a note on the unification of the akaike information criterion to measure the distance between a robust function evaluated under the true regression model and under a fitted model we propose generalized kullback leibler information using this generalization we have developed three robust model selection criteria aicr aiccr and aiccr that allow the selection of candidate models that not only fit the majority of the data but also take into account non normally distributed errors the aicr and aiccr criteria can unify most existing akaike information criteria three examples of such unification are given simulation studies are presented to illustrate the relative performance of each criterion

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

minimum mean squared estimation of location and scale parameters under misspecification of the model summary the paper is concerned with estimating location and scale parameters by estimators minimizing a mean squared distance between the empirical distribution function and a conveniently chosen parameterized distribution function within the true parametric family the location estimator has the same asymptotic distribution as that of hodges lehmann if the underlying distribution is not a member of the assumed parametric family the questions concerning bias rather than variance are dominant and most of the paper is concerned with this situation numerical results are given for the case of contaminated normal distributions

a simple proof of the transposed qr algorithm this paper presents a simple proof of the transposed qr algorithm which permits the singular value decomposition of a matrix to be introduced to a first course in matrix algebra in the context of iterative procedures

asymptotic behavior of robust estimators of regression and scale parameters with fixed carriers for the linear regression model y i mathbf x i mathbf beta varepsilon i with fixed mathbf x i s the asymptotic normality of hat mathbf beta hat sigma which minimizes the huber dutter loss function sum sigma rho y i mathbf x i mathbf beta sigma a n sigma is established under rather general conditions

generalized m estimators for errors in variables regression this paper discusses robust estimation for structural errors in variables ev linear regression models such models have important applications in many areas under certain assumptions including normality the maximum likelihood estimates for the ev model are provided by orthogonal regression or which minimizes the orthogonal distance from the regression line to the data points instead of the vertical distance used in ordinary regression or is very sensitive to contamination and thus efficient robust procedures are needed this paper examines the theoretical properties of bounded influence estimators for univariate gaussian ev models using a generalized m estimate approach the results include fisher consistency most b robust estimators and the or version of hampel s optimality problem

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

the role of critical exponents in blow up theorems the sequel abstract in 27 fujita showed that for positive solutions the initial value problem in r n for u t u u p with p 1 exhibited the following behavior if p p c 1 2 n then the initial value problem does not have any nontrivial non negative solution existing on r n 0 a global solution whereas if p p c there exist global small data positive solutions as well as solutions which are non global we call such a result a blow up theorem of fujita type in 50 levine discussed the various theorems of this type that had appeared in the literature prior to 1990 in this paper we revisit the literature since 1990

nonsmooth analysis and frechet differentiability of m functionals abstract a necessary requirement for existence of the frechet derivative is that the defining psi function is uniformly bounded and this naturally excludes those nonrobust estimators such as the maximum likelihood estimator in normal parametric models in this paper the methods of nonsmooth analysis described in the book by f h clarke 1983 are introduced to the theory of statistical expansions and are used here in the proofs of weak continuity and frechet differentiability of m functionals subsequently the conditions for frechet differentiability given in clarke 1983 can be relaxed to include most popular m functionals additional keywords distribution functions m estimators robustness gross error sensitivity weak continuity asymptotic expansions asymptotic normality selection functional local uniqueness author

breakdown points of t type regression estimators summary to bound the influence of a leverage point generalised m estimators have been suggested however the usual generalised m estimator of regression has a breakdown point that is less than the inverse of its dimension this paper shows that dimension independent positive breakdown points can be attained by a class of well defined generalised mestimators with redescending scores the solution can be determined through optimisation of t type likelihood applied to properly weighted residuals the highest breakdown point of is attained by cauchy score these bounded influence and high breakdown estimators can be viewed as a fully iterated version of the one step generalised m estimates of simpson ruppert carroll 1992 with the two advantages of easier interpretability and avoidance of undesirable roots to estimating equations given the design dependent weights they can be computed via em algorithms empirical investigations show that they are highly competitive with other robust estimators of regression

