robust bounded influence tests in general parametric models abstract we introduce robust tests for testing hypotheses in a general parametric model these are robust versions of the wald scores and likelihood ratio tests and are based on general m estimators their asymptotic properties and influence functions are derived it is shown that the stability of the level is obtained by bounding the self standardized sensitivity of the corresponding m estimator furthermore optimally bounded influence tests are derived for the wald and scores type tests applications to real and simulated data sets are given to illustrate the tests performance

reweighted ls estimators converge at the same rate as the initial estimator the problem of combining high efficiency with high breakdown properties for regression estimators has piqued the interest of statisticians for some time one proposal specifically suggested by rousseeuw and leroy is to use the least median of squares estimator omit observations whose residuals are larger than some constant cut off value and apply least squares to the remaining observations although this proposal does retain high breakdown point it actually converges no faster than the initial estimator in fact the reweighted least squares estimator is asymptotically a constant times the initial estimator if the initial estimator converges at a rate strictly slower than n 1 2

optimum robust testing in linear models robust tests for linear models are derived via wald type tests that are based on asymptotically linear estimators for a robustness criterion the maximum asymptotic bias of the level of the test for distributions in a shrinking contamination neighborhood is used by also regarding the asymptotic power of the test admissible robust tests and most efficient robust tests are derived for the greatest efficiency the determinant of the covariance matrix of the underlying estimator is minimized also most robust tests are derived it is shown that at the classical d optimal designs the most robust tests and the most efficient robust tests have a very simple form moreover the d optimal designs provide the highest robustness and the highest efficiency under robustness constraints across all designs so d optimal designs are also the optimal designs for robust testing two examples are considered for which the most robust tests and the most efficient robust tests are given

bounded influence and high breakdown point testing procedures in linear models abstract three classes of testing procedures based on one step high breakdown point bounded influence estimators for testing subhypotheses in linear models are developed these are drop in dispersion wald type and score type tests the asymptotic distributions of these testing procedures are obtained under the null hypothesis and under contiguous alternatives their stability properties are studied in terms of their influence functions and breakdown points it is shown that the tests have bounded influence functions for the wald type tests the level and power breakdowns are determined by the breakdown point of the parameter estimate and the associated variance covariance matrix the drop in dispersion test exhibits high level breakdown but not high power breakdown point similar behavior is exhibited by the score type tests but slight modifications can be made in the construction of the test statistics to ensure high breakdown points in terms of both level and power an example is given to illustrat

robust wald type tests of one sided hypotheses in the linear model abstract let us consider the linear model y x e in the usual matrix notation where the errors are iid our main objective is to develop robust wald type tests for a large class of hypotheses on by robust we mean robustness in terms of size and power against long tailed error distributions first assume that the error distribution is symmetric about the origin let l and be the least squares and a robust estimator of assume that they are asymptotically normal about with covariance matrices 2 xtx 1 and 2 xtx 1 respectively so could be an m estimator or a high breakdown point estimator robust wald type tests based on denoted by rw are studied here for testing a large class of one sided hypotheses on it is shown that the asymptotic null distribution of rw and that of the usual wald type statistic based on l denoted by w are the same this is a useful result since the critical values and procedures for computing the p values for w are directly applicable to rw as well a more impo

strong factorization of operators on spaces of vector measure integrable functions and unconditional convergence of series in this paper we characterize the space of multiplication operators from an lp space into a space l1 m of integrable functions with respect to a vector measure m as the subspace l 1 rm p mu bf m defined by the functions that have finite p semivariation we prove several results concerning the banach lattice structure of such spaces we obtain positive results for instance they are always complete and we provide counterexamples to prove that other properties are not satisfied for example simple functions are not in general dense we study the operators that factorize through l 1 rm p mu bf m and we prove an optimal domain theorem for such operators we use our characterization to generalize the bennet maurey nahoum theorem on decomposition of functions that define an unconditionally convergent series in l1 0 1 to the case of 2 concave banach function spaces

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

robust alternatives to the f test in mixed linear models based on mm estimates mixed linear models are commonly used to analyze data in many settings these models are generally fitted by means of restricted maximum likelihood techniques relying heavily on normality the sensitivity of the resulting estimators and related tests to this underlying assumption has been identified as a weakness that can even lead to wrong interpretations very recently a highly robust estimator based on a scale estimate that is an s estimator has been proposed for general mixed linear models it has the advantage of being easy to compute and allows the computation of a robust score test however this proposal cannot be used to define a likelihood ratio type test that is certainly the most direct route to robustify an f test as the latter is usually a key tool of hypothesis testing in mixed linear models we propose two new robust estimators that allow the desired extension they also lead to resistant wald type tests useful for testing contrasts and covariate effects we study their properties theoretically and by means of simulations the analysis of a real data set illustrates the advantage of the new approach in the presence of outlying observations

factorizing operators on banach function spaces through spaces of multiplication operators abstract in order to extend the theory of optimal domains for continuous operators on a banach function space x over a finite measure we consider operators t satisfying other type of inequalities than the one given by the continuity which occur in several well known factorization theorems for instance pisier factorization theorem through lorentz spaces p th power factorable operators we prove that such a t factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for t our extended optimal domain technique does not need necessarily the equivalence between and the measure defined by the operator t and by using rings is allowed to be infinite classical and new examples and applications of our results are also given including some new results on the hardy operator and a factorization theorem through hilbert spaces

the use and interpretation of residuals based on robust estimation abstract residual plots and diagnostic techniques are important tools for examining the fit of a regression model in the case of least squares fits plots of residuals provide a visual assessment of the adequacy of various aspects of the fitted model an important question is whether plots of robust residuals can be interpreted in the same manner as their least squares counterparts this article addresses this problem for two popular classes of robust estimates m estimates and gm estimates of the mallows and schweppe types first order properties of the residuals and fitted values are derived under correct and misspecified models these properties are insightful on the general interpretability of robust residual plots and on their ability to detect curvature in misspecified models the results of a simulation study consisting of tests for randomness and curvature in residual plots supports these theoretical properties standardization of robust residuals is also presented

