influence curves of estimators for directional data much recent research has centred on the robustness of estimators particularly estimators of location and scale for linear data however the robustness of estimators for directional data has not been investigated comprehensively mardia 1975 and discussants mentioned some aspects of robustness collett 1980 investigated outliers in circular data in this note we will use the influence curve introduced by hampel 1974 as a heuristic tool in evaluating the robustness of some estimators of location for directional data discussed by mardia 1972 similar calculations can be used to obtain influence curves for scale estimators for the location model we consider a circular distribution f which is unimodal and symmetric about the unknown direction p let p be the known concentration parameter the sample circular mean is the most commonly used measure of location we express the circular population mean in functional form as

complex elliptically symmetric distributions survey new results and applications complex elliptically symmetric ces distributions have been widely used in various engineering applications for which non gaussian models are needed in this overview circular ces distributions are surveyed some new results are derived and their applications e g in radar and array signal processing are discussed and illustrated with theoretical examples simulations and analysis of real radar data the maximum likelihood ml estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness and for convergence of the iterative fixed point algorithm are established specific ml estimators for several ces distributions that are widely used in the signal processing literature are discussed in depth including the complex t distribution k distribution the generalized gaussian distribution and the closely related angular central gaussian distribution a generalization of ml estimators the m estimators of the scatter matrix are also discussed and asymptotic analysis is provided applications of ces distributions and the adaptive signal processors based on ml and m estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for direction of arrival doa estimation and beamforming furthermore experimental validation of the usefulness of ces distributions for modelling real radar data is given

the complex bingham quartic distribution and shape analysis summary the complex bingham distribution was introduced by kent as a tractable model for landmark based shape analysis it forms an exponential family with a sufficient statistic which is quadratic in the data however the distribution has too much symmetry to be widely useful in particular under high concentration it behaves asymptotically as a normal distribution but where the covariance matrix is constrained to have complex symmetry to overcome this limitation and to provide a full range of asymptotic normal behaviour we introduce a new complex bingham quartic distribution by adding a selection of quartic terms to the log density in the simplest case this new distribution corresponds to kent s fb5 distribution asymptotic and saddlepoint methods are developed for the normalizing constant to facilitate maximum likelihood estimation examples are given to show the usefulness of this new distribution

a generalized gronwall inequality and its application to a fractional differential equation this paper presents a generalized gronwall inequality with singularity using the inequality we study the dependence of the solution on the order and the initial condition of a fractional differential equation

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

computing class fields via the artin map based on an explicit representation of the artin map for kummer extensions we present a method to compute arbitrary class fields as in the proofs of the existence theorem the problem is first reduced to the case where the field contains sufficiently many roots of unity using kummer theory and an explicit version of the artin reciprocity law we show how to compute class fields in this case we conclude with several examples

computing ray class groups conductors and discriminants we use the algorithmic computation of exact sequences of abelian groups to compute the complete structure of z k m for an ideal m of a number field k as well as ray class groups of number fields and conductors and discriminants of the corresponding abelian extensions as an application we give several number fields with discriminants less than previously known ones

fast median filtering for phase or orientation data median filtering is among the most utilized tools for smoothing real valued data as it is robust edge preserving value preserving and yet can be computed efficiently for data living on the unit circle such as phase data or orientation data a filter with similar properties is desirable for these data there is no unique means to define a median so we discuss various possibilities the arc distance median turns out to be the only variant which leads to robust edge preserving and value preserving smoothing however there are no efficient algorithms for filtering based on the arc distance median here we propose fast algorithms for filtering of signals and images with values on the unit circle based on the arc distance median for non quantized data we develop an algorithm that scales linearly with the filter size the runtime of our reference implementation is only moderately higher than the matlab implementation of the classical median filter for real valued data for quantized data we obtain an algorithm of constant complexity w r t the filter size we demonstrate the performance of our algorithms for real life data sets phase images from interferometric synthetic aperture radar planar flow fields from optical flow and time series of wind directions