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

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

uniform asymptotics for robust location estimates when the scale is unknown most asymptotic results for robust estimates rely on regularity conditions that are dicult to verify and that real data sets rarely satisfy moreover these results apply to fixed distribution functions in the robustness context the distribution of the data remains largely unspecified and hence results that hold uniformly over a set of possible distribution functions are of theoretical and practical interest in this paper we study the problem of obtaining verifiable and realistic conditions that suce to obtain uniform consistency and uniform asymptotic normality for location robust estimates when the scale of the errors is unknown we study m location estimates calculated with an s scale and we obtain uniform asymptotic results over contamination neighbourhoods there is a trade o between the size of these neighbourhoods and the breakdown point of the scale estimate we also show how to calculate the maximum size of the contamination neighbourhoods where these uniform results hold

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

frechet differentiability p variation and uniform donsker classes classically a stastistical functional is defined on a space of distribution f on the real line with the supremum norn the values may be real or themselves functions such as the quantile function f 1 nonlinear functionals are studied via their derivatives this paper and a related one dudley 1991a will show that the differentiability properties originally proved by reeds 1976 also treated in fernholz 1983 can be improved substantially

tests against qualitative interaction exact critical values and robust tests summary consider a study to evaluate treatment a with a placebo in two or more groups of patients if treatment a is beneficial to one group of patients and harmful to another then we say that there is qualitative interaction or crossover interaction between patient groups and the treatments gail and simon 1985 biometrics41 361 372 developed a large sample procedure for this testing problem their test has received favorable coverage in the literature in this article we obtain corresponding exact finite sample results for normal error distribution and provide a table of critical values the test statistic is similar to the familiar f ratio and its p value is equal to a weighted sum of tail areas of f distributions the computations to implement this are simple a simulation study shows that the exact critical values provided here for normal error distribution are preferable to the asymptotic critical values for a wide range of error distributions we also develop tests that are power robust against long tailed error distributions our robust test uses m estimators instead of the least squares estimators we show that the efficiency robustness of the m estimator translates to power robustness of the corresponding test therefore our robust tests are better if outliers are expected a simulation study illustrates the substantial power advantages of our robust tests

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

a review on robust estimators applied to regression credibility the lack of robustness of regression credibility estimators leads to the development of this paper the appearance of outlier events including large claims catastrophic events can offset the result of the ordinary least squares regression technique and perturb the credibility premium estimation our proposal is to apply robust statistical procedures to the regression credibility estimation which are insensitive to the occurrence of outlier events in the data a review of robust estimators that appeared in the literature is provided including robust estimators that simultaneously attain maximum breakdown point and full asymptotic efficiency under normal errors these estimators are the following l1 estimators m estimators gm estimators lms estimators lts estimators s estimators mm estimators and robust rewls estimators applications of these estimators to regression credibility with two data sets are also presented herein

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

life span and a new critical exponent for a degenerate parabolic equation abstract in this paper we consider the positive solution of the cauchy problem for the equation u t u p u u q p 1 q 1 and give a secondary critical exponent of the behavior of initial value at infinity for the existence of global and nonglobal solutions of the cauchy problem furthermore the life span of solutions are also studied

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

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

a novel dynamic model of pseudo random number generator an interesting hierarchy of random number generators is introduced in this paper based on the review of random numbers characteristics and chaotic functions theory the main objective of this paper is to produce an ergodic dynamical system which can be implemented in random number generators in order to check the efficacy of pseudo random number generators based on this map we have carried out certain statistical tests on a series of numbers obtained from the introduced hierarchy the results of the tests were promising as the hierarchy passed the tests satisfactorily and offers a great capability to be employed in a pseudo random number generator

