march 1998 to the case of the above curve

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

small sample results for the kaplan meier estimator abstract whereas much is known about the asymptotic properties of the kaplan meier 1958 estimator kme of a survival function exact results for small samples have been difficult to obtain in this article we obtain an exact expression for the th moment 0 of the kme under a model of proportional hazards this enables us under proportional hazards to a study the bias of the kme and b compare the exact variance of the kme to its asymptotic variance

minimum hellinger distance estimation for multivariate location and covariance abstract the hellinger distance between a nonparametric density estimator and a model family is minimized to produce estimates of location and covariance in multivariate data with suitable restrictions on the density estimators and the model family these minimum hellinger distance estimators mhde s are shown to be affine invariant consistent and asymptotically normal the robustness of the mhde as measured by the breakdown point compares favorably against the previously studied m estimators monte carlo results suggest that the mhde s are an attractive robust alternative to the usual sample means and covariance matrix

minimum distance estimators of scale with censored data abstract this article examines how the level of censoring affects the performance of estimators of scale that minimize a weighted cramervon mises distance between the kaplan meier product limit estimator of the survival distribution function and an assumed model the article shows that under certain conditions these estimators are asymptotically normal if the survival function is a member of the assumed scale family weights that minimize the asymptotic variance under these conditions also are found under stronger conditions the minimum distance estimators mde s are asymptotically normal even if the survival function is not a member of the assumed scale family asymptotic coverage probabilities of confidence intervals are found for percentiles constructed from exponential mde s of scale under departures from the exponential distribution and what happens in the limiting cases of no censoring and complete censoring is examined using the asymptotic results and monte carlo experiments several exponenti

a note on hermitian codes over gf q sup 2 properties of codes defined by goppa s algebraic geometric method using hermitian curves are discussed generator and parity check matrices for such codes are derived explicitly results concerning the minimum distance and the weight distribution of the codes are provided

when are two geometric goppa codes equal sufficient and necessary conditions are obtained for which the geometric goppa codes c d g and c d h are equal for two divisors g and h in particular it is proven that if g and h are two effective divisors of the same degree smaller than n 1 then c d g and c d h are equal if and only if g h

semiparametric inference in the proportional odds regression model abstract for fitting the proportional odds regression model with right censored survival times we introduce some weighted empirical odds functions these functions are solutions of some self consistency equations and have a nice martingale representation from these functions several classes of new regression estimators such as the pseudo maximum likelihood estimator martingale residual based estimators and minimum distance estimators are derived these estimators have desirable properties such as easy computation asymptotic normality via a martingale analysis and reliable asymptotic covariance estimation in closed form extensive numerical studies show that the minimum l 2 distance estimators have very good finite sample behaviors compared to existing methods results of some simulation studies and applications to a real dataset are given the weighted odds function based approach also provides inference on the baseline odds function and some measures for lack of fit analysis

predictive likelihood inference with applications summary conditional and marginal predictive likelihood are defined and used in predicting un observed or missing data a distinction is made between predictive and estimative goals of a data analysis such methodology and distinctions are illustrated in the context of outlier theory and mixed model ano va the prediction of unobserved or missing data is considered from a likelihood perspective a general predictive likelihood is proposed for such purposes and contrasted with parametric likelihood the resulting maximum likelihood predictor is also compared with the limiting imputation from the em algorithm the new methodology is illustrated and contrasted with estimative likelihood analysis in the context of outlier theory and mixed model ano va predictions are proposed that are based on conditional and marginal predictive likelihoods for the unobserved data which are predictive adaptations of the methods developed by kalbfleisch and sprott 1970 1973 these authors were concerned with the elimination of nuisance parameters in order to make inference about another set of structural parameters prediction in the frame work of parametric distributions involves the elimination of all parameters in such a way that inference about the unobserved data is possible therefore in the predictive setting all para meters are nuisance parameters and the unobserved data now play a role similar to that of the structural parameters various predictive approaches to statistical inference have been advocated principally by geisser 1971 1975 stone 1974 aitchison and dunsmore 1975 and guttman 1970 a survey is provided by geisser 1980 and a brief overview in a parametric setting is presented in the next section following this a conditional likelihood approach is developed and shown to be closely related to the lauritzen 1974 hinkley 1979 likelihood method but specified in a manner allowing for a wider range of applicability a marginal predictive likelihood procedure is also specified which when applicable essentially agrees with the conditional likelihood procedure for the examples considered for a general account of likelihood based inference see fraser

