transformations in regression a robust analysis we consider two approaches to robust estimation for the box cox power transformation model one approach maximizes weighted modified likelihoods a second approach bounds a measure of gross error sensitivity among our primary concerns is the performance of these estimators on actual data in examples that we study there seem to be only minor differences between these two robust estimators but they behave rather differently than the maximum likelihood estimator or estimators that bound only the influence of the residuals these examples show that model selection determination of the transformation parameter and outlier identification are fundamentally interconnected

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

high breakdown point and high efficiency robust estimates for regression a class of robust estimates for the linear model is introduced these estimates called mm estimates have simultaneously the following properties i they are highly efficient when the errors have a normal distribution and ii their breakdown point is 0 5 the mm estimates are defined by a three stage procedure in the first stage an initial regression estimate is computed which is consistent robust and with high breakdown point but not necessarily efficient in the second stage an m estimate of the errors scale is computed using residuals based on the initial estimate finally in the third stage an m estimate of the regression parameters based on a proper redescending psi function is computed consistency and asymptotical normality of the mm estimates assuming random carriers are proved a convergent iterative numerical algorithm is given finally the asymptotic biases under contamination of optimal bounded influence estimates and mm estimates are compared

exact and operator rational approximate solutions of the helmholtz weyl composition equation in underwater acoustics the quadratic profile the application of phase space and path integral methods to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one way helmholtz equation is briefly reviewed the role of operator symbols is stressed and their properties briefly discussed the construction of the operator symbol requires the exact or approximate solution of the helmholtz weyl composition equation in the weyl pseudodifferential operator calculus the exact symbols for several quadratic profiles are presented and briefly analyzed the symbols exactly corresponding to a family of operator rational approximations are also presented for the quadratic case these results are used to illustrate several points pertinent to wide angle propagation modeling and the refractive index profile reconstruction problem in underwater acoustics

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

the identification of multiple outliers abstract one approach to identifying outliers is to assume that the outliers have a different distribution from the remaining observations in this article we define outliers in terms of their position relative to the model for the good observations the outlier identification problem is then the problem of identifying those observations that lie in a so called outlier region methods based on robust statistics and outward testing are shown to have the highest possible breakdown points in a sense derived from donoho and huber but a more detailed analysis shows that methods based on robust statistics perform better with respect to worst case behavior a concrete outlier identifier based on a suggestion of hampel is given

breakdown properties of location estimates based on halfspace depth and projected outlyingness we describe multivariate generalizations of the median trimmed mean and w estimates the estimates are based on a geometric construction related to projection pursuit they are both affine equivariant coordinate free and have high breakdown point the generalization of the median has a breakdown point of at least 1 d 1 in dimension d and the breakdown point can be as high as 1 3 under symmetry in contrast various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than 1 d 1 in dimension d

ell _ p music robust direction of arrival estimator for impulsive noise environments a family of algorithms named lp music for direction of arrival doa estimation in impulsive noise is proposed the lp music estimator adopts the lp norm 1 p 2 of the residual fitting error matrix as the objective function for subspace decomposition rather than the frobenius norm that is used in the conventional music method although the matrix lp norm minimization based subspace decomposition will lead to a nonconvex optimization problem two iterative algorithms are designed for achieving efficient solutions the first algorithm is the iteratively reweighted singular value decomposition ir svd where the svd of a reweighted data matrix is performed in each iteration the second algorithm solves the nonconvex matrix lp norm minimization by alternating convex optimization two complex valued newton s methods with optimal step size in each iteration are devised to solve the resulting convex problem the convergence of the iterative procedure is also proved numerical results verify that the lp music methodology outperforms the standard music scheme and several existing outlier resistant doa estimation approaches in terms of resolution capability and estimation accuracy

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

the masking breakdown point of multivariate outlier identification rules abstract in this article we consider simultaneous outlier identification rules for multivariate data generalizing the concept of so called outlier identifiers as presented by davies and gather for the case of univariate samples such multivariate outlier identifiers are based on estimators of location and covariance therefore it seems reasonable that characteristics of the estimators influence the behavior of outlier identifiers several authors mentioned that using estimators with low finite sample breakdown point is not recommended for identifying outliers to give a formal explanation we investigate how the finite sample breakdown points of estimators used in these identification rules influence the masking behavior of the rules

