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

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

stability comparisons of estimators this paper investigates a property of estimators called stability the stability exponent of an estimator is a measure of the magnitude of the effect of any single observation in the sample on the realized value of the estimator a number of reasons related to robustness suggest that often it is desirable for an estimator to be relatively insensitive to any particular observation in the sample i e to have high stability in addition it is useful for diagnostic purposes to have knowledge of the stability exponents of different estimators in order to know which estimators are likely to rely more heavily on some single observation the paper is organized as follows section 1 introduces the basic idea contained in the paper motivates it and summarizes the results in an informal manner section 2 presents definitions assumptions and the general results for purposes of illustration the linear regression model with the least squares estimator is used as a running example throughout this section section 3 discusses numerous additional applications of the general results an appendix contains proofs of

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

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

robust autoregressive estimates using quadratic programming abstract the robust estimation of the autoregressive parameters is formulated in terms of the quadratic programming problem this article s main contribution is to present an estimator that down weights both types of outliers in time series and improves the forecasting results new robust estimates are yielded by combining optimally two weight functions suitable for innovation and additive outliers in time series the technique which is developed here is based on an approach of mathematical programming applications to ip approximation the behavior of the estimators are illustrated numerically under the additive outlier generating model monte carlo results show that the proposed estimators compared favorably with respect to m estimators and bounded influence estimators based on these results we conclude that one can improve the robust properties of ar p estimators using quadratic programming

high breakdown point robust regression with censored data in this paper we propose a class of high breakdown point estimators for the linear regression model when the response variable contains censored observations these estimators are robust against high leverage outliers and they generalize the lms least median of squares s mm and estimators for linear regression an important contribution of this paper is that we can define consistent estimators using a bounded loss function or equivalently a redescending score function since the calculation of these estimators can be computationally costly we propose an efficient algorithm to compute them we illustrate their use on an example and present simulation studies that show that these estimators also have good finite sample properties

some new estimation methods for weighted regression where there are possible outliers the problem considered is the robust estimation of the variance parameter in a heteroscedastic linear model we treat the situation in which the variance is a function of the explanatory variables to estimate robustly the variance in this case it is necessary to guard against the influence of outliers in the design as well as outliers in the response by analogy with the homoscedastic regression case we propose two estimators that do this their performances are evaluated on a number of data sets we had considerable success with estimators that bound the self influence that is the influence an observation has on its own fitted value we conjecture that in other situations e g homoscedastic regression bounding the selfinfluence will lead to estimators with good robustness properties

robust regression with asymmetric heavy tail noise distributions in the presence of a heavy tail noise distribution regression becomes much more difficult traditional robust regression methods assume that the noise distribution is symmetric and they downweight the influence of so called outliers when the noise distribution is asymmetric these methods yield biased regression estimators motivated by data mining problems for the insurance industry we propose a new approach to robust regression tailored to deal with asymmetric noise distribution the main idea is to learn most of the parameters of the model using conditional quantile estimators which are biased but robust estimators of the regression and to learn a few remaining parameters to combine and correct these estimators to minimize the average squared error in an unbiased way theoretical analysis and experiments show the clear advantages of the approach results are on artificial data as well as insurance data using both linear and neural network predictors

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

the standardized influence matrix and its applications abstract in this article we introduce the standardized influence matrix sim of parameter estimators as a conjugate of the sample covariance matrix of standardized influence functions sifs evaluated at the data points we propose principal component analysis for the sim and its complement sim analysis as a diagnostic tool for regression or other statistical inference and provide theoretical insight to the local influence defined by cook we show that sim analysis reveals the multivariate structure of outlying and or influential points specifically sim analysis uncovers hidden structures of influence such as clustering that cannot be identified by the lengths of the standardized influences finally examples in linear regression show that a diagnostic method using sim is more effective if robust parameter estimates are used in calculating the sample sim

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

infinitesimal robustness for autoregressive processes we define the influence function and construct optimal robust estimators for autoregressive processes we show that the asymptotic bias caused by small contaminations of the marginals can be written as the integral of a certain function with respect to the contamination this function is called the influence function it is unique only up to an equivalence relation but there is a natural unique version which describes the limiting influence of an additional observation given the previous observations moreover with this version the asymptotic variance at the true model can be expressed in a simple form optimal robust estimators minimize this asymptotic variance under a constraint on the influence function as in the i i d case they are found by truncating a multiple of the influence function of the maximum likelihood estimator