parametric statistical modeling by minimum integrated square error the likelihood function plays a central role in parametric and bayesian estimation as well as in nonparametric function estimation via local polynomial modeling however integrated square error has enjoyed a long tradition as the goodness of fit criterion of choice in nonparametric density estimation in this article i investigate the use of integrated square error or l2 distance as a theoretical and practical estimation tool for a variety of parametric statistical models i show that the asymptotic inefficiency of the parameters estimated by minimizing the integrated square error or l2estimation l2e criterion versus the maximum likelihood estimator is roughly that of the median versus the mean i demonstrate by example the well known result that minimum distance estimators including l2e are inherently robust however l2e does not require specification of any tuning factors found in robust likelihood algorithms l2e is particularly appropriate for analyzing massive datasets in which data cleanin

regression analysis with randomly right censored data this paper proposes a new estimator of the parameter vector in a linear regression model when the observations are randomly censored on the right and when the error distribution is unknown this estimator is explicitly defined and easily computable the paper contains sufficient conditions under which this estimator is mean square consistent and asymptotically normal a numerical example is given

robust estimation of mixture complexity in many applications it is important to find the mixture with fewest number of components known as the mixture complexity that provides a satisfactory fit to the data this article focuses on developing an estimator of mixture complexity that is consistent when the form of component densities are unknown but are postulated to be members of some parametric family and is simultaneously robust against model misspecification we treat the estimation of mixture complexity as a model selection problem and construct an estimator of mixture complexity as a byproduct of minimizing a hellinger information criterion this estimator is shown to be consistent for any parametric family of mixtures when the model is correctly specified monte carlo simulations for a wide variety of normal mixtures show that our estimator is very competitive with several others in the literature in correctly identifying the true mixture complexity the basic construction being firmly rooted in the minimum hellinger distance approach

robust estimation via robust gradient estimation we provide a new computationally efficient class of estimators for risk minimization we show that these estimators are robust for general statistical models under varied robustness settings including in the classical huber e contamination model and in heavy tailed settings our workhorse is a novel robust variant of gradient descent and we provide conditions under which our gradient descent variant provides accurate estimators in a general convex risk minimization problem we provide specific consequences of our theory for linear regression and logistic regression and for canonical parameter estimation in an exponential family these results provide some of the first computationally tractable and provably robust estimators for these canonical statistical models finally we study the empirical performance of our proposed methods on synthetic and real data sets and we find that our methods convincingly outperform a variety of baselines

rates of convergence of minimum distance estimators and kolmogorov s entropy on construit des estimateurs robustes et on montre que le taux de convergence resultant depend naturellement d une fonction entropie

on automorphism groups of the hermitian codes we determine the exact automorphism groups of the goppa geometric codes from the hermitian curves over gf q sup 2

higher order infinitesimal robustness using the von mises expansion we study the higher order infinitesimal robustness of a general m functional and characterize its second order properties we show that second order robustness is equivalent to the boundedness of both the estimator s estimating function and its derivative with respect to the parameter it implies at the same time i variance robustness and ii robustness of higher order saddlepoint approximations to the estimator s finite sample density the proposed construction of second order robust m estimators is fairly general and potentially useful in a variety of relevant settings besides the theoretical contributions we discuss the main computational issues and provide an algorithm for the implementation of second order robust m estimators finally we illustrate our theory by monte carlo simulation and in a real data estimation of the maximal losses of nikkei 225 index returns our findings indicate that second order robust estimators can improve on other widely applied robus