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

on the uniqueness of s functionals and m functionals under nonelliptical distributions the s functionals of multivariate location and scatter including the mve functionals are known to be uniquely defined only at unimodal elliptically symmetric distributions the goal of this paper is to establish the uniqueness of these functionals under broader classes of symmetric distributions we also discuss some implications of the uniqueness of the functionals and give examples of striclty unimodal and symmetric distributions for which the mve functional is not uniquely defined the uniqueness results for the s functionals are obtained by embedding them within a more general class of functionals which we call the m functionals with auxiliary scale the uniqueness results of this paper are then obtained for this class of multivariate functionals besides the s functionals the class of multivariate m functionals with auxiliary scale include the constrained m functionals recently introduced by kent and tyler as well as a new multivariate generalization of yohai s mm functionals

the construction of exact d optimum experimental designs with application to blocking response surface designs summary the properties of optimum designs are investigated when there are both qualitative and quantitative factors algorithms for the construction of exact designs are described which are of general applicability the case when the qualitative factors represent blocking variables is given special treatment a modified algorithm provides the opportunity to divide the experimental trials into blocks of specified sizes examples are given of second order models in the quantitative factors divided into two or three blocks of a variety of sizes use of the adjustment algorithm to improve such designs generated from a list of candidate points is demonstrated

designing for a response transformation parameter summary we investigate d optimum designs for experiments in which a linear model holds after an unknown power transformation of the univariate response variable this is a departure from standard d optimal design in which an appropriate scale for the response is assumed known before data collection the design problem that we formulate is intrinsically nonlinear requiring characterization of likely parameter values several applications are presented to illustrate the importance of recognizing the role of response transformations at the design stage

robustness and efficiency properties of scatter matrices the robustness and efficiency properties of likelihood ratio tests for functions of the population covariance matrix are studied an alternative class of tests based upon affineinvariant m estimates of scatter is proposed whenever the function of the covariance matrix is invariant under a common scale change for such inferences it is shown that a single scalar valued index of efficiency is sufficient

factorization of a dissipative wave equation and the green functions technique for axially symmetric fields in a stratified slab the time domain scattering problem for axially symmetric fields and a dissipative stratified slab is considered explicit representations are derived for the wave splitting operators that factorize the dissipative hankel transformed wave equation the green functions approach associated with this splitting is developed partial differential equations pdes for the green functions are derived and used to derive the imbedding equations for the reflection and transmission kernels numerical results for the internal fields are given in the direct problem and the phase velocity is reconstructed in the inverse problem

bounded influence estimation in the mixed linear model abstract bounded influence estimation also known as generalized m or gm estimation in the regression model is reviewed the definitions of bounded influence estimation proposed by mallows and schweppe are then extended to the mixed linear model this is achieved by applying appropriate weight functions to maximum likelihood and restricted maximum likelihood estimating equations the asymptotic properties of the new estimators are obtained and the estimators are applied to an artificial dataset the article concludes with an extension of the example into a small simulation study designed to test some properties of the estimators in samples of moderate size

on wielandt s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix a relatively obscure eigenvalue inequality due to wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix the asymptotic distributions are obtained under a fairly general setting an application of the general theory to the bootstrap distribution of the eigenvalues of the sample covariance matrix is given 1 introduction and summary the derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix arises in many papers in multivariate analysis although the main idea behind most of the derivations is quite basic i e the expansion of the sample roots about the population roots the derivations themselves are often quite involved these complications are primarily due to the mathematical rather than statistical nature of the eigenvalue problem one of the main objectives of this paper is to introduce a simple method for obtaining the asymptotic distribution of the eigenvalue of random symmetric matrices the method is based upon a relatively obscure eigenvalue inequality

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

asymptotic behaviour of s estimates of multivariate location parameters and dispersion matrices it is shown under appropriate conditions that rousseeuw s minimum volume estimator and other s estimators of multivariate location and dispersion parameters are consistent under certain differentiability conditions the estimates are asymptotically normally distributed with a norming factor of n 1 2

breakdown and groups the concept of breakdown point was introduced by hampel ph d dissertation 1968 univ california berkeley ann math statist 42 1971 1887 1896 and developed further by among others huber robust statistics 1981 wiley new york and donoho and huber in a festschrift for erich l lehmann 1983 157 184 wadsworth belmont ca it has proved most successful in the context of location scale and regression problems attempts to extend the concept to other situations have not met with general acceptance in this paper we argue that this is connected to the fact that in the location scale and regression problems the translation and affine groups give rise to a definition of equivariance for statistical functionals comparisons in terms of breakdown points seem only useful when restricted to equivariant functionals and even here the connection between breakdown and equivariance is a tenuous one institute of mathematical statistics 2005

optimal rank based tests for homogeneity of scatter we propose a class of locally and asymptotically optimal tests based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations contrary to the existing parametric procedures these tests remain valid without any moment assumptions and thus are perfectly robust against heavy tailed distributions validity robustness nevertheless they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all efficiency robustness in particular their normal score version outperforms traditional gaussian likelihood ratio tests and their pseudo gaussian robustifications under a very broad range of non gaussian densities including for instance all multivariate student and power exponential distributions

fast very robust methods for the detection of multiple outliers abstract a few repeats of a simple forward search from a random starting point are shown to provide sufficiently robust parameter estimates to reveal masked multiple outliers the stability of the patterns obtained is exhibited by the stalactite plot the robust estimators used are least median of squares for regression and the minimum volume ellipsoid for multivariate outliers the forward search also has potential as an algorithm for calculation of these parameter estimates for large problems parallel computing provides appreciable reduction in computational time