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

instrumental variable estimation based on conditional median restriction abstract we develop a method named the l 1 iv estimator to estimate structural equations based on the conditional median restriction imposed on the error terms we study its asymptotic behavior and show how to estimate its asymptotic covariance matrix we also discuss the point identification in the l 1 iv estimation and propose an over identifying restriction test we further demonstrate the performance of the l 1 iv estimator in comparison with the familiar two stage least squares estimator the proposed method is applied to estimate the labor supply and wage offer functions for working married women

mallows type bounded influence regression trimmed means abstract the influence functions of the regression trimmed mean estimators proposed by koenker and bassett 1978 and welsh 1987 are bounded in the dependent variable space but not in the independent variable space this article follows the approach of mallows 1973 1975 and modifies these estimators so that the resulting estimators have bounded influence functions the large sample behavior of these estimators is studied and it is shown that they have the same asymptotic distribution the small sample behaviors of the ordinary regression and bounded influence regression trimmed means are then investigated by means of a monte carlo study and by applying the estimators to water salinity data see ruppert and carroll 1980 based on these results we conclude that one can potentially gain much by using bounded influence regression trimmed means over ordinary regression trimmed means however there does not seem to be a clear choice between the koenker bassett and welsh versions

qualitative robustness for stochastic processes in this paper we generalize hampel s concept of qualitative robustness of a sequence of estimators to the case of stochastic processes with non i i d observations defining appropriate metrics between samples we also present a different approach to qualitative robustness which formalizes the notion of resistance we give two definitions based on this approach strong and weak resistance we show that for estimating a finite dimensional real parameter pi robustness is equivalent to weak resistance and in the i i d case is also equivalent to strong resistance finally we prove the strong resistance of a class of estimators which includes common gm estimates for linear models and autoregressive processes

bounded influence regression via local minimax mean squared error abstract the class of general m estimators for the stochastic linear regression model is considered and influence and change of variance functions are used to approximate the main term of the mean squared error mse the estimator is then found in the class that minimizes the maximum of that approximate mse over the contamination neighborhood this approach allows for additional flexibility of designing estimators optimal for prediction in specific regions of the factor space and it provides a practical way of choosing estimators tuning constants as functions of a fraction of contamination and sample size

efficient bounded influence regression estimation abstract the least squares estimator for in the classical linear regression model is strongly efficient under certain conditions however in the presence of heavy tailed errors and or anomalous data the least squares efficiency can be markedly reduced in this article we propose an estimator that limits the influence of any small subset of the data and show that it satisfies a first order condition for strong efficiency subject to the constraint we then show that the estimator is asymptotically normal the article concludes with an outline of an algorithm for computing a bounded influence regression estimator and with an example comparing least squares robust regression as developed by huber and the estimator proposed in this article

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

estimation following a sequentially designed experiment abstract we investigate the design of experiments for the nonparametric estimation of the root of an unknown regression function approaches to this problem include the robbins monro 1951 venter 1967 and lai robbins 1979 1981 stochastic approximation procedures and wu s 1985 1986 sequential maximum likelihood estimators because the regression function is not assumed to belong to a parametric family only experimentation near to the root is informative and the sequential design should converge to the root after the sequential design has been generated there are at least two distinct methods for estimation of the root a estimate the root by the last design point or b fit a parametric model except for ruppert 1988 and simulation studies by bodt 1985 and bodt and tingey 1990 all studies known to us of stochastic approximation procedures used the last design point as the estimator wu 1985 1986 fit a generalized linear model when the last design point is the estimator then cle

the breakdown point of simultaneous general m estimates of regression and scale abstract this article deals with the breakdown point of simultaneous general m estimates of the regression parameter vector and scale parameter of a linear model a general expression for is derived and its numerical values are computed for the case in which the carriers have a normal distribution the numerical results show that is in general much lower than the minimum of the breakdown points of with known and of with known the values of for some real and simulated data sets are also given

outlier induced collinearities when an observation in a regression analysis has very large values on two or more predictor variables artificial collinearities can be induced the effects of such collinearities on a regression analysis are not well documented although they can be shown to be similar in many respects to those resulting from approximate linear dependencies among the columns of predictor variable values the purpose of this article is to explore the effects of outlier induced collinearities on the estimation of regression coefficients