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

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

a robust test for non nested hypotheses we propose a robust version of cox type test statistics for the choice between two non nested hypotheses we first show that the influence of small amounts of contamination in the data on the test decision can be very large secondly we build a robust test statistic by using the results on robust parametric tests that are available in the literature and show that the level of the robust test is stable finally we show numerically not only the robustness of this new test statistic but also that its asymptotic distribution is a good approximation of its sample distribution unlike for the classical test statistic we apply our results to the choice between a pareto and an exponential distribution as well as between two competing regressors in the simple linear regression model without intercept

detecting the direction of a signal on high dimensional spheres non null and le cam optimality results we consider one of the most important problems in directional statistics namely the problem of testing the null hypothesis that the spike direction pmb theta of a fisher von mises langevin distribution on the p dimensional unit hypersphere is equal to a given direction pmb theta 0 after a reduction through invariance arguments we derive local asymptotic normality lan results in a general high dimensional framework where the dimension p n goes to infinity at an arbitrary rate with the sample size n and where the concentration kappa n behaves in a completely free way with n which offers a spectrum of problems ranging from arbitrarily easy to arbitrarily challenging ones we identify various asymptotic regimes depending on the convergence divergence properties of kappa n that yield different contiguity rates and different limiting experiments in each regime we derive le cam optimal tests under specified kappa n and we compute from the le cam third lemma asymptotic powers of the classical watson test under contiguous alternatives we further establish lan results with respect to both spike direction and concentration which allows us to discuss optimality also under unspecified kappa n to investigate the non null behavior of the watson test outside the parametric framework above we derive its local asymptotic powers through martingale clts in the broader semiparametric model of rotationally symmetric distributions a monte carlo study shows that the finite sample behaviors of the various tests remarkably agree with our asymptotic results

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

a propensity score adjustment for multiple group structural equation modeling in the behavioral and social sciences quasi experimental and observational studies are used due to the difficulty achieving a random assignment however the estimation of differences between groups in observational studies frequently suffers from bias due to differences in the distributions of covariates to estimate average treatment effects when the treatment variable is binary rosenbaum and rubin 1983a proposed adjustment methods for pretreatment variables using the propensity score

a directional mixed effects model for compositional expenditure data abstractcompositional data are vectors of proportions defined on the unit simplex and this type of constrained data occur frequently in government surveys it is also possible for the compositional data to be correlated due to the clustering or grouping of the observations within small domains or areas we propose a new class of the mixed model for compositional data based on the kent distribution for directional data where the random effects also have kent distributions one useful property of the new directional mixed model is that the marginal mean direction has a closed form and is interpretable the random effects enter the model in a multiplicative way via the product of a set of rotation matrices and the conditional mean direction is a random rotation of the marginal mean direction in small area estimation settings the mean proportions are usually of primary interest and these are shown to be simple functions of the marginal mean direction for estimation we apply a quasi likelihood method whic

computing residue class rings and picard groups of orders abstract let k be a global field and o be an order of k we develop algorithms for the computation of the unit group of residue class rings for ideals in o as an application we show how to compute the unit group and the picard group of o provided that we are able to compute the unit group and class group of the maximal order o of k

on locally uniformly linearizable high breakdown location and scale functionals this article gives two constructions of a weighted mean which has a large domain is affinely equivariant has a locally high breakdown point and is locally uniformly linearizable one construction is based on m functionals with smooth defining psi and chi functions which are used to control the weighting the second construction involves a locally uniformly linearizable reduction of the data to a finite set of points this construction has the advantage of computational speed and opens up the possibility of allowing the weighting to take the shape of the original data set into account its disadvantage lies in its inability to deal with large atoms the aim of the locally uniform linearizability is to provide a stable analysis based on uniform asymptotics or uniform bootstrapping the stability of the first construction is exhibited using different stochastic models and different data sets its performance is compared with three other functionals which are not locally uniformly linearizable