robust inference with gmm estimators the local robustness properties of generalized method of moments gmm estimators and of a broad class of gmm based tests are investigated in a unified framework gmm statistics are shown to have bounded influence if and only if the function defining the orthogonality restrictions imposed on the underlying model is bounded since in many applications this function is unbounded it is useful to have procedures that modify the starting orthogonality conditions in order to obtain a robust version of a gmm estimator or test we show how this can be obtained when a reference model for the data distribution can be assumed we develop a flexible algorithm for constructing a robust gmm rgmm estimator leading to stable gmm test statistics the amount of robustness can be controlled by an appropriate tuning constant we relate by an explicit formula the choice of this constant to the maximal admissible bias on the level or and the power of a gmm test and the amount of contamination that one can reasonably assume given some information on the data finally we illustrate the rgmm methodology with some simulations of an application to rgmm testing for conditional heteroscedasticity in a simple linear autoregressive model in this example we find a significant instability of the size and the power of a classical gmm testing procedure under a non normal conditional error distribution on the other side the rgmm testing procedures can control the size and the power of the test under non standard conditions while maintaining a satisfactory power under an approximatively normal conditional error distribution

the riemann green function and the invariant imbedding equations for hyperbolic systems of first order abstract in the present paper we deal with a boundary value problem for a hyperbolic system of first order in one space variable an explicit integral representation for the riemann green function is first obtained from which a set of discontinuities at both the initial time and along certain characteristics are calculated next the reflection and transmission functions are then defined by simply evaluating this function at certain boundary points and finally a set of integro partial differential equations satisfied by the reflection and transmission functions is derived there are the time dependent invariant imbedding equations for the systems we consider here and these extend the results obtained by corones and krueger for second order hyperbolic equations

breakdown theory for bootstrap quantiles a general formula for computing the breakdown point in robustness for the tth bootstrap quantile of a statistic t n is obtained the answer depends on t and the breakdown point of t n since the bootstrap quantiles are vital ingredients of bootstrap confidence intervals the theory has implications pertaining to robustness of bootstrap confidence intervals for certain l and m estimators a robustification of bootstrap is suggested via the notion of winsorization

on the relation between s estimators and m estimators of multivariate location and covariance we discuss the relation between s estimators and m estimators of multivariate location and covariance as in the case of the estimation of a multiple regression parameter s estimators are shown to satisfy first order conditions of m estimators we show that the influence function if x s f of s functionals exists and is the same as that of corresponding m functionals also we show that s estimators have a limiting normal distribution which is similar to the limiting normal distribution which is similar to the limiting normal distribution of m estimators finally we compare asymptotic variances and breakdown point of both types of estimators

d optimal split plot designs with given numbers and sizes of whole plots the design of split plot experiments has received considerable attention during the last few years the goal of this article is to provide an efficient algorithm to compute d optimal split plot designs with given numbers of whole plots and given whole plot sizes the algorithm is evaluated and applied to a protein extraction experiment in addition it is shown that two level factorial and fractional factorial designs are d optimal for estimating first order response surface models for specific numbers and sizes of whole plots

an electromagnetic inverse problem for dispersive media the dispersion of transient electromagnetic waves in a homogeneous medium can be characterized by expressing either the complex permittivity as a function of frequency or the susceptibility kernel as a function of time in this paper a time domain technique is used to derive a nonlinear integrodifferential equation which relates the susceptibility kernel for a one dimensional homogeneous slab to the reflection operator for the medium thus the susceptibility kernel which is a function of time can be determined from reflection data a numerical implementation of this technique is shown the more general case of a medium consisting of a stack of homogeneous dispersive layers is also addressed

a note on tyler s modification of the mad for the stahel donoho estimator in 1994 tyler proposed a modification of the mad in order to achieve the maximal possible finite sample breakdown point for the stahel donoho estimator based upon the median and the modified mad a proof is not given however and as far as we know has not been published yet moreover there seems to be an error concerning the exact definition of the modified mad which is needed to achieve this goal for the sake of theoretical completeness and in order to correct this error we calculate the finite sample breakdown point of modifications of the mad and point out which of these modifications lead to a stahel donoho estimator with maximal finite sample breakdown point

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

finite sample breakdown points of projection based multivariate location and scatter statistics finite sample breakdown points are obtained for two classes of projection based multivariate location and scatter statistics the stahel donoho statistics and the maronna yohai statistics the definition of these multivariate statistics are dependent on the value of location and scale statistics for all univariate projections of the data and consequently their properties depend on the nature of the corresponding univariate location and scale statistics used on the projected data the finite sample breakdown points of the multivariate statistics though are not directly related to those of the corresponding univariate location and scale statistics a uniform finite sample breakdown point concept is needed the median and the median absolute deviation about the median m a d are one possible choice for the univariate location and scale statistics respectively for sparse data sets in high dimensions though it is not recommended that the m a d be used as the univariate scale statistic for the projected data since its uniform finite sample breakdown point is shown to be much less than optimum a simple modification to the m a d however is shown to alleviate this problem for various reasons one may wish to consider univariate location and scale statistics other than the median and the m a d respectively a very broad and natural class of univariate location and scale statistics to consider for the projected data is the simultaneous m estimates of location and scale new results on their breakdown properties are given in this paper implicit formulas for the breakdown points of monotonic simultaneous m estimates of location and scale are known and they tend to imply rather low breakdown points for smooth choices of the defining weight functions it is shown here that this phenomenon does not occur for a large class of nonmonotonic simultaneous m estimates of location and scale furthermore explicit rather than implicit expressions for the uniform finite sample breakdown points are given for these nonmonotonic m estimates