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

on non linear serial dependencies in stock returns abstract this paper considers the h p tests of linearity and gaussianity of daily stock returns and their implications for non linear models of those returns

infinitesimal robustness for diffusions we develop infinitesimally robust statistical procedures for the general diffusion processes we first prove the existence and uniqueness of the times series influence function of conditionally unbiased m estimators for ergodic and stationary diffusions under weak conditions on the martingale estimating function used we then characterize the robustness of m estimators for diffusions and derive a class of conditionally unbiased optimal robust estimators to compute these estimators we propose a general algorithm which exploits approximation methods for diffusions in the computation of the robust estimating function monte carlo simulation shows a good performance of our robust estimators and an application to the robust estimation of the exchange rate dynamics within a target zone illustrates the methodology in a real data application

robust detection of straight and circular road segments in noisy aerial images this paper treats the problem of detecting straight or circular pieces of road in noisy low resolution aerial images it first uses a local operator to detect pixels whose neighborhoods are line like and then applies robust estimation techniques to find sets of such pixels that lie on or near straight or circular loci an unbiased ordinary least squares estimator cannot handle outlying data even for straight loci on the other hand conventional robust techniques for fitting circular arcs are severely affected by digitization effects and the fact that circular road segments are typically short and shallow we therefore introduce an estimator that is both robust and statistically efficient we also present a simple ad hoc technique that achieves comparable results and can handle road segments that are either straight or circular

influence functions for proportional hazards regression summary influence functions for the regression parameters in the proportional hazards model are presented it is suggested that empirical influence functions computed for each observation and each covariate can be useful in an informal way to identify influential observations this is illustrated on the stanford heart transplant data and two other examples

semiparametric robust estimation of truncated and censored regression models many estimation methods of truncated and censored regression models such as the maximum likelihood and symmetrically censored least squares scls are sensitive to outliers and data contamination as we document therefore we propose a semiparametric general trimmed estimator gte of truncated and censored regression which is highly robust but relatively imprecise to improve its performance we also propose data adaptive and one step trimmed estimators we derive the robust and asymptotic properties of all proposed estimators and show that the one step estimators e g one step scls are as robust as gte and are asymptotically equivalent to the original estimator e g scls the finite sample properties of existing and proposed estimators are studied by means of monte carlo simulations

robust regression based on infinitesimal neighbourhoods we study robust estimation in the general normal regression model with random carriers permitting small departures from the model the framework is that of bickel 1981 we obtain solutions of huber 1982 kraskerhampel 1980 and krasker welsch 1982 as special cases as well as some new procedures our calculations indicate that the optimality properties of these estimates are more limited than suggested by krasker and welsch

minimax aspects of bounded influence regression abstract bounded influence regression in the sense of hampel mallows krasker welsch is examined critically from the point of view of finite sample minimax robust estimation theory the main conclusion is that the influence of position of very high leverage points that is of points where the diagonal element of the hat matrix h x xtx 1 xt exceeds a certain bound should be cut down selectively this bound should be chosen roughly between 2 and 5 for the upper of these values the resulting estimate can be approximated very simply by scaling the residuals by their own estimated standard deviation a conflict between decision theoretic estimation and data analytic diagnostic viewpoints is pointed out and briefly discussed

resistant estimation for simultaneous equations models using weighted instrumental variables in this paper we present a weighted instrumental variables estimator that is resistant2 to heavy tailed errors aberrant data in either the endogenous or exogenous variables and certain other model failures the estimator is analogous to the weighted least squares approach to robustness proposed by krasker and welsch 21 for ordinary regression we will discuss the theory that motivated this estimator derive some of its properties describe our computational algorithm and using an empirical example illustrate the estimator s utility as a tool for data analysis in structural models the evolution of robust estimators for simultaneous equations models has closely resembled the development of robust methods for ordinary regression in both cases research focused at first on the well documented effects of long tailed error distributions on the classical procedures for ordinary regression models statisticians have studied the properties of least absolute deviations lad estimators see bassett and koenker 2 and amemiya 1 and maximum likelihood type estimators called m estimators see huber 14 for simultaneous equations models lad can be generalized in several different ways to modify two stage least squares or instrumental variables amemiya 1 has presented these estimators in a unified framework and powell 31 has proven their asymptotic normality under weak assumptions fair 8 has compared two stage lad estimates of a u s macroeconomic model with two stage least squares and full information maximum likelihood estimates the m estimator concept can also be generalized to simultaneous equations for example prucha and kelejian 32 have considered maximum likelihood estimation under the assumption that the disturbance vector is multivariate student t although lad and m estimators maintain a high efficiency when the error distribution is heavy tailed they are not robust in the stronger sense of hampel 10 which roughly speaking requires an estimator to have a limited sensitivity to any small fraction of the data 3 lad and m estimators fail under this criterion because a single observation whose values for the right side variables are highly anomalous can have an arbitrarily large effect on lad estimates or m estimates just as on the classical procedures consequently these estimators do not provide insurance against the sort of gross errors that occur in some data sets nor do they serve as reliable diagnostics for departures from linearity occurring in extreme regions of the space of right side variables for linear models the estimators that satisfy the strong hampel robustness criterion have become known as bounded influence estimators several such estimators have been