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

breakdown points of affine equivariant estimators of multivariate location and covariance matrices finite sample replacement breakdown points are derived for different types of estimators of multivariate location and covariance matrices the role of various equivariance properties is illustrated the breakdown point is related to a measure of performance based on large deviations probabilities finally we show that one step reweighting preserves the breakdown point

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

addendum to the discussion of breakdown and groups in his discussion of davies and gather ann statist 33 2005 977 1035 tyler pointed out that the theory developed there could not be applied to the case of directional data he related the breakdown of directional functionals to the problem of definability in this addendum we provide a concept of breakdown defined in terms of definability and not in terms of bias if a group of finite order k acts on the sample space we show that the breakdown point can be bounded above by k 1 k in the case of directional data there is a group of order k 2 giving an upper bound of 1 2

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

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

spatial median and directional data summary we introduce the normalized spatial median as an estimator of location for rotationally symmetric distributions on the hypersphere we investigate some of its asymptotic properties and use them to obtain confidence regions for the modal direction of a distribution on the hypersphere these results are then applied to the von mises fisher distribution and to a contamination model it is seen that the normalized spatial median can perform more efficiently than the normalized mean in presence of outliers

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

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

analogues on the sphere of the affine equivariant spatial median robust estimation of location for data on the unit sphere sp 1 is an important problem in directional statistics even though the influence functions of the sample mean direction and other location

finite time stability of neutral fractional order time delay systems with lipschitz nonlinearities abstract this paper is concerned with finite time stability of neutral fractional order time delay systems with lipschitz nonlinearities by use of the method of steps and the generalized gronwall inequality a new criterion on finite time stability is obtained two numerical examples are given to illustrate the effectiveness of our main results

on median type estimators of direction for the von mises fisher distribution summary we study three median type estimators of direction the normalized spatial median l1 estimator on the sphere and spherical median their asymptotic distributions efficiencies influence functions and sensitivities are given at the von mises fisher model a simulation study is also carried out to compare their finite sample performance in terms of mean squared errors we conclude that overall the normalized spatial median is the preferred estimator

exploring delayed mittag leffler type matrix functions to study finite time stability of fractional delay differential equations abstract in this paper we introduce a concept of delayed two parameters mittag leffler type matrix function which is an extension of the classical mittag leffler matrix function with the help of the delayed two parameters mittag leffler type matrix function we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations via the variation of constants method in addition we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations thereafter we present finite time stability results of nonlinear fractional delay differential equations under mild conditions on nonlinear term finally an example is presented to illustrate the validity of the main theorems

majorization minimization algorithms in signal processing communications and machine learning this paper gives an overview of the majorization minimization mm algorithmic framework which can provide guidance in deriving problem driven algorithms with low computational cost a general introduction of mm is presented including a description of the basic principle and its convergence results the extensions acceleration schemes and connection to other algorithmic frameworks are also covered to bridge the gap between theory and practice upperbounds for a large number of basic functions derived based on the taylor expansion convexity and special inequalities are provided as ingredients for constructing surrogate functions with the pre requisites established the way of applying mm to solving specific problems is elaborated by a wide range of applications in signal processing communications and machine learning

a multivariate sign ewma control chart nonparametric control charts are useful in statistical process control spc when there is a lack of or limited knowledge about the underlying process distribution especially when the process measurement is multivariate this article develops a new multivariate spc methodology for monitoring location parameters it is based on adapting a powerful multivariate sign test to online sequential monitoring the weighted version of the sign test is used to formulate the charting statistic by incorporating the exponentially weighted moving average control ewma scheme which results in a nonparametric counterpart of the classical multivariate ewma mewma it is affine invariant and has a strictly distribution free property over a broad class of population models that is the in control ic run length distribution can attain or is always very close to the nominal one when using the same control limit designed for a multivariate normal distribution moreover when the process