robust and efficient one way manova tests we propose robust tests as alternatives to the classical wilks lambda test in one way manova the robust tests use highly robust and efficient multisample multivariate s estimators or mm estimators instead of the empirical covariances the properties of several robust test statistics are compared under the null hypothesis the distribution of the test statistics is proportional to a chi square distribution as an alternative to the asymptotic distribution we develop a fast robust bootstrap method to estimate the distribution under the null hypothesis we show when it is asymptotically correct to estimate the null distribution in this way and we use simulations to verify the performance of the bootstrap based tests in finite samples we also investigate the power of the new tests as well as their robustness against outliers finally we illustrate the use of these robust test statistics on a real data example some additional results are provided as supplemental material

computer aided blocking of factorial and response surface designs we consider the problem of blocking response surface and factorial designs when block sizes are prespecified often rendering standard approaches inapplicable a generalization of harville s 1974

general weighted optimality of designed experiments the standard approach to finding optimal experimental designs employs conventional measures of design efficacy such as the a e and d criterion that assume equal interest in all estimable functions of model parameters this paper develops a general theory for weighted optimality allowing precise design selection according to expressed relative interest in different functions in the estimation space the approach employs a very general class of matrix specified weighting schemes that produce easily interpretable weighted optimality criteria in particular for any set of estimable functions and any selected corresponding weights analogs of standard optimality criteria are found that guide design selection according to the weighted variances of estimators of those particular functions the results are applied to solve the a optimal design problem for baseline factorial effects in unblocked experiments

computable robust estimation of multivariate location and shape in high dimension using compound estimators abstract estimation of multivariate shape and location in a fashion that is robust with respect to outliers and is affine equivariant represents a significant challenge the use of compound estimators that use a combinatorial estimator such as rousseeuw s minimum volume ellipsoid mve or minimum covariance determinant mcd to find good starting points for high efficiency robust estimators such as s estimators has been proposed in this article we indicate why this scheme will fail in high dimension due to combinatorial explosion in the space that must be searched for the mve or mcd we propose a meta algorithm based on partitioning the data that enables compound estimators to work in high dimension we show that even when the computational effort is restricted to a linear function of the number of data points the algorithm results in an estimator with good asymptotic properties extensive computational experiments are used to confirm that significant benefits accrue in finite samples as well we also g

the behavior of the stahel donoho robust multivariate estimator abstract the stahel donoho estimators t v of multivariate location and scatter are defined as a weighted mean and a weighted covariance matrix with weights of the form w r where w is a weight function and r is a measure of outlyingness obtained by considering all univariate projections of the data it has a high breakdown point for all dimensions and order n consistency the asymptotic bias of v for point mass contamination for suitable weight functions is compared with that of rousseeuw s minimum volume ellipsoid mve estimator a simulation shows that for a suitable w t and v exhibit high efficiency for both normal and cauchy distributions and are better than their competitors for normal data with point mass contamination the performances of the estimators for detecting outliers are compared for both a real and a synthetic data set

a numerical solution to the zakharov shabat inverse scattering problem a novel numerical method to solve the inverse scattering problem associated with the zakharov shabat coupled mode equations is developed in this problem the coupling coefficient for two propagating modes is determined from reflection data both codirectional and contradirectional coupling can be accommodated as well as the presence of bound states analytical results based on an independent analytical method are derived in order to demonstrate the accuracy of the proposed method which is applicable to a broad class of reflection data

the usefulness of optimum experimental designs optimum experimental designs were originally developed by kiefer mainly for response surface models this survey of recent developments emphasizes potential or actual usefulness for linear models the construction of exact designs particularly over irregular design regions is stressed as is the blocking of response surface designs other important areas include systematic designs that are robust against trend and designs for mixtures with irregular design regions several industrial examples are mentioned both d and c optimum designs are found for a non linear model of the economic response of cereal production to fertilizer level the c optimum design being for the conditions of maximum economic return locally optimum and bayesian designs are both described similar results for generalized linear models lead to designs for the ld 95 in a logistic model in which male and female subjects respond differently designs with structure in the variance suggest alternatives to the potentially wasteful product designs of taguchi designs for sequential clinical trials to include random balance are presented the last section outlines some applications including life testing and models in which time is a factor

