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

nearly vertical hopf bifurcation for a passively q switched microchip laser passively q switched microchip lasers generate strongly pulsating intensity oscillations that emerge from a hopf bifurcation point we show that this bifurcation is nearly vertical and explain why strongly pulsating oscillations are immediately observed as we pass the hopf bifurcation point the laser dynamical problem is mathematically a singular perturbation problem which we investigate the leading order problem is conservative and corresponds to lotka volterra equations

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

estimation in linear models with censored data summary three methods for linear regression with censored data are considered that of buckley james 1979 a proposed simpler nonparametric method and a normal model for censored data a new estimator for the variance of the error term in buckley james s 1979 procedure is proposed and simulations comparing the three methods are described many methods have been proposed for handling regression problems in which the dependent variable may be censored we focus on three methods for the linear model y xt3 e two methods are nonparametric so that the distribution of e is unspecified while the third assumes that e is normally distributed the nonparametric methods are analogous to the expectation maximization em algorithm introduced by dempster laird rubin 1977 which is broadly applicable for computing maximum likelihood estimates in 2 we discuss each of these methods and propose a new estimator of the variance of e for the buckley james method in 3 we give simulation results which compare the three methods

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

censored regression local linear approximations and their applications abstract various statistical tools are available for modeling the relationship between response and covariate if the data are fully observable in the situation of censored data however those tools are no longer directly applicable this article provides an easily implemented methodology for modeling the association based on censored data the form of the regression relationship will be completely determined by the data no assumptions are made about this form basic ideas behind the methodology are to transform the observed data in an appropriate simple way and then to apply a locally weighted least squares regression the proposed estimator involves a variable bandwidth that automatically adapts to the design of the data points that the methodology is very easy to implement is illustrated by several examples including simulation studies and an analysis of the stanford heart transplant data and the primary biliary cirrhosis data several theoretical considerations are reflected in the examples fina

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

large sample theory of a modified buckley james estimator for regression analysis with censored data this approach we introduce herein a slight modification of the buckley james estimator to get around the difficulties caused by the instability at the upper tail of the associated kaplan meier estimate of the underlying error distribution and show that the modified buckleyjames estimator is consistent and asymptotically normal under certain regularity conditions a simple formula for the asymptotic variance of the modified buckley james estimator is also derived and is used to study the asymptotic efficiency of the estimator extensions of these results to the multiple regression model are also given

on estimating equations with censored data summary general estimating equations based on independent identically distributed random variables which may be right censored are considered the equations which appear to be natural analogues of the uncensored equations require full distributional assumptions and a distribution free modification based on the product limit estimator is suggested it follows from self consistency that the resulting estimating functions are simply means obtained from the product limit estimator the asymptotic variances of the estimators obtained from the equations are derived and compared

a method for computing phase shifts for scattering abstract the method developed by papageorgiou and raptis for computing the eigenvalues is now applied to scattering problems in particular a new approach is developed for the computation of the phase shifts of the one dimensional schrodinger equation the algorithm is very simple to code and very efficient for computational purposes the method is also general and can be applied to any kind of potential functions coulomb type etc

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

rescue of the quasi steady state approximation in a model for oscillations in an enzymatic cascade a three variable model describing the oscillatory activity of a cascade of enzyme reactions is analyzed a quasi steady state approximation reduces the three equations to a system of two equations which admits only a stable steady state this apparent failure of the quasi steady state approximation to describe the limit cycle oscillations observed in the full three variable system is analyzed in detail we first show that the oscillations occur in the full system provided the michaelis constants are sufficiently small we then develop a method for determining the correct limit for application of the quasi steady state approximation the leading problem consists of two equations for a conservative oscillator and a higher order analysis is required in order to determine the amplitude of the limit cycle oscillations finally we observe a good agreement when comparing exact numerical and approximate bifurcation diagrams

renovated scatterplots for censored data summary in simple linear regression where the response variable has been subject to right censoring censored points in the scatterplot of observed data are moved vertically to create a renovated scatterplot where the bias of censoring is removed the relationship to least squares estimation and the effect on regression diagnostics are examined in the multiple regression context

