right indefinite half linear sturm liouville problems we study the regular half linear sturm liouville equation p f r y q f r y lw f r y on j a b where f r u u r 1u r 0 p 1 r q w l a b and p 0 a e on j let n l denote the number of zeros in j of a nontrivial solution of the equation asymptotic formulas are found for n l when w 0 a e and w changes sign respectively as a consequence the existence and asymptotics of real eigenvalues are established for the half linear sturm liouville problem consisting of the above equation and a separated boundary condition when w changes sign our results cover the work of atkinson and mingarelli on second order linear equations as a special case the generalized prufer transformation plays a key role in the proofs

robust fitting of the binomial model we consider the problem of robust inference for the binomial m pi model the discreteness of the data and the fact that the parameter and sample spaces are bounded mean that standard robustness theory gives surprising results for example the maximum likelihood estimator mle is quite robust it cannot be improved on for m 1 but can be for m 1 we discuss four other classes of estimators m estimators minimum disparity estimators optimal mgp estimators and a new class of estimators which we call e estimators we show that e estimators have a non standard asymptotic theory which challenges the accepted relationship between robustness concepts and thereby provided new perspectives on these concepts

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

robust inference in the negative binomial regression model with an application to falls data summary a popular way to model overdispersed count data such as the number of falls reported during intervention studies is by means of the negative binomial nb distribution classical estimating methods are well known to be sensitive to model misspecifications taking the form of patients falling much more than expected in such intervention studies where the nb regression model is used we extend in this article two approaches for building robust m estimators of the regression parameters in the class of generalized linear models to the nb distribution the first approach achieves robustness in the response by applying a bounded function on the pearson residuals arising in the maximum likelihood estimating equations while the second approach achieves robustness by bounding the unscaled deviance components for both approaches we explore different choices for the bounding functions through a unified notation we show how close these approaches may actually be as long as the bounding functions are chosen and tuned appropriately and provide the asymptotic distributions of the resulting estimators moreover we introduce a robust weighted maximum likelihood estimator for the overdispersion parameter specific to the nb distribution simulations under various settings show that redescending bounding functions yield estimates with smaller biases under contamination while keeping high efficiency at the assumed model and this for both approaches we present an application to a recent randomized controlled trial measuring the effectiveness of an exercise program at reducing the number of falls among people suffering from parkinsons disease to illustrate the diagnostic use of such robust procedures and their need for reliable inference

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 robustness of tests abstract power and level breakdown functions of a test statistic are introduced to obtain a unified analysis of local stability as typified by the influence function and global stability as illustrated by the breakdown point the power breakdown function gives the amount of contamination of each alternative distribution that can carry the test statistic to a null value the level breakdown function gives the amount of contamination of a null distribution that can carry the test statistic to each value in the alternative space in many cases the breakdown function bears an inverse relationship to the maximum bias as a function of the contamination as the maximum bias is not invariant to one to one transformations however equivalent test statistics can have different scales for bias a distinct advantage of the inverse scale is the invariance of the breakdown functions to such transformations the resulting stability curves depend on the parameters in a way analogous to the power curve the slopes o

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

weighted likelihood equations with bootstrap root search abstract we discuss a method of weighting likelihood equations with the aim of obtaining fully efficient and robust estimators we discuss the case of continuous probability models using unimodal weighting functions these weighting functions downweight observations that are inconsistent with the assumed model at the true model therefore the proposed estimating equations behave like the ordinary likelihood equations we investigate the number of solutions of the estimating equations via a bootstrap root search the estimators obtained are consistent and asymptotically normal and have desirable robustness properties an extensive simulation study and real data examples illustrate the operating characteristics of the proposed methodology

a robust version of mallows s c p abstract we present a robust version of mallows s c p for regression models it is defined by rc p w p 2 u p v p where w p i 2 i r 2 i is a weighted residual sum of squares computed from a robust fit of model p 2 is a robust and consistent estimator of 2 in the full model and u p and v p are constants depending on the weight function and the number of parameters in model p good subset models are those with rc p close to v p or smaller than v p when the weights are identically 1 w p becomes the residual sum of squares of a least squares fit and rc p reduces to mallows s c p the robust model selection procedure based on rc p allows us to choose the models that fit the majority of the data by taking into account the presence of outliers and possible departures from the normality assumption on the error distribution together with the classical c p the robust version suggests several models from which we can choose

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

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

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

robustified maximum likelihood estimation in generalized partial linear mixed model for longitudinal data summary in this article we study the robust estimation of both mean and variance components in generalized partial linear mixed models based on the construction of robustified likelihood function under some regularity conditions the asymptotic properties of the proposed robust estimators are shown some simulations are carried out to investigate the performance of the proposed robust estimators just as expected the proposed robust estimators perform better than those resulting from robust estimating equations involving conditional expectation like sinha 2004 journal of the american statistical association99 451 460 and qin and zhu 2007 journal of multivariate analysis98 1658 1683 in the end the proposed robust method is illustrated by the analysis of a real data set