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

fast exact bootstrap principal component analysis for p 1 million many have suggested a bootstrap procedure for estimating the sampling variability of principal component analysis pca results however when the number of measurements per subject p is much larger than the number of subjects n calculating and storing the leading principal components from each bootstrap sample can be computationally infeasible to address this we outline methods for fast exact calculation of bootstrap principal components eigenvalues and scores our methods leverage the fact that all bootstrap samples occupy the same n dimensional subspace as the original sample as a result all bootstrap principal components are limited to the same n dimensional subspace and can be efficiently represented by their low dimensional coordinates in that subspace several uncertainty metrics can be computed solely based on the bootstrap distribution of these low dimensional coordinates without calculating or storing the p dimensional bootstrap components fast bootstrap pca is applied to a dataset of sleep electroencephalogram recordings p 900 n 392 and to a dataset of brain magnetic resonance images mris p 3 million n 352 for the mri dataset our method allows for standard errors for the first 3 principal components based on 1000 bootstrap samples to be calculated on a standard laptop in 47 minutes as opposed to approximately 4 days with standard methods

high breakdown inference for mixed linear models mixed linear models are used to analyze data in many settings these models have a multivariate normal formulation in most cases the maximum likelihood estimator mle or the residual mle reml is usually chosen to estimate the parameters however the latter are based on the strong assumption of exact multivariate normality welsh and richardson have shown that these estimators are not robust to small deviations from multivariate normality this means that in practice a small proportion of data even only one can drive the value of the estimates on their own because the model is multivariate we propose a high breakdown robust estimator for very general mixed linear models that include for example covariates this robust estimator belongs to the class of s estimators from which we can derive asymptotic properties for inference we also use it as a diagnostic tool to detect outlying subjects we discuss the advantages of this estimator compared with other robust estimators proposed previously and i

highly efficient estimators of multivariate location with high breakdown point we propose an affine equivariant estimator of multivariate location that combines a high breakdown point and a bounded influence function with high asymptotic efficiency this proposal is basically a location m estimator based on the observations obtained after scaling with an affine equivariant high breakdown covariance estimator the resulting location estimator will inherit the breakdown point of the initial covariance estimator and within the location covariance model only the m estimator will determine the type of influence function and the asymptotic behaviour we prove consistency and asymptotic normality and obtain the breakdown point and the influence function

bootstrapping robust estimates of regression we introduce a new computer intensive method to estimate the distribution of robust regression estimates the basic idea behind our method is to bootstrap a reweighted representation of the estimates to obtain a bootstrap method that is asymptotically correct we include the auxiliary scale estimate in our reweighted representation of the estimates our method is computationally simple because for each bootstrap sample we only have to solve a linear system of equations the weights we use are decreasing functions of the absolute value of the residuals and hence outlying observations receive small weights this results in a bootstrap method that is resistant to the presence of outliers in the data the breakdown points of the quantile estimates derived with this method are higher than those obtained with the bootstrap we illustrate our method on two datasets and we report the results of a monte carlo experiment on confidence intervals for the parameters of the linear model

the breakdown points of the mean combined with some rejection rules in the past methods for rejection of outliers have been investigated mostly without regard to the quantitative consequences for subsequent estimation or testing procedures moreover although rejection of outliers with subsequent application of least squares methods is one of the oldest and most widespread classes of robust procedures until recently no comparison was made with other robust methods in this article the simplest situation namely estimation of a location parameter in the potential presence of outliers is treated by means of a monte carlo study this study yields monte carlo variances of the arithmetic mean after rejection of outliers according to several classical and recent formal rules the results are also compared with those for other robust estimators of location parameters it turns out that a simple summary and theoretical explanation of the monte carlo results is provided by the breakdown points of the combined rejection estimation procedures as a by product the concept of br

improved split plot and multistratum designs many industrial experiments involve some factors whose levels are harder to set than others the best way to deal with these is to plan the experiment carefully as a split plot or more generally a multistratum design several different approaches for constructing split plot type response surface designs have been proposed in the literature since 2001 which has allowed experimenters to make better use of their resources by using more efficient designs than the classical balanced ones one of these approaches the stratum by stratum strategy has been shown to produce designs that are less efficient than locally d optimal designs an improved stratum by stratum algorithm is given which though more computationally intensive than the old one makes better use of the advantages of this approach that is it can be used for any structure and does not depend on prior estimates of the variance components this is shown to be almost as good as the locally optimal designs in terms of their own criteria and more

acoustic impulse responses for nonuniform media abstract the acoustic impulse responses of a family of nonuniform lossless one dimensional media are determined the nonuniform portions of the media are confined to layers of either finite or infinite thicknesses the acoustic impedances of the nonuniform layers as functions of travel time are assumed known the impulse responses are determined by evaluating the fourier transforms of their reflection coefficients using infinite series expansions and contour integrations

approximate model spaces for model robust experiment design optimal designs depend upon a prespecified model form a popular and effective model robust alternative is to design with respect to a set of models instead of just one however model spaces associated with experiments of interest are often prohibitively large and so algorithmically generated designs are infeasible here we present a simple method that largely eliminates this problem by choosing a small set of models that approximates the full set and finding designs that are explicitly robust for this small set we build our procedure on a restricted columnwise pairwise algorithm and explore its effectiveness for two model spaces in the literature for smaller full model spaces we find that the designs constructed with the new method compare favorably with robust designs that use the full model space with construction times reduced by orders of magnitude we also construct designs that heretofore have been unobtainable due to the size of their model spaces supplementary material available online