robust bounded influence tests in linear models abstract a robust test that we call an aligned generalized m test for testing subhypotheses in general linear models is developed and its asymptotic properties are studied the test is a robustification of the well known f test and it is an elegant alternative to ronchetti s 1982 class of tests p values associated with it can be approximated readily using existing chi square tables the test is based on an appropriately constructed quadratic form and uses the generalized m estimators of the parameters in the reduced model under the null hypothesis the asymptotic distribution is a central chi square and under contiguous alternatives it is a noncentral chi square with the same degrees of freedom the test can be viewed as a generalization of sen s 1982 m test for linear models the influence function of the test is bounded the bound not only applies to the influence of residuals but to the influence of position in the factor space as well on the other hand sen s test has bounded influence only

robust inference for generalized linear models by starting from a natural class of robust estimators for generalized linear models based on the notion of quasi likelihood we define robust deviances that can be used for stepwise model selection as in the classical framework we derive the asymptotic distribution of tests based on robust deviances and we investigate the stability of their asymptotic level under contamination the binomial and poisson models are treated in detail two applications to real data and a sensitivity analysis show that the inference obtained by means of the new techniques is more reliable than that obtained by classical estimation and testing procedures

robust tests in regression models with omnibus alternatives and bounded influence a robust approach for testing the parametric form of a regression function versus an omnibus alternative is introduced this generalizes existing robust methods for testing subhypotheses in a regression model the new test is motivated by developments in modern smoothing based testing procedures and can be viewed as a robustification of a smoothing based conditional moment test it is asymptotically normal under both the null hypothesis and local alternatives the robustified test retains the omnibus property of the corresponding smoothing test that is it is consistent for any fixed smooth alternative in an infinite dimensional space it is shown that the bias of the asymptotic level under shrinking local contamination is bounded only if the second order hampel s influence function is bounded the test s performance is demonstrated through both monte carlo simulations and application to an agricultural dataset

on one step gm estimates and stability of inferences in linear regression abstract the folklore on one step estimation is that it inherits the breakdown point of the preliminary estimator and yet has the same large sample distribution as the fully iterated version as long as the preliminary estimate converges faster than n 1 4 where n is the sample size we investigate the extent to which this folklore is valid for one step gm estimators and their associated standard errors in linear regression we find that one step gm estimates based on newton raphson or scoring inherit the breakdown point of high breakdown point initial estimates such as least median of squares provided the usual weights that limit the influence of extreme points in the design space are based on location and scatter estimates with high breakdown points moreover these estimators have bounded influence functions and their standard errors can have high breakdown points the folklore concerning the large sample theory is correct assuming the regression errors are symmetrically distributed and homoscedastic

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

a bounded influence high breakdown efficient regression estimator abstract we consider the multiple linear regression model y i x i e i i 1 2 n with random carriers and focus on the estimation of this article s main contribution is to present an estimator that is affine regression and scale equivariant has both a high breakdown point and a bounded influence function and has an asymptotic efficiency greater than 95 versus least squares under gaussian errors we give conditions under which the estimator a one step general m estimator that uses schweppe weights and is based on a high breakdown initial estimator satisfies these properties the major conditions necessary for the estimator are a it must be based on a n consistent initial estimator with a 50 breakdown point and b it must be based on a psi function that is odd bounded and strictly increasing the advantage of this estimator over previous approaches is that it does not downweight high leverage points without first considering how they fit the bulk of the data methods of computi

the integration theory of banach space valued measures and the tonelli fubini theorems ii pettis integration in this part of the paper we turn to the concepts of gelfand and pettis integrability and integration of scalar valued functions with respect to a banach space valued measure the problem before us and our method for solving it have been outlined in section 1 of part i of this paper in what follows we adhere to the notation of part i and continue the trend set in that part all references with numbers such as 1 2 2 3 and a5 are to the items in sections 1 and 2 and appendix a in part i

a hybrid finite integration finite volume scheme a hybridized scheme for the numerical solution of transient electromagnetic field problems is presented the scheme combines the finite integration technique fit and the finite volume method fvm in order to profit from the computational efficiency of the fit while taking advantage of the superior dispersive properties of the fvm the scheme is based on the longitudinal transverse lt splitting of the discrete curl operator the fit is employed for discretizing the two dimensional subproblem while the one dimensional problem is discretized according to the fvm the scheme offers benefits for the simulation of multiscale setups where the size of the computational domain along one preferred direction is electrically much larger than along the others in such situations the accumulation of dispersion errors within hundreds of thousands of time steps usually deteriorates the solution accuracy the hybrid scheme is applied in combination with adaptive mesh refinement yielding an efficient scheme for multiscale applications

asymptotic behavior of general m estimates for regression and scale with random carriers let xini y i be a sequence of independent identically distributed random variables where x i r p and y i r and let r p be an unknown vector such that y i x i u i where u i is independent of x i and has distribution function f u where 0 is an unknown parameter this paper deals with a general class of m estimates of regression and scale defined as solutions of the system sum limits i phi text x i r i x i 0 sum limits i chi r i 0 where r y i xi1 with r p r r and r r this class contains estimators of proposed by huber mallows and krasker and welsch the consistency and asymptotic normality of the general m estimators are proved assuming general regularity conditions on and and assuming the joint distribution of x i y i to fulfill the model only approximately

on m tests in linear models summary for subhypothesis testing in some general linear models some tests based on residuals using m estimators are considered and their asymptotic properties are studied