bounded influence estimators for the sure model abstract this paper considers robust estimation of the seemingly unrelated regression equations sure model following the infinitesimal approach to robustness we characterize the class of optimal bounded influence estimators and we propose two computationally simple estimators we also present the results of a set of monte carlo experiments carried out to study the behavior of these estimators

a theory of passive linear systems with no assumptions we present two linked theorems on passivity the passive behavior theorem parts 1 and 2 part 1 provides necessary and sufficient conditions for a general linear system described by a set of high order differential equations to be passive part 2 extends the positive real lemma to include uncontrollable and unobservable state space systems

robust ridge regression for high dimensional data ridge regression being based on the minimization of a quadratic loss function is sensitive to outliers current proposals for robust ridge regression estimators are sensitive to bad leverage observations cannot be employed when the number of predictors p is larger than the number of observations n and have a low robustness when the ratio p n is large in this article a ridge regression estimate based on repeated m estimation mm estimation is proposed it is a penalized regression mm estimator in which the quadratic loss is replaced by an average of where ri are the residuals and the residual scale from an initial estimator which is a penalized s estimator and is a bounded function the mm estimator can be computed for p n and is robust for large p n a fast algorithm is proposed the advantages of the proposed approach over its competitors are demonstrated through both simulated and real data supplemental materials are available online

procedures for the identification of multiple outliers in linear models abstract we consider the problem of identifying and testing multiple outliers in linear models the available outlier identification methods often do not succeed in detecting multiple outliers because they are affected by the observations they are supposed to identify we introduce two test procedures for the detection of multiple outliers that appear to be less sensitive to this problem both procedures attempt to separate the data into a set of clean data points and a set of points that contain the potential outliers the potential outliers are then tested to see how extreme they are relative to the clean subset using an appropriately scaled version of the prediction error the procedures are illustrated and compared to various existing methods using several data sets known to contain multiple outliers also the performances of both procedures are investigated by a monte carlo study the data sets and the monte carlo indicate that both procedures are effective in the detection of multiple outliers

robust nonparametric methods the terms nonparametric statistics and distributionfree methods have historically referred to a collection of statistical tests whose null distributions do not depend on the underlying distribution of the data the area has been expanded to include estimates and confidence intervals derived from the tests and includes asymptotically distribution free tests for complex models as well it is now generally recognized that these statistical methods are highly efficient over large sets of possible models and are robust as well this was not always the case the earliest work was by hotelling and pabst in 1936 references to works in this paragraph can be found in the books cited in the next paragraph on rank correlation and by friedman in 1937 on rank tests in a two way design followed in 1945 with the introduction of the signed rank and rank sum tests by wilcoxon in 1947 mann and whitney extended the ideas of wilcoxon all of these rank tests were considered quick and dirty but not competitive with the presumably more efficient t tests this perspective was about to change in 1948 pitman developed efficiency concepts in a set of lecture notes that although never published were widely disseminated during the 1950s and 1960s lehmann working with hodges and with his students at berkeley showed that rank tests are surprisingly efficient and robust in fact the wilcoxon rank tests are essentially efficient at the normal model and can be much more efficient than least squares methods when the underlying distribution of the data has heavy tails in the early 1960s hodges and lehmann derived estimates and confidence intervals from rank test statistics and also introduced aligned rank tests for use in the regression model in 1967 hajek and sid published their seminal book on the theory of rank tests it is impossible to mention all of the contributions to this early development hence we cite several key mono

change of variance sensitivities in regression analysis the change of variance function is defined for estimators of regression coefficients both an unstandardized and a standardized form of the change of variance sensitivity are introduced and their relation with the corresponding gross error sensitivities is investigated the problems of optimal robustness lead to the hampel krasker and the krasker welsch estimators at the same time also the scale parameter has to be estimated robustly by means of the change of variance sensitivity optimal robust redescending scale estimators are constructed