some guidelines for constructing exact d optimal designs on convex design spaces some problems unique to the construction of n point d optimal designs on convex design spaces are considered multiple point augmentation and exchange algorithms are shown to be more costly and less efficient than the analogous single point procedures moreover some recommendations for improving single point methods are given finally empirical evidence is found that supports the galil and kiefer option for constructing initial designs and powell s optimization method for design augmentation

inverse scattering for discrete transmission line models this paper presents several methods for the identification of the impedance and reflection coefficient profile of a nonuniform discrete transmission line from its response to a given forcing function this problem is the prototype for a wealth of one dimensional inverse scattering problems arising in various fields a unified straightforward approach is presented providing all known inversion procedures and some novel ones the derivations readily follow from causality of signal propagation on the transmission line it is shown that a direct exploitation of the signal propagation model leads to recursive computationally efficient and reliable inversion algorithms in particular the relationships between layer peeling schur type difference equation and layer adjoining levinson type integral equation algorithms are clearly displayed

robustness properties of s estimators of multivariate location and shape in high dimension author s rocke david abstract for the problem of robust estimation of multivariate location and shape defilllng s estimators using scale transformations of a fixed p function regardless of the dimension as is usually done leads to a perverse outcome estimators in high dimension can have a breakdown point approaching 50 but still fail to reject as outliers points that are large distances from the main mass of points this leads to a form of nonrobustness that has important practical consequences in this paper estimators are defined that improve on known s estimators in having alll of the following properties 1 maximal breakdown for the given sample size and dimension 2 ability completely to reject as outliers points that are far from the main mass of points 3 convergence to good solutions with a modest amount of computation from a nonrobust starting point for large though not near 50 contamination however to attain maximal breakdown these estimates like other known maximal breakdown estimators require large amounts of computational effort this greater ability of the new estimators to reject outliers comes at a modest cost in efficiency and gross error sensitivity and at a greater but finite cost in local shift sensitivity

optimal design of blocked and split plot experiments for fixed effects and variance component estimation in general modeling data from blocked and split plot response surface experiments requires the use of generalized least squares and the estimation of two variance components the literature on the optimal design of blocked and split plot response surface experiments however focuses entirely on the precise estimation of the fixed factor effects and completely ignores the necessity to estimate the variance components as well to overcome this problem we propose a new bayesian optimal design criterion which focuses on both the variance components and the fixed effects a novel feature of the criterion is that it incorporates prior information about the variance components through log normal or beta prior distributions the resulting designs allow for a more powerful statistical inference than traditional optimal designs in our algorithm for generating optimal blocked and split plot designs we implement efficient quadrature approaches for the numerical approximation of the new optimal design criterion

unmasking multivariate outliers and leverage points abstract detecting outliers in a multivariate point cloud is not trivial especially when there are several outliers the classical identification method does not always find them because it is based on the sample mean and covariance matrix which are themselves affected by the outliers that is how the outliers get masked to avoid the masking effect we propose to compute distances based on very robust estimates of location and covariance these robust distances are better suited to expose the outliers in the case of regression data the classical least squares approach masks outliers in a similar way also here the outliers may be unmasked by using a highly robust regression method finally a new display is proposed in which the robust regression residuals are plotted versus the robust distances this plot classifies the data into regular observations vertical outliers good leverage points and bad leverage points several examples are discussed

response surface designs for quantitative and qualitative variables when some variables are quantitative and some qualitative in a response surface context standard designs may not be suitable the reasons for this are illuminated some alternative designs are discussed every n point design provides n linearly independent estimation contrasts some of these p say are needed to estimate the p parameters of the postulated model the remaining n p linearly independent estimation contrasts are available to estimate pure error if used and to test for lack of fit either overall or in particular ways the key to choosing a good design is to use the available degrees of freedom well given certain assumptions about the model to be fitted when there is also uncertainty about the model assumptions dogmatic design advice is not possible sound guidelines are available however and these are presented and illustrated

projection pursuit approach to robust dispersion matrices and principal components primary theory and monte carlo abstract this article proposes and discusses a type of new robust estimators for covariance correlation matrices and principal components via projection pursuit techniques the most attractive advantage of the new procedures is that they are of both rotational equivariance and high breakdown point besides they are qualitatively robust and consistent at elliptic underlying distributions the monte carlo study shows that the best of the new estimators compare favorably with other robust methods they provide as good a performance as m estimators and somewhat better empirical breakdown properties