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

energy levels for a double well potential in three dimensional system using hill determinant approach abstract energy levels of the schrodinger equation for the double well potential v x y z z 2 1 2 z 2 x 2 y 2 z 2 1 2 a xx x 4 a yy y 4 a zz z 4 2a xy x 2 y 2 2a xz x 2 z 2 2a yz y 2 z 2 in a three dimensional system are calculated using the hill determinant approach for several eigenstates and large values of the perturbation parameters and z2 emphasis is specially placed on the larger values of z2 because the eigenvalues for different states have almost degenerate eigenvalues some of the results calculated by the hill determinant approach are compared with the results produced by other techniques

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

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

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

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

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

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

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

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

a distribution free least squares estimator for censored linear regression models abstract this paper describes a method for estimating simultaneously the parameter vector of the systematic component and the distribution function of the random component of a censored linear regression model the estimator is obtained by minimizing the sum of the squares of the differences between the observed values of the dependent variable and the corresponding expected values of this variable according to the estimated parameter vector and distribution function the resulting least squares parameter estimator incorporates information on the distribution of the random component of the regression model that is available from the estimation sample hence it may often be more efficient than are parameter estimators that do not use such information the results of numerical experiments with the least squares estimator tend to support this hypothesis

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

an application of imputation to an estimation problem in grouped lifetime analysis tanner and wang in press have introduced the data augmentation algorithm for the analysis of parametric missing data problems in this article this paradigm is used to develop an algorithm for the nonparametric estimation of the hazard function from grouped and censored lifetime data this algorithm makes use of the notions of cross validation and multiple imputation to prescribe the appropriate degree of smoothing for the nonparametric hazard estimate a procedure for estimating the variance of the estimator is also proposed the nonparametric hazard estimate and corresponding variance formula are shown to perform well in a simulation study the algorithm is illustrated with a numerical example

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

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

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

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

a minimax bias property of the least alpha quantile estimates a natural measure of the degree of robustness of an estimate mathbf t is the maximum asymptotic bias b mathbf t varepsilon over an varepsilon contamination neighborhood martin yohai and zamar have shown that the class of least alpha quantile regression estimates is minimax bias in the class of m estimates that is they minimize b mathbf t varepsilon with alpha depending on varepsilon in this paper we generalize this result proving that the least alpha quantile estimates are minimax bias in a much broader class of estimates which we call residual admissible and which includes most of the known robust estimates defined as a function of the regression residuals e g least median of squares least trimmed of squares s estimates tau estimates m estimates signed r estimates etc the minimax results obtained here likewise the results obtained by martin yohai and zamar require that the carriers have elliptical distribution under the central model

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

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

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

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

a missing information principle and m estimators in regression analysis with censored and truncated data a general missing information principle is proposed for constructing mestimators of regression parameters in the presence of left truncation and right censoring on the observed responses by making use of martingale central limit theorems and empirical process theory the asymptotic normality of m estimators is established under certain assumptions asymptotically efficient m estimators are also developed by using data dependent score functions in addition robustness properties of the estimators are discussed and formulas for their influence functions are derived for the robustness analysis

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

energy levels of a nonpolynomial oscillator using finite difference technique abstract the energy levels of a one dimensional system are calculated for the rational potentials v 1 x n g x 2 x 2n 1 gx 2 and v 2 x i n m g x 2i gx 2n 1 g x 2m using the finite difference approach over wide range of values of the perturbation parameters g i n m and for various state numbers n the numerical results are compared with previous numerical data and with available exact values

a nonparametric approach to the truncated regression problem abstract a description is given of a new method of estimating the regression parameters in the linear regression model from data where the dependent variable is subject to truncation the residual distribution is allowed to be unspecified the method is iterative and involves estimation of the residual distribution under the truncated sampling scheme the technique can be interpreted as an iterative bias adjustment of the observations in order to correct the regression relationship in the sampled population to match that of the model a simulation study compares the performance of various estimators including one suggested by bhattacharya chernoff and yang 1983 this truncation regression problem arises in many contexts of scientific and social research in economics tobin 1958 analyzed household expenditure on durable goods using a regression model that took account of the fact that the expenditure is always nonnegative a more general situation was studied by hausman and wise 1976 1977 in conn