optimally bounded score functions for generalized linear models with applications to logistic regression summary we study optimally bounded score functions for estimating regression parameters in a generalized linear model our work extends results obtained by krasker welsch 1982 for the linear model and provides a simple proof of krasker welsch s first order condition for strong optimality the application of these results to logistic regression is studied in some detail with an example given comparing the bounded influence estimator with maximum likelihood where h q and c are known functions and 0 is a vector of regression parameters models of this type include logistic and probit regression poisson regression linear regression with known variance and certain models for lifetime data our motivation for seeking robust estimators is the same as that in the linear model maximum likelihood estimation is sometimes sensitive to outlying data for logistic regression pregibon 1981 1982 has documented the nonrobustness of the maximum likelihood estimator and expounded the benefits of diagnostics as well as robust or resistant fitting procedures see also johnson 1985 much of the work on robust estimation concerns finding estimators which sacrifice little efficiency at the assumed model while providing protection against outliers and model violations we follow this course finding bounded influence estimators minimizing certain functionals of the asymptotic covariance matrix related work includes that of hampel 1978 krasker 1980 and krasker welsch 1982 when fitting models to data two important issues are identification of outliers and influential cases and accommodation of these observations frequently when influential cases are present the fitted model is not representative of the bulk of the data to rectify this one can simply delete influential cases and refit via standard methods but this

diagnostics and robust estimation when tranforming the regression model and the response in regression analysis the response is often transformed to remove heteroscedasticity and or skewness when a model already exists for the untransformed response then it can be preserved by applying the same transform to both the model and the response this methodology which we call transform both sides has been applied in several recent papers and appears highly useful in practice when a parametric transformation family such as the power transformations is used then the transformation can be estimated by maximum likelihood the maximum likelihood estimator however is very sensitive to outliers in this article we propose diagnostics to indicate cases influential for the transformation or regression parameters we also propose a robust bounded influence estimator similar to the krasker welsch regression estimator

on the bounded influence regression estimator of krasker and welsch abstract recently krasker and welsch 1982 considered a class of bounded influence regression estimators they showed that within this class the so called krasker welsch estimator is the only solution to a first order necessary condition for strong optimality that is for minimizing in the sense of positive definiteness the asymptotic covariance matrix however whether any strongly optimal estimator in fact exists remained an open question in this article an example is given where no strongly optimal estimator exists moreover the practical significance of the lack of a strongly optimal estimator is discussed

positive real lemma necessary and sufficient conditions for the existence of solutions under virtually no assumptions in this note the celebrated positive real lemma equations are considered with the purpose of obtaining necessary and sufficient conditions for the existence of solutions under the mildest system theoretic assumptions more precisely necessary and sufficient conditions are established under the very weak assumption of sign controllability it is then shown that the order of the equations may be suitably reduced by restricting attention to a subspace related to a certain observable subsystem this reduction beside being interesting per se permits us to weaken the assumption of sign controllability finally the assumptions are further weakened as to include the case when the state matrix has uncontrollable eigenvalues on the imaginary axis

moduli of regular holonomic d modules with normal crossing singularities this paper solves the global moduli problem for regular holonomic dmodules with normal crossing singularities on a nonsingular complex projective variety this is done by introducing a level structure which gives rise to pre d modules and then introducing a notion of semi stability and applying geometric invariant theory to construct a coarse moduli scheme for semistable pre d modules a moduli is constructed also for the corresponding perverse sheaves and the riemann hilbert correspondence is represented by an analytic morphism between these moduli spaces

moduli of semistable logarithmic connections under the riemann hilbert correspondence which is an equivalence of categories local systems on a nonsingular complex projective variety x correspond to pairs e s v where is a locally free sheaf on x and v 8 q1 is an integrable connection on f generalizing this correspondence to noncomplete quasi projective varieties deligne see d showed that the correct algebro geometric objects which correspond to local systems on such a variety y are the so called regular connections on y the condition of regularity on algebraic connections on y can be expressed as follows fix a hironaka completion x of y that is nonsingular projective variety x which contains y as an open subvariety such that the complement s x y is a smooth divisor with normal crossings a logarithmic connection f y v on x with singularities over s is by definition a torsion free coherent sheaf y on x together with a map v 9 21 log s 9 which satisfies c linearity and the leibniz rule where i14 logs is the sheaf of rational 1 forms on x with logarithmic poles on s the curvature of the connection v is assumed to be zero finally an algebraic connection e f v on y is regular if and only if there exists a logarithmic connection f on x with singularities over s which extends e if such an extension f exists the underlying sheaf f can be chosen to be locally free and will be called a logarithmic lattice for e a logarithmic lattice for a given regular connection is of course not unique but there is a canonical way of choosing it called deligne s construction which only depends on the choice of a set theoretic section of the map exp c c as such a section is not even continuous the deligne lattice does not behave well in families of regular connections simpson has given in s a construction of a moduli scheme for nonsingular connections on a projective variety x however a simple example shows that it would be unreasonable to expect in general the existence of a moduli scheme for regular connections on a quasi projective variety y for this let y be the affine line minus the origin with coordinate x the differential equation dy dx tylx 0 with parameter t defines a family of regular connections on the sheaf y parametrized by the affine line t with coordinate t we can see that t1 and t2 parametrize isomorphic connections if and only if t1 t2