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

the second critical exponent for a semilinear nonlocal parabolic equation abstract this article considers the cauchy problem for a semilinear nonlocal parabolic equation u t u r n k y u q y t d y p 1 q u r 1 in r n 0 t where p q 1 r 0 and p r 1 we study the second critical exponent i e describing the critical smallness of initial data required by global solutions non global solutions via the decay rates of the initial data at spatial infinity differently from other parabolic equations the second critical exponent is related to n in this problem when k l 1 r n

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

exploratory projection pursuit abstract a new projection pursuit algorithm for exploring multivariate data is presented that has both statistical and computational advantages over previous methods a number of practical issues concerning its application are addressed a connection to multivariate density estimation is established and its properties are investigated through simulation studies and application to real data the goal of exploratory projection pursuit is to use the data to find low one two or three dimensional projections that provide the most revealing views of the full dimensional data with these views the human gift for pattern recognition can be applied to help discover effects that may not have been anticipated in advance since linear effects are directly captured by the covariance structure of the variable pairs which are straightforward to estimate the emphasis here is on the discovery of nonlinear effects such as clustering or other general nonlinear associations among the variables although arbitrary

robust asymptotic statistics contents preface notation chapter 1 von mises functionals chapter 2 log likelihoods chapter 3 asymptotic statistics chapter 4 nonparametric statistics chapter 5 optimal influence curves chapter 6 stable constructions chapter 7 robust regression appendix a weak convergence of measures appendix b some functional analysis appendix c complements bibliography index

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

nonlinear transformation on the transfer entropy of financial time series abstract transfer entropy te now is widely used in the data mining and economic field however te itself demands that time series intend to be stationary and meet markov condition naturally we are interested in investigating the effect of the nonlinear transformation of the two series on the te therefore the paper is designed to study the te of five nonlinear volatile transformations based on the data which are generated by the linear modeling and the logistic maps modeling as well as the dataset that come from financial markets with only one of the te of nonlinear transformations fluctuating around the te of original series the te of others all have increased with different degrees

uniqueness and frechet differentiability of functional solutions to maximum likelihood type equations solutions of simultaneous equations of the maximum likelihood type or m estimators can be represented as functionals existence and uniqueness of a root in a local region of the parameter space are proved under conditions that are easy to check if only one root of the equation exists the resulting statistical functional is frechet differentiable and robust when several solutions exists conditions on the loss criterion used to select the root for the statistic ensure frechet differentiability an interesting example of a frechet differentiable functional is the solution of the maximum likelihood equations for location and scale parameters in a cauchy distribution the estimator is robust and asymptotically efficient

bi iteration svd subspace tracking algorithms we present a class of fast subspace tracking algorithms that arise from a straightforward extension of bauer s 1957 classical bi iteration to the sequential processing case the bi iteration concept has an unexpected potential in subspace tracking our new bi svd subspace trackers are well structured and show excellent convergence properties they outperform the tqr svd subspace tracking algorithm detailed comparisons confirm our claims an application to rank and data adaptive signal reconstruction is also discussed

chaos induced true randomness abstract we investigate functions of type x n p z n where p t is a periodic function and z are real parameters we show that these functions produce truly random sequences we prove that a class of autonomous dynamical systems containing nonlinear terms described by periodic functions of the variables can generate random dynamics we generalize these results to dynamical systems with nonlinearities in the form of noninvertible functions several examples are studied in detail we discuss how the complexity of the dynamics depends on the kind of nonlinearity we present real physical systems that can produce random time series we report the results of real experiments using nonlinear circuits with noninvertible i v characteristics in particular we show that a josephson junction coupled to a chaotic circuit can generate unpredictable dynamics

estimation of variance components with high breakdown point and high efficiency summary for the one way random effects model three noniterative robust variance component estimators are investigated for these estimators block breakdown points measurement breakdown points and efficiency are derived it is shown that the block breakdown points are approximating 50 and that the measurement breakdown points are not smaller than 25 by deriving the asymptotic normal distribution of these estimators we show that two of the new estimators have an asymptotic efficiency greater than 82 and under certain circumstances up to 96 competing estimators have lower breakdown points and lower efficiencies a simulation study shows that the high efficiency also appears for finite samples the use of the estimators for interlaboratory tests is demonstrated in a practical example