estimation in a linear regression model with censored data we consider the semiparametric linear regression model with censored data and with unknown error distribution we describe estimation equations of the buckley james type that admit n consistent and asymptotically normal solutions the derived estimator is efficient at a particular error distribution we show the equivalence between this type of estimator and an estimator based on a linear rank test suggested by tsiatis an extension to an estimator that is efficient everywhere is discussed

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

linear regression with doubly censored data linear regression with doubly censored responses is considered buckley james ritov type estimators are proposed semiparametric information and projective scores are discussed an expansion of the estimating equations is obtained under fairly general assumptions sufficient conditions are given for the asymptotic consistency and normality of the estimators

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

tests for location with k samples and censored data summary tests which are specific for location are developed in the k sample situation with right censored data the tests are based on general estimating functions and do not require full parametric assumptions for large samples but in general they are not based only on ranks one class of tests allows shapes to differ in the k populations and includes as a special case the median test of brookmeyer crowley 1982 if the distributions are assumed to differ at most in location modified tests are suggested in which survival function estimates are obtained by pooling the data from all samples prentice s 1978 median test with minor modifications is included in this class

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

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

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 comparison of estimators for regression with a censored response variable summary several methods of estimating parameters in a linear regression when the response variable is censored are compared the estimator of buckley james 1979 is preferred further it is shown that the strength of the regression the support and distribution of the censoring variable and the magnitude of the slope all have an effect on the relative performance of the estimators

some inner product techniques for computing eigenvalues for three dimensional anharmonic oscillators with quartic and sextic perturbations abstract an algorithm previously used by the author for calculating the energy eigenvalues for a three dimensional oscillator with limited range of perturbation parameters eigenstates and accuracy is revisited and implemented in a more effective way to give excellent accuracy even for higher values of and nx ny nz x y z fifty coefficients of renormalized perturbation series are calculated in contrast to the previous works which were limited to 20 coefficients of the perturbation series due to computational facilities some of the results calculated by the inner product technique are compared with results produced by other techniques

on plotting renovated samples in this note we use the buckley james method for censored regression in the p sample problem where the samples are subject to right censoring the samples are reconstructed so as to remove the effect of censoring and graphical procedures based on quantiles such as boxplots may then be used as a standard data analytic tool to describe the variable being measured

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

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

high breakdown point estimates of regression by means of the minimization of an efficient scale abstract a new class of robust estimates estimates is introduced the estimates have simultaneously the following properties a they are qualitatively robust b their breakdown point is 5 and c they are highly efficient for regression models with normal errors they are defined by minimizing a new scale estimate applied to the residuals asymptotically a estimate is equivalent to an m estimate with a function given by a weighted average of two functions one corresponding to a very robust estimate and the other to a highly efficient estimate the weights are adaptive and depend on the underlying error distribution we prove consistency and asymptotic normality and give a convergent iterative computing algorithm finally we compare the biases produced by gross error contamination in the estimates and optimal bounded influence estimates

sungular hopf bifurcation to relaxation oscillations this paper is a continuation of our study of singular hopf bifurcations to relaxation oscillations siam journal on applied mathematics 46 1986 pp 721 739 in our previous analysis a system of two first order nonlinear equatins is investigated and it is shown how the bifurcation problem can be reduced to a weakly perturbed conservative system of two first order equations in this paper a method based on analyzing nearly conservative quantities was developed to find the bifurcation diagram of the periodic solutions emerging from singular hopf bifurcation points specifically the subcritical hopf bifurcation of the fitzhugh nagumo equations is analyzed the extension from the earlier bifurcation analysis is achieved by performing a perturbation analysis from the first integral of the leading order nonlinear oscillator at a critical value of a parameter the first integral exists up to an additional order in the perturbed equation this condition permits the resolution of features determined by h