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

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

a class of locally and globally robust regression estimates abstract we present a new class of regression estimates called generalized estimates these estimates are defined by minimizing the scale of the weighted residuals with weights that penalize high leverage observations like the estimates the generalized estimates utilize for their definition two loss functions 1 and 2 which together with the weights can be chosen to achieve simultaneously high breakdown point finite gross error sensitivity and high efficiency we recommend however choosing these functions so as to control the bias behavior of the estimate for a large range of possible contaminations and then boosting the efficiency by a simple least squares reweighting step the generalized estimate with loss functions 1 and 2 is related to the hill ryan gm estimate with a loss function which is a linear combination of 1 and r in fact both estimates have the same influence function and asymptotic distribution under the central model we show that a certain generalized estima

m estimation for discrete data asymptotic distribution theory and implications abstract the asymptotic distribution of an m estimator is studies when the underlying distribution is discrete asymptotic normality is shown to hold quite generally within the assumed parametric family when the specification of the model is inexact however it is demonstrated that an m estimator with a non smooth score function e g a huber estimator has a non normal limiting distribution at certain distributions resulting in unstable inference in the neighborhood of such distributions consequently smooth score functions are proposed for discrete data keywords robust estimation and discrete parametric models author

transformations and influential observations in minimum sum of absolute errors regression the box cox power transformation model is considered for minimum sum of absolute errors msae regression a long tailed error distribution specifically the laplace distribution is included and could accommodate observations that in the normal case are outlying or unduly influencing a choice of transformation the log likelihood procedure of box and cox 1964 for obtaining the optimal transformation parameter is adapted for laplace errors and msae regression graphical methods for detecting the influence of individual observations on the choice of transformation are described application to examples illustrates that this approach can provide valuable additional information to the data analyst

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 structure of simultaneous equation estimators a generalization towards nonnormal disturbances a general linear simultaneous equation system with a multivariate student t disturbance vector is considered the normal equations of the corresponding maximum likelihood estimator are used as estimator generating equations to introduce a new class of estimators properties of large subclasses of these estimators are determined for disturbance vectors other than the multivariate student t

on b robust instrumental variable estimation of the linear model with panel data the aim of this paper is to demonstrate how to acquire robust consistent estimates of the linear model when the fundamental orthogonality condition is not fulfilled with this end in view we develop two estimation procedures two stage generalized m 2sgm and robust generalized method of moments rgmm both estimators are b robust i e their associated influence function is bounded consistent and asymptotic normally distributed our simulation results indicate that the relatively efficient rgmm estimator in regressions with heteroskedastic and or autocorrelated errors provides accurate parameter estimates of a panel data model with all variables subject to measurement errors even if a substantial portion of the data is contaminated with aberrant observations the traditional estimation techniques such as 2sls and gmm break down when outliers corrupt the data

location of several outliers in multiple regression data using elemental sets the outlying tendency of any case in a multiple regression of p predictors may be estimated by drawing all subsets of size p from the remaining cases and fitting the model each such subset yields an elemental residual for the case in question and a suitable summary statistic of them can be used as an estimate of the case s outlying tendency we propose two such summary statistics an unweighted median which is of bounded influence and a weighted median which is more efficient but less robust the computational load of the procedure is reduced by using random samples in place of the full set of subsets of size p as a byproduct the method yields useful information on the influence or leverage of cases and the mutual masking of high leverage points

conditionally unbiased bounded influence estimation in general regression models with applications to generalized linear models abstract in this article robust estimation in generalized linear models for the dependence of a response y on an explanatory variable x is studied a subclass of the class of m estimators is defined by imposing the restriction that the score function must be conditionally unbiased given x within this class of conditionally fisher consistent estimators optimal bounded influence estimators of regression parameters are identified and their asymptotic properties are studied the estimators studied in this article and the efficient bounded influence estimators studied by stefanski carroll and ruppert 1986 depend on an auxiliary centering constant and nuisance matrix the centering constant can be given explicitly for the conditionally fisher consistent estimators and thus they are easier to compute than the estimators studied by stefanski et al 1986 in addition estimation of the nuisance matrix has no effect on the asymptotic distribution of the conditionally fisher consistent estimators the sam