interplay of degree correlations and cluster synchronization we study the evolution of coupled chaotic dynamics on networks and investigate the role of degree degree correlation in the networks cluster synchronizability we find that an increase in the disassortativity can lead to an increase or a decrease in the cluster synchronizability depending on the degree distribution and average connectivity of the network networks with heterogeneous degree distribution exhibit significant changes in cluster synchronizability as well as in the phenomena behind cluster synchronization as compared to those of homogeneous networks interestingly cluster synchronizability of a network may be very different from global synchronizability due to the presence of the driven phenomenon behind the cluster formation furthermore we show how degeneracy at the zero eigenvalues provides an understanding of the occurrence of the driven phenomenon behind the synchronization in disassortative networks the results demonstrate the importance of degree degree correlations in determining cluster synchronization behavior of complex networks and hence have potential applications in understanding and predicting dynamical behavior of complex systems ranging from brain to social systems

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

robust singular value decompositions a new approach to projection pursuit abstract robust location and covariance estimators are developed via general m estimation for covariance matrix eigenvectors and eigenvalues the solution to this gm estimation problem is obtained by transforming it into a series of robust regression problems based on a new algorithm for the singular value decomposition it is shown here that the singular value decomposition can be represented as an iteration of two steps a least squares regression fit of the data matrix followed by a rotation to the regression hyperplanes an algorithm to obtain the solution to this gm estimation problem is presented along with results of a monte carlo study and examples of its application in addition it is shown how the output of this algorithm can be used to numerically search for multivariate outliers which is especially useful in exploratory data analysis with high dimensional data and large sample sizes where standard graphical techniques are difficult to implement because the algorithm computes robust estima

second critical exponent and life span for pseudo parabolic equation this paper studies the second critical exponent and life span of solutions for the pseudo parabolic equation ut k ut u up in rn 0 t with p 1 k 0 it is proved that the second critical exponent i e the decay order of the initial data required by global solutions in the coexistence region of global and non global solutions is independent of the pseudo parabolic parameter k nevertheless it is revealed that the viscous term k ut relaxes restrictions on the amplitude of the initial data required by the global solutions moreover it is observed that the life span of the non global solutions will be delayed by the third order viscous term finally some numerical examples are given to illustrate all these results

logistic map a possible random number generator the logistic map is one of the simple systems exhibiting order to chaos transition in this work we have investigated the possibility of using the logistic map in the chaotic regime logmap for a pseudorandom number generator to this end we have performed certain statistical tests on the series of numbers obtained from the logmap we find that the logmap passes these tests satisfactorily and therefore it possesses many properties required of a pseudorandom number generator

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

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

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

model robust designs for quantile regression we give methods for the construction of designs for regression models when the purpose of the investigation is the estimation of the conditional quantile function and the estimation method is quantile regression the designs are robust against misspecified response functions and against unanticipated heteroscedasticity the methods are illustrated by example and in a case study in which they are applied to growth charts

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

model checks for regression an innovation process approach in the context of regression analysis it is known that the residual cusum process may serve as a basis for the construction of various omnibus smooth and directional goodness of fit tests since a deeper analysis requires the decomposition of the cusums into their principal components and this is difficult to obtain we propose to replace this process by its innovation martingale it turns out that the resulting tests are asymptotically distribution free under composite null models and may be readily performed a simulation study is included which indicates that the distributional approximations already work for small to moderate sample sizes

m estimators and gnostical estimators for indentification of a regression model abstract this paper shows that the gnostical estimators of kovanic 1986 automatica 22 657 674 for identification of a regression model with imprecisely known output and input derived from non statistical considerations are equivalent to the m estimators for a regression model with random carriers and scale parameters estimated separately both the theoretical and practical importance of this fact for identification is established

robust estimation of variance components outliers may occur with respect to any of the random components in the mixed linear model a procedure for limiting the influence of these outliers on the estimates of the model parameters is described given the variances or estimates of them the model effects are estimated by augmenting the original observations with auxiliary observations that contain the prior information represented by the variances large residuals among either the original or the auxiliary observations are interpreted as outlying random errors or outlying random effects as appropriate and winsorized the robust estimation of the variances is obtained by modifying the defining equations for the restricted maximum likelihood estimates under normality along the lines of huber s proposal 2 a numerical example illustrates the use of the methodology both as a diagnostic and as an estimation tool