discrete inverse method for viscoelastic medium with complete data the discrete inverse scattering problem for viscoelastic medium is studied in this paper it is assumed that the relaxation modulus varies only with time t the object of this paper is to develop a method to reconstruct the relaxation modulus with less measurement data than before the propagation operators of the viscoelastic medium are defined first and the imbedding equations governing the behavior of the propagation operators are derived with the invariant imbedding techniques using the finite difference method these equations can be discretized to obtain a system of linear algebraic equations about the propagation operators and the material modulus for the inverse scattering problem it is assumed that the reflection data obtained from the scattering experiments are only available on one side of the medium and for one round trip through the viscoelastic slab to reconstruct the unknown relaxation modulus an inversion procedure is developed using this set of data that are complete in the sense that they can be extended to arbitrary time t and the other scattering and propagation operators can also be determined by the inversion procedure described in this paper the inversion algorithm is implemented numerically on several examples at the end of the paper it can be seen that the obtained curves of the material modulus coincide with the original relaxation modulus very well

constrained m estimation for multivariate location and scatter consider the problem of estimating the location vector and scatter matrix from a set of multivariate data two standard classes of robust estimates are m estimates and s estimates the m estimates can be tuned to give good local robustness properties such as good efficiency and a good bound on the influence function at an underlying distribution such as the multivariate normal however m estimates suffer from poor breakdown properties in high dimensions on the other hand s estimates can be tuned to have good breakdown properties but when tuned in this way they tend to suffer from poor local robustness properties in this paper a hybrid estimate called a constrained m estimate is proposed which combines both good local and good global robustness properties

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

optimal rank based testing for principal components this paper provides parametric and rank based optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families the parametric tests extend the gaussian likelihood ratio tests of anderson 1963 and their pseudo gaussian robustifications by davis 1977 a nd tyler 1981 1983 the rank based tests address a much broader class of problems where covariance matrices need not exist and principal components are associated with more general scatter matrices the proposed tests are shown to outperform daily practice both from the point of view of validity as from the point of view of efficiency this is achieved by utilizing the le cam theory of locally asymptotically normal experiments in the nonstandard context however of a curved parametrization the results we derive for curved experiments are of independent interest and likely to apply in other contexts 1 introduction this fairly detailed introduction aims at providing a comprehensive and nontechnical overview of the paper including its asymptotic theory aspects and a rough description of some of the rank based test statistics to be derived it is expected to be accessible to a broad readership it should be sufficiently informative for the reader not interested in the technical aspects of asymptotic theory to proceed to sections 5 gaussian and pseudo gaussian tests and 6 rank based tests where the proposed testing procedures are described and for the reader mainly interested in asymptotics to decide whether he she is interested in the treatment of a lan family with curved parametrization developed in sections 3 and 4

inverse scattering for inhomogeneous viscoelastic media in this paper the inverse scattering problems for the full inhomogeneous viscoelastic medium are studied via the invariant imbedding technique special attention is paid to the propagation operators of the viscoelastic medium and the imbedding equations for these operators are derived for the inverse scattering problems it is shown that the reflection data can be extended from one round trip through the iscoelastic slab to arbitrary time with the help of the propagation operators hence the reconstruction of the relaxation modulus is sufficient to be considered only in one round trip it is also shown that only one side measurement reflection data are not sufficient to reconstruct the relaxation modulus and the density of the medium simultaneously the corresponding numerical examples are presented for the case that the relaxation modulus of the medium is modeled by two independent functions an iterative inversion procedure is proposed to recover the relaxation modulus and the density simultaneously with the input two side normally reflection data

principal components analysis based on multivariate mm estimators with fast and robust bootstrap we consider robust principal components analysis pca based on multivariate mm estimators we first study the robustness and efficiency of these estimators particularly in terms of eigenvalues and eigenvectors we then focus on inference procedures based on a fast and robust bootstrap for mm estimators this method is an alternative to the approach based on the asymptotic distribution of the estimators and can also be used to assess the stability of the principal components a formal consistency proof for the bootstrap method is given and its finite sample performance is investigated through simulations we illustrate the use of the robust pca and the bootstrap inference on a real dataset

transient waves in nonstationary media this paper treats propagation of transient waves in nonstationary media which has many applications in for example electromagnetics and acoustics the underlying hyperbolic equation is a general homogeneous linear first order 2 2 system of equations the coefficients in this system depend on one spatial coordinate and time furthermore memory effects are modeled by integral kernels which in addition to the spatial dependence are functions of two different time coordinates these integrals generalize the convolution integrals frequently used as a model for memory effects in the medium specifically the scattering problem for this system of equations is addressed this problem is solved by a generalization of the wave splitting concept originally developed for wave propagation in media which are invariant under time translations and by an imbedding or a green s functions technique more explicitly the imbedding equation for the reflection kernel and the green s functions propagator kernels equations are derived special attention is paid to the problem of nonstationary characteristics a few numerical examples illustrate this problem

finding an unknown number of multivariate outliers we use the forward search to provide robust mahalanobis distances to detect the presence of outliers in a sample of multivariate normal data theoretical results on order statistics and on estimation in truncated samples provide the distribution of our test statistic we also introduce several new robust distances with associated distributional results comparisons of our procedure with tests using other robust mahalanobis distances show the good size and high power of our procedure we also provide a unification of results on correction factors for estimation from truncated samples