two stage least absolute deviations estimators1 in this paper the method of least absolute deviations is applied to the estimation of the parameters of a structural equation in the simultaneous equations model a class of estimators called two stage least absolute deviations estimators is defined their asymptotic properties are derived and the problem of finding the optimal member of the class is considered in this paper we apply the method of least absolute deviations to the estimation of the parameters of a structural equation in the simultaneous equations model we define a class of estimators called two stage least absolute deviations estimators 2slad and derive their asymptotic properties they are so named as their relationship to the two stage least squares estimator 2sls is analogous to the relationship of the least absolute deviations estimator lad to the least squares estimator ls in the standard regression model the lad estimation has been extensively studied in the context of the standard regression model and its usefulness is universally recognized in this paper we show that the advantage of 2slad over 2sls in the simultaneous equations model can be as great as that of lad over ls in the standard regression model if 2slad is properly defined this last clause is very important since the results of this paper indicate that the lad analogue of 2sls that has been considered before in the literature is not an appropriate method

robust regression using repeated medians abstract the repeated median algorithm is a robustified u statistic in which nested medians replace the single mean unlike many generalizations of the univariate median repeated median estimates maintain the high 50 breakdown value and can resist the effects of outliers even when they comprise nearly half of the data because they are calculated directly not iteratively repeated median procedures can be used as starting values for iterative robust estimation methods for bivariate linear regression with symmetric errors repeated median estimates are unbiased and fisher consistent and their efficiency under gaussian sampling can be comparable to the efficiency of the univariate median author

a natural robustification of the ordinary instrumental variables estimator summary instrumental variables estimators are designed to provide consistent parameter estimates for linear regression models when some covariates are correlated with the error term we propose a new robust instrumental variables estimator riv which is a natural robustification of the ordinary instrumental variables estimator oiv specifically we construct riv using a robust multivariate location and scatter s estimator to robustify the solution of the estimating equations that define oiv riv is computationally inexpensive and readily available for applications through the r library riv it has attractive robustness and asymptotic properties including high resilience to outliers bounded influence function consistency under weak distributional assumptions asymptotic normality under mild regularity conditions and equivariance we further endow riv with an iterative algorithm which allows for the estimation of models with endogenous continuous covariates and exogenous dummy covariates we study the performance of riv when the data contains outliers using an extensive monte carlo simulation study and by applying it to a limited access dataset from the framingham heart study cohort to estimate the effect of long term systolic blood pressure on left atrial size

robust estimation with many instruments abstract linear instrumental variables models are widely used in empirical work but often associated with low estimator precision this paper proposes an estimator that is robust to outliers and shows that the estimator is minimax optimal in a class of estimators that includes the limited maximum likelihood estimator liml intuitively this optimal robust estimator combines liml with winsorization of the structural residuals and the winsorization leads to improved precision under thick tailed error distributions consistency and asymptotic normality of the estimator are established under many instruments asymptotics and a consistent variance estimator which allows for asymptotically valid inference is provided

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

robust spectral regression on considere le probleme de l estimation de la regression lineaire lorsque les perturbations suivent un processus stationnaire avec sa densite spectrale connue pour etre seulement dans un voisinage d une densite spectrale specifiee

efficient robust regression via two stage generalized empirical likelihood large and finite sample efficiency and resistance to outliers are the key goals of robust statistics although often not simultaneously attainable we develop and study a linear regression estimator that comes close efficiency is obtained from the estimator s close connection to generalized empirical likelihood and its favorable robustness properties are obtained by constraining the associated sum of weighted squared residuals we prove maximum attainable finite sample replacement breakdown point and full asymptotic efficiency for normal errors simulation evidence shows that compared to existing robust regression estimators the new estimator has relatively high efficiency for small sample sizes and comparable outlier resistance the estimator is further illustrated and compared to existing methods via application to a real dataset with purported outliers

flat parabolic vector bundles on elliptic curves we describe the moduli space of logarithmic rank 2 connections on elliptic curves with 2 poles

an observation on the positive real lemma abstract we give a necessary and a sufficient condition that the transfer function of an exponentially stable linear finite dimensional system be a real positive matrix the condition does not assume controllability observability properties