a one way random effects model for trimmed means the random effects anova model plays an important role in many psychological studies but the usual model suffers from at least two serious problems the first is that even under normality violating the assumption of equal variances can have serious consequences in terms of type i errors or significance levels and it can affect power as well the second and perhaps more serious concern is that even slight departures from normality can result in a substantial loss of power when testing hypotheses jeyaratnam and othman 1985 proposed a method for handling unequal variances under the assumption of normality but no results were given on how their procedure performs when distributions are nonnormal a secondary goal in this paper is to address this issue via simulations as will be seen problems arise with both type i errors and power another secondary goal is to provide new simulation results on the rust fligner modification of the kruskal wallis test the primary goal is to propose a generalization of the usual random effects model based on trimmed means the resulting test of no differences among j randomly sampled groups has certain advantages in terms of type i errors and it can yield substantial gains in power when distributions have heavy tails and outliers this last feature is very important in applied work because recent investigations indicate that heavy tailed distributions are common included is a suggestion for a heteroscedastic winsorized analog of the usual intraclass correlation coefficient

bias robust estimation of scale in this paper we consider the problem of robust estimation of the scale of the location residuals when the underlying distribution of the data belongs to a contamination neighborhood of a parametric location scale family we define the class of m estimates of scale with general location and show that under certain regularity assumptions these scale estimates converge to their asymptotic functionals uniformly with respect to the underlying distribution and with respect to the m estimate defining score function chi we establish expressions for the maximum asymptotic bias of m estimates of scale over the contamination neighborhood as a function of the fraction of contamination using these expressions we construct asymptotically min max bias robust estimates of scale in particular we show that a scaled version of the madm median of absolute residuals about the median is approximately min max bias robust within the class of huber s proposal 2 joint estimates of location and scale we also consider the larger class of m estimates of scale with general location and show that a scaled version of the shorth the shortest half of the data is approximately min max bias robust in this class finally we present the results of a monte carlo study showing that the shorth has attractive finite sample size mean squared error properties for contaminated gaussian data

asymptotically robust detection and estimation for very heavy tailed noise the asymptotically robust detection of signals and the analogous problem of estimation of location are considered for noise distributions with heavier than exponential tails the solution to both problems is identical and is obtained when using efficacy or asymptotic variance as the performance measure the density of the noise is assumed to belong to a e contamination class where the contaminant is either 1 from the class of unimodal densities 2 from the class of densities limited in support to an interval about the mode of the nominal density or 3 a mixture of a member of 1 with a member of 2 the robust detector or estimator is similar to a censored version of the nonlinearity used for maximum likelihood estimation or locally optimal detection in the presence of noise with the nominal distribution

generalised weighted cramer von mises distance estimators summary generalised weighted cramer von mises distance estimators in an arbitrary model with a k dimensional parameter vector are investigated the distance function is defined as a function g of model based residuals for a specified target model f and the empirical cumulative distribution function f over the real line invoking a weight function w it is shown that the estimator is fisher consistent asymptotically multivariate normal and nearly efficient with desirable robustness properties if the true model is equal to the target model the residual function g does not affect the limiting distribution the weight function w controls the asymptotic distribution and the robustness of the estimator three different classes of the weight functions are introduced for different outlier patterns these weight functions produce estimators asymptotically as efficient as the maximum likelihood estimators at the true model an alternative way of calculating the estimators is considered simulation results indicate that asymptotic results are useful for moderate sample sizes and that the estimators are stable at the neighbourhood of the target model

lower bounds for contamination bias globally minimax versus locally linear estimation we study how robust estimators can be in parametric families obtaining a lower bound on the contamination bias of an estimator that holds for a wide class of parametric families this lower bound includes as a special case the bound used to establish that the median is bias minimax among location equivariant estimators and it is tight or nearly tight in a variety of other settings such as scale estimation discrete exponential families and multiple linear regression the minimum variation distance estimator has contamination bias within a dimension free factor of this bound a second lower bound applies to locally linear estimates and implies that such estimates cannot be bias minimax among all fisher consistent estimates in higher dimensions in linear regression this class of estimates includes the familiar m estimates gm estimates and s estimates in discrete exponential families yet another lower bound implies that the proportion of zeros estimate has minimax bias if the median of the distribution is zero a common situation in some fields this bound also implies that the information standardized sensitivity of every fisher consistent estimate of the poisson mean and of the binomial proportion is unbounded