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

an examination of the effect of heterogeneity on the estimation of population size using capture recapture data summary part of the folklore of capture recapture experiments is that ignoring heterogeneity of capture probabilities results in a downward bias this has been based on experience and simulation studies but is often interpreted as being due to individuals with lower capture probabilities here estimating equation arguments are used to show that the effect on horvitz thompson type estimators of ignoring heterogeneity in capture recapture experiments is to introduce a downward bias the arguments are extended to continuous time experiments and to an influence function constructed to determine the effect of a small number of individuals with heterogeneous capture probabilities in an otherwise homogeneous population and the influence function is shown to be negative the downward bias holds even if the small number of heterogeneous individuals have capture probabilities larger than the homogeneous majority and this is confirmed by simulations

approximate likelihood ratios for general estimating functions summary the method of estimating functions godambe 1991 is commonly used when one desires to conduct inference about some parameters of interest but the full distribution of the observations is unknown however this approach may have limited utility due to multiple roots for the estimating function a poorly behaved wald test or lack of a goodness of fit test this paper presents approximate likelihood ratios that can be used along with estimating functions when any of these three problems occurs we show that the approximate likelihood ratio provides correct large sample inference under very general circumstances including clustered data and misspecified weights in the estimating function two methods of constructing the approximate likelihood ratio one based on the quasi likelihood approach and the other based on the linear projection approach are compared and shown to be closely related in particular we show that quasi likelihood is the limit of the projection approach we illustrate the technique with two applications

robust analysis of generalized linear mixed models the method of maximum likelihood ml is widely used for analyzing generalized linear mixed models glmm s a full maximum likelihood analysis requires numerical integration techniques for calculation of the log likelihood and to avoid the computational problems involving irreducibly high dimensional integrals several maximum likelihood algorithms have been proposed in the literature to estimate the model parameters by approximating the log likelihood function although these likelihood algorithms are useful for fitting the glmm s efficiently under strict model assumptions they can be highly influenced by the presence of unusual data points in this article the author develops a technique for finding robust maximum likelihood rml estimates of the model parameters in glmm s which appears to be useful in downweighting the influential data points when estimating the parameters the asymptotic properties of the robust estimators are investigated under some regularity conditions small simulations are

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

the primitive topology of a scheme abstract we define a grothendieck topology on the category of schemes whose associated sheaf theory coincides in many cases with that of the zariski topology we also give some indications of possible advantages this new topology has over the zariski topology

least absolute deviations fits for generalized linear models reference stap article 1992 004view record in web of science record created on 2006 04 04 modified on 2016 08 08

robust regression for clustered data with application to binary responses summary generalized estimating equations gee can be highly influenced by the presence of unusual data points a generalization of the gee procedure which yields parameter estimates and fitted values that are resistant to influential data is introduced resistant generalized estimating equations regee include weights in the estimating equations to downweight influential observations or clusters influential observations are downweighted according to their leverage or residual in an example of correlated binary regression applied to 137 urinary incontinent elderly patients from 38 medical practices

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

thomason s theorem for varieties over algebraically closed fields we present a novel proof of thomason s theorem relating bott inverted algebraic k theory with finite coefficients and etale cohomology for smooth varieties over algebraically closed ground fields our proof involves first introducing a new theory which we term algebraic k homology and proving it satisfies etale descent with finite coefficients on the category of normal cohen macaulay varieties then we prove algebraic k homology and algebraic k theory each taken with finite coefficients coincide on smooth varieties upon inverting the bott element

marginally specified generalized linear mixed models a robust approach la modelisation des donnees longitudinales est compliquee par la necessite de prendre en compte correctement la correlation entre les observations faites chez le meme individu en partant d une version anterieure non robuste proposee par heagerty 1999 biometrics 55 688 698 notre modele lineaire generalise mixte robuste marginalement robms glmm fournit methode une methode efficace pour traiter de telles donnees ce modele est un des premiers a permettre a la fois des inferences moyennee sur la population et specifique a l individu de plus il adopte la flexibilite et l interpretation des modeles lineaires generalises mixtes pour introduire la dependance mais construit la structure de regression pour l esperance marginale permettant une application valide avec des covariables dependantes du temps exogenes et independantes du temps ces nouveaux estimateurs sont obtenus comme les solutions d une equation de vraisemblance rendue robuste basee sur la distribution la moins favorable de huber et une collection de poids la distribution la moins favorable de huber produit tics estimations qui sont resistantes a certaines deviations des hypotheses de distribution des effets aleatoires des strategies innovantes de ponderation permettent an robs glmm de bien fonctionner quand on est confronte a des observations eloignees a la fois dans la reponse et les covariables nous illustrons la methodologie avec une analyse d une etude longitudinale prospective de l intubation laryngoscopique endotracheale une methode que de nombreux professionnels de sante doivent acquerir le but principal de notre recherche est d obtenir une inference robuste dans les analyses longitudinales