robust estimation of dispersion matrices and principal components abstract this paper uses monte carlo methods to compare the performances of several robust procedures for estimating a correlation matrix and its principal components the estimators are formed either from separate bivariate analyses or by simultaneous manipulation of all variables by using techniques such as multivariate trimming and m estimation the m estimators stand up exceptionally well they and the multivariate trimming procedure are especially effective at estimating the principal components including a near singularity however the m estimators can break down relatively easily when the dimensionality is large and the outliers are asymmetric with missing data the element wise approach becomes more attractive

robust estimation of multivariate location and scatter in the presence of missing data two main issues regarding data quality are data contamination outliers and data completion missing data these two problems have attracted much attention and research but surprisingly they are seldom considered together popular robust methods such as s estimators of multivariate location and scatter offer protection against outliers but cannot deal with missing data except for the obviously inefficient approach of deleting all incomplete cases we generalize the definition of s estimators of multivariate location and scatter to simultaneously deal with missing data and outliers we show that the proposed estimators are strongly consistent under elliptical models when data are missing completely at random we derive an algorithm similar to the expectation maximization algorithm for computing the proposed estimators this algorithm is initialized by an extension for missing data of the minimum volume ellipsoid we assess the performance of our proposal by monte carlo simulation and give some real dat

s estimators for functional principal component analysis principal component analysis is a widely used technique that provides an optimal lower dimensional approximation to multivariate or functional datasets these approximations can be very useful in identifying potential outliers among high dimensional or functional observations in this article we propose a new class of estimators for principal components based on robust scale estimators for a fixed dimension q we robustly estimate the q dimensional linear space that provides the best prediction for the data in the sense of minimizing the sum of robust scale estimators of the coordinates of the residuals we also study an extension to the infinite dimensional case our method is consistent for elliptical random vectors and is fisher consistent for elliptically distributed random elements on arbitrary hilbert spaces numerical experiments show that our proposal is highly competitive when compared with other methods we illustrate our approach on a real dataset where the robust estimator discovers atypi

mixed response surface models with heterogeneous within block error variances the experimental error variance in a response surface model with a block effect has traditionally been assumed to be constant in some experimental situations however this variance may be different for the different blocks that compose the associated design this article discusses the analysis of a response surface model with a random block effect and heterogeneous error variances among the blocks the model includes interactions between the fixed polynomial effects and the block effect procedures for estimating and testing the fixed effects and the random effects in the model are described in particular anova based procedures for testing equality of the error variances and testing the random effects are introduced

blocking orthogonal designs with mixed integer linear programming we present a mixed integer linear programming approach to orthogonally block two level multilevel and mixed level orthogonal designs the approach involves an exact optimization technique which guarantees an optimal solution it can be applied to many problems where combinatorial methods for blocking orthogonal designs cannot be used by means of 54 run and 64 run examples we demonstrate that our approach outperforms two benchmark techniques in terms of the number of estimable two factor interaction contrasts and in terms of the d efficiency for models with main effects and some two factor interaction contrasts we demonstrate the generic nature of our approach by applying it to the most challenging instances in a catalog of all orthogonal designs of strength 3 with up to 81 runs as well as a small catalog of strength 4 designs the approach can also be applied to strength 2 designs but for these cases alternative methods described in the literature may perform equally well supplementary materials for this article are available online 2015 american statistical association and the american society for quality

direct and inverse scattering in the time domain via invariant imbedding equations this paper examines the implementation of a new approach to direct and inverse scattering problems in the time domain which is applied here to the case of a one dimensional lossless medium the method is based on an integro differential equation which is satisfied by the reflection kernel and has been derived elsewhere using invariant imbedding techniques an example of an exact solution to this equation is given numerical schemes for solving the direct and inverse problems are derived by discretizing this equation the behavior of the resulting algorithms is then tested on several examples the results suggest that the inversion method is stable in the presence of additive noise

reconstruction of the velocity and density in a stratified acoustic half space using a short pulse point source the reconstruction of the velocity and density in a stratified half space using the reflection data generated by a short pulse point source is considered transverse spatial hankel transform is applied to the transient reflected and incident pressures on the surface the reflection kernel in the hankel transformed space is obtained through a deconvolution the time domain green function approach based on the wave splitting is then used to reconstruct the parameters this time domain approach to the inverse problem is exact and noniterative numerical results for some synthetical examples are presented to illustrate the hankel transform deconvolution and the reconstruction the finite difference time domain method is used to generate the transient reflection data in the physical space in a numerical simulation the reconstruction using obliquely incident plane waves is also considered in an appendix

new tools for residual analysis techniques are presented for stepwise selection of observations in order of discrepancy from a linear model and for assigning a p value to each selected observation a general class of criteria for discrepancy is defined and leading members of the class are discussed two ways to define the sequence of p values for a general selection criterion are discussed and their relative advantages are compared the techniques are illustrated on several data sets using only the computationally simpler of the two approaches to p values