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

rates of convergence of estimates kolmogorov s entropy and the dimensionality reduction principle in regression l 1 optimal minimum distance estimators are provided for a projection pursuit regression type function with smooth functional components that are either additive or multiplicative in the presence of or without interactions the obtained rates of convergence of the estimate to the true parameter depend on kolmogorov s entropy of the assumed model and confirm stone s heuristic dimensionality reduction principle rates of convergence are also obtained for the error in estimating the derivatives of a regression type function

min max bias robust regression abstract this paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over epsilon contamination neighborhoods for the joint distribution of the response and carriers two classes of estimates are treated 1 m estimates with bounded function rho applied to the scaled residuals using a very general class of scale estimates and 2 bounded influence function type generalized m estimates estimates in the first class are obtained as the solution of a minimization problem while estimates in the second class are specified by an estimating equation the first class of m estimates is sufficiently general to include both huber proposal 2 simultaneous estimates of regression coefficients and residuals scale and rousseeuw yohai s estimates of regression it is shown than an s estimate based on a jump function type rho solves the min max bias problem for the class of m estimates with very general scale this estimate is obtained by the minimization of the alpha quantile of the squared residuals where alpha epsilon depends on the fraction of contamination epsilon when epsilon approaches limit of 5 alpha epsilon approaches limit of 5 and the min max estimator approaches the least median of squared residuals estimator introduced by rousseeuw for the bounded influence class of gm estimates it is shown the a sign type nonlinearity yields the min max estimate this estimate coincides with the minimum gross error sensitivity gm estimate

on the automorphism group of generalized hermitian codes we determine the full automorphism group of the generalized hermitian curve denoted by gh which generalizes the hermitian curve the automorphism group of a code is important in coding theory and in this way we determine completely the automorphism group of the one point ag codes over the gh curve

robustness infinitesimal neighborhoods and moment restrictions this paper is concerned with robust estimation under moment restrictions a moment restriction model is semiparametric and distribution free therefore it imposes mild assumptions yet it is reasonable to expect that the probability law of observations may have some deviations from the ideal distribution being modeled due to various factors such as measurement errors it is then sensible to seek an estimation procedure that is robust against slight perturbation in the probability measure that generates observations this paper considers local deviations within shrinking topological neighborhoods to develop its large sample theory so that both bias and variance matter asymptotically the main result shows that there exists a computationally convenient estimator that achieves optimal minimax robust properties it is semiparametrically efficient when the model assumption holds and at the same time it enjoys desirable robust properties when it does not

minimum hellinger distance estimation for finite mixture models abstract minimum hellinger distance estimates are considered for finite mixture models when the exact forms of the component densities are unknown in detail but are thought to be close to members of some parametric family minimum hellinger distance estimates are asymptotically efficient if the data come from a member of the parametric family and are robust to certain departures from the parametric family a new algorithm is introduced that is similar to the em algorithm a specialized adaptive density estimate is also introduced standard measures of robustness are discussed some difficulties are noted the robustness and asymptotic efficiency of the estimators are illustrated using simulations

inconsistency of resampling algorithms for high breakdown regression estimators and a new algorithm because high breakdown estimators hbes are impractical to compute exactly in large samples approximate algorithms are used the algorithm generally produces an estimator with a lower consistency rate and breakdown value than the exact theoretical estimator this discrepancy grows with the sample size with the implication that huge computations are needed for good approximations in large high dimension samples the workhorse for hbes has been the elemental set or basic resampling algorithm this turns out to be completely ineffective in high dimensions with high levels of contamination however enriching it with a concentration step turns it into a method that can handle even high levels of contamination provided that the regression outliers are located on random cases it remains ineffective if the regression outliers are concentrated on high leverage cases we focus on the multiple regression problem but several of the broad conclusions notably those of the inadequacy of fixed numbers of