robust estimators for simultaneous equations models this paper presents a class of robust estimators for linear and non linear simultaneous equations models which are a direct generalization of the maximum likelihood estimator the new estimators are obtained as solutions of a generalized likelihood equation they are resistant to deviations from the model distribution to outlying observations and to some model misspecifications an optimality principle leads to the construction of an optimal robust estimator which is the best trade off between efficiency at the model and robustness

bounded influence rank regression a one step estimator based on wilcoxon scores abstract the krasker welsch 1982 approach to bounding influence is merged with rank regression i propose a one step estimator that is analogous to bickel s 1975 one step m estimator of type 1 but uses weights that depend on the design vector and the residuals to reduce the influence of outliers in fact the standardized sensitivity of the estimator is made equal to a prechosen constant it is based however on a second derivative approximation to a dispersion surface that is not convex this one step variant avoids the problem of multiple roots the estimator is shown to be consistent and asymptotically multivariate normal an example shows that it yields results similar to those of the krasker welsch estimator

robust estimation in heteroscedastic linear models we consider a heteroscedastic linear model in which the variances are given by a parametric function of the mean responses and a parameter theta we propose robust estimates for the regression parameter beta and show that as long as a reasonable starting estimate of theta is available our estimates of beta are asymptotically equivalent to the natural estimate obtained with known variances a particular method for estimating theta is proposed and shown by monte carlo to work quite well especially in power and exponential models for the variances we also briefly discuss a feedback estimate of beta

bounded error identification of time varying parameters by rls techniques the performance of the recursive least squares algorithm with constant forgetting factor in the identification of time varying parameters is studied in a stochastic framework it is shown that the mean square tracking error keeps bounded if and only if the so called covariance matrix of the algorithm is l sup 1 bounded then a feasibility range for the forgetting factor is worked out in correspondence of which the covariance matrix and therefore the tracking error keeps bounded

a robust gm estimator for the automated detection of external defects on barked hardwood logs and stems the ability to detect defects on hardwood trees and logs holds great promise for the hardwood forest products industry at every stage of wood processing there is a potential for improving value and recovery with knowledge of the location size shape and type of log defects this paper deals with a new method that processes hardwood laser scanned surface data for defect detection the detection method is based on robust circle fitting applied to scanned cross section data sets recorded along the log length it can be observed that these data sets have missing data and include large outliers induced by loose bark that dangles from the log trunk because of that and because of the nonlinearity of the circle model which presents both additive and nonadditive errors we initiated a new robust generalized m estimator for which the residuals are standardized via scale estimates calculated by means of projection statistics and incorporated in the huber objective function yielding a bounded influence method our projection statistics are based on the 2 d radial vectors instead of the row vectors of the jacobian matrix as advocated in the literature dealing with linear regression these radial distances allow us to develop algorithms aimed at pinpointing large surface rises and depressions from the contour image levels and thereby locating severe external defects having at least a height of 0 5 in and a diameter of 5 in

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

resistant fits for some commonly used logistic models with medical application logistic regression type models are used in many applications some examples include the classical dose response experiment prospective and retrospective studies of disease incidence with and without matching and the analysis of ordinal data in most instances the model is fitted by the method of maximum likelihood which like least squares is sensitive to atypical observations an alternative to maximum likelihood is proposed and illustrated by examples

bounded influence estimators for the tobit model abstract this paper introduces a class of bounded influence estimators for the gaussian censored regression or tobit model these estimators can be interpreted as weighted ml estimators with weights chosen to attain the best trade off between efficiency and robustness an empirical example illustrates the feasibility and usefulness of these estimators as well as their performance vis a vis the tobit ml clad and scls estimators

the accuracy of elemental set approximations for regression abstract the elemental set algorithm involves performing many fits to a data set each fit made to a subsample of size just large enough to estimate the parameters in the model elemental sets have been proposed as a computational device to approximate estimators in the areas of high breakdown regression and multivariate location scale estimation where exact optimization of the criterion function is computationally intractable although elemental set algorithms are used widely and for a variety of problems the quality of the approximation they give has not been studied this article shows that they provide excellent approximations for the least median of squares least trimmed squares and ordinary least squares criteria it is suggested that the approach likely will be equally effective in the other problem areas in which exact optimization of a criterion is difficult or impossible

on the solvability of the positive real lemma equations abstract in this paper we consider the classical equations of the positive real lemma under the sole assumption that the state matrix a has unmixed spectrum a a without any other system theoretic assumption observability reachability stability etc we derive a necessary and sufficient condition for the solvability of the positive real lemma equations

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