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

robust tests of inequality constraints and one sided hypotheses in the linear model summary for the linear regression model with independent and identically distributed errors robust tests of various hypotheses on the regression parameter are developed by robust we mean robustness of size and power against long tailed error distributions which may not be symmetric the proposed test statistic denoted by fm resembles the usual f statistic the two main results are i under the null hypothesis the asymptotic distribution of fm and that of the least squares based f statistic are the same and ii the pitman asymptotic efficiency of fm relative to f is equal to the asymptotic efficiency of the corresponding m estimator relative to the least squares estimator an example and simulation results illustrate that fm has desirable robustness properties compared to f

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

robust estimation of parameters in a mixed unbalanced model this paper describes a method of robust estimation of shift and scale parameters in a mixed unbalanced interlaboratory model estimators presented result from easily computable fr echet differentiable functionals which enjoy some optimal properties in a small neighborhood of the model a rigorous treatment of their asymptotic behaviour under departures from the model assumptions and a simulation study are given 1 introduction a method of robust analysis of random effects in mixed models is introduced in the paper by rocke 1991 see also iglewicz 1983 and a general treatment of robust estimators of variance components is presented by fellner 1986 rocke proposes estimators of variance components applying robust scale estimators to residuals and laboratory effects in the rst step he uses huber s method to robustly assess location parameters a similar point of view is taken by lischer 1994 who improves breakdown properties of the estimators he also gives a comprehensive practical motivation for the use of robust methods in interlaboratory experiments the above mentioned estimation technique may lead to high dimensional nonlinear equations and it is inconvenient for assessing variability of estimators a possible improvement may be in the simultaneous estimation of xed effects and components of variation where problems of a rigorous asymptotic treatment smoothness of estimators and their optimality become more feasible in a preliminary study by bednarski zmy slony and zontek 1992 it is shown that fr echet differentiability of statistical functionals easily leads to reasonable robust estimators of variance components and treatment fixed effects in a simple interlaboratory model the basic idea there is to construct a smooth fisher consistent functional t for the parameter de 1 a ee t at the model

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

a class of partially adaptive one step m estimators for the non linear regression model with dependent observations in this paper we consider a class of partially adaptive one step m estimators for the non linear regression model with dependent observations those estimators adapt themselves with respect to a measure of the tailthickness of the disturbance distribution as well as to a measure of the scale the large sample behavior of those estimators is examined theoretically for general disturbance distributions and numerically for various specific ones the estimators considered are motivated by the student r maximum likelihood estimator given appropriate specifications of the adaptation parameter the estimators are asymptotically efficient on the family of student t distributions including the normal distribution

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

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

bivariate location scale models for regression analysis with applications to lifetime data summary the literature on multivariate linear regression includes multivariate normal models models that are used in survival analysis and a variety of models that are used in other areas such as econometrics the paper considers the class of location scale models which includes a large proportion of the preceding models it is shown that for complete data the maximum likelihood estimators for regression coefficients in a linear location scale framework are consistent even when the joint distribution is misspecified in addition gains in efficiency arising from the use of a bivariate model as opposed to separate univariate models are studied a major area of application for multivariate regression models is to clustered parallel lifetime data so we also study the case of censored responses estimators of regression coefficients are no longer consistent under model misspecification but we give simulation results that show that the bias is small in many practical situations gains in efficiency from bivariate models are also examined in the censored data setting the methodology in the paper is illustrated by using lifetime data from the diabetic retinopathy study