on automorphisms of geometric goppa codes the algebraic geometric codes which were introduced by v d goppa in 1977 7 8 we call them geometric goppa codes can be used to prove the existence of long linear codes which are better than the gilbert varshamov bound there are many good expositions of this important result ll 13 15 181 in that context one has to consider curves of high genus which admit many rational points on the other hand new interesting codes were found among the geometric goppa codes by looking at special curves e g elliptic curves hermite curves or artin schreier curves 3 5 8 13 16 171 in coding theory one is often interested in codes admitting many automorphisms e g cyclic codes goppa ls already observed that automorphisms of the underlying algebraic curve induce automorphisms of the associated geometric codes since goppa s paper is not quite correct we shall give a detailed exposition of this connexion in section 2 section 3 is the central part of our paper it is devoted to the case of geometric goppa codes associated with a curve of genus 0 so called rational goppa codes these codes are particularly interesting since many well known classes of codes e g bch codes classical goppa codes reed solomon codes cauchy codes 6 can be represented as rational goppa codes or subfield subcodes of them cl our main result is theorem 3 1 it states that in general all automorphisms of a rational goppa code are induced by automorphisms of the rational curve i e by projective transformations of the projective line

a characterization of all single integral non kernel divergence estimators divergence measures have been used for a long time for different purposes in information theory and statistics in particular density based minimum divergence estimation is a popular tool in the statistical literature given the sampled data and a parametric model we estimate the model parameter by choosing the member of the model family that is closest to the data distribution in terms of the given divergence in the absolutely continuous set up when the distributions from the model family and the unknown data generating distribution are assumed to have densities the application of kernel based non parametric smoothing is sometimes unavoidable to get an estimate of the true data density the use of kernels or other non parametric smoothing techniques makes the estimation process considerably more complex as now one has to impose necessary conditions not just on the model but also on the kernel and its bandwidth in higher dimensions the efficiency of the kernel density estimator kde often becomes too low for the minimum divergence procedure to be practically useful it can therefore lead to a significant advantage to have a divergence which allows minimum divergence estimation bypassing the use of non parametric smoothing for the same reason characterizing the class of such divergences would be a notable achievement in this work we provide a characterization of the class of divergences that bypasses the use of non parametric smoothing in the construction of divergences providing a solution to this very important problem

cycle map for strictly decomposable cycles we introduce a class of cycles called nondegenerate strictly decomposable cycles and show that the image of each cycle in this class under the refined cycle map to an extension group in the derived category of arithmetic mixed hodge structures does not vanish this class contains certain cycles in the kernel of the abel jacobi map the construction gives a refinement of nori s argument in the case of a self product of a curve as an application we show that a higher cycle which is not annihilated by the reduced higher abel jacobi map produces uncountably many indecomposable higher cycles on the product with a variety having a nonzero global 1 form

linear models random censoring and synthetic data summary estimators for the linear model in the presence of censoring are available a new extension of the least squares estimator to censored data is equivalent to applying the ordinary least squares estimator to synthetic times time constructed by magnifying the gaps between successive order statistics under suitable regularity conditions the synthetic data estimator is fisher consistent and asymptotically normal examples facilitate com parison of the synthetic data estimator with estimators proposed by buckley james 1979 and by koul susarla van ryzin 1981 the entire lifetime of a person a machine a plant or an animal is not always observable some lifetimes may be censored in that only a lower bound for the lifetime is recorded statisticians are often interested in modelling the distribution of the true lifetimes as a function of covariates when censoring is present a proportional hazards model is often used when there is no censoring linear models are often used for ad hoc modelling of dependent variables miller halpern 1982 survey the methods that have been proposed for the linear model in the presence of censoring this paper describes a closed form method which is consistent and asymptotically normal this method incorporates censor ing in a natural way section 2 formulates classical least squares estimation in a manner that generalizes to accommodate random right censoring section 3 gives some theoretical results for the simplest nontrivial linear model the two sample case section 4 contains examples and the final section discusses the role of this estimator the covariate vector of the ith person will be denoted by xi and usually includes one as the first component the true lifetime yi follows a linear model if its conditional distribution given xi is

the percentage bend correlation coefficient a well known result is that the usual correlation coefficient is highly nonrobust very slight changes in only one of the marginal distributions can alter by a substantial amount there are a variety of methods for correcting this problem this paper identifies one particular method which is useful in psychometrics and provides a simple test for independence it is not recommended that the new test replace the usual test ofh 0 0 but the new test has important advantages over the usual test in terms of both type i errors and power

the role of beliefs in inference for rational expectations models abstract this paper discusses inference for rational expectations models estimated via minimum distance methods by characterizing the probability beliefs regarding the data generating process dgp that are compatible with given moment conditions the null hypothesis is taken to be rational expectations and the alternative hypothesis to be distorted beliefs this distorted beliefs alternative is analyzed from the perspective of a hypothetical semiparametric bayesian who believes the model and uses it to learn about the dgp this interpretation provides a different perspective on estimates test statistics and confidence regions in large samples particularly regarding the economic significance of rejections in rational expectations models a suggestive application to the equity premium puzzle is examined

bayesian nonparametric estimation of the median part ii asymptotic properties of the estimates for data theta varepsilon i i 1 ldots n where varepsilon i are i i d sim f with the median of f equal to 0 but f otherwise unknown it is desired to estimate theta in doss 1985 priors are put on the pair f theta the marginal posterior distribution of theta is computed and the mean of the posterior is taken as the estimate of theta in the present paper a frequentist point of view is adopted the consistency properties of the bayes estimates computed in doss 1985 are investigated when the prior on f is of the dirichlet type any f whose median is 0 is in the support of these priors it is shown that if the varepsilon i are i i d from a discrete distribution then the bayes estimates are consistent however if the distribution of the varepsilon is is continuous the bayes estimates can be inconsistent

improvements on parameters of one point ag codes from hermitian curves by choosing specific divisors we show that some algebraic geometry ag codes from the hermitian curves have better parameters than one point hermitian codes the general lower bound is discussed

minimum distance methods based on quadratic distances for transforms in simple linear regression models minimum distance estimators and test statistics based on quadratic distances for transforms are introduced in simple linear regression models asymptotic properties of quadratic distance estimators and tests for the error distributions are derived the modified quadratic distance estimator is shown to be efficient provided that the error distribution is symmetric and the derivative of the logarithm of the error density is a linear combination of the transforms

minimum distance estimation in a linear regression model this paper discusses a class of minimum distance cramer von mises type estimators of the slope parameter in a linear regression model these estimators are obtained by minimizing an integral of squared difference between weighted empiricals of the residuals and their expectations with respect to a large class of integrating measures the estimator corresponding to the weights proportional to the design variable is shown to be asymptotically efficient within the class at a given error distribution the paper also discusses the asymptotic null distribution of a class of minimum cramer von mises type goodness of fit test statistics

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

a lower bound on the error in nonparametric regression type problems let x1 y1 xn yn be a sample denote the conditional density of ylxi xi as f ylxi 0 xi and 0 an element of a metric space 9 d a lower bound is provided for the d error in estimating 0 the order of the bound depends on the local behavior of the kullback information of the conditional density as an application we consider the case where 1 is the space of q smooth functions on 0 i d metrized with the lr distance 1 r oo

an rkhs approach to robust l sup 2 estimation and signal detection the application of rkhs reproducing kernel hilbert space theory to the problems of robust signal detection and estimation is investigated it is shown that this approach provides a general and unified framework in which to analyze the problems of l sup 2 estimation matched filtering and quadratic detection in the presence of uncertainties regarding the second order structure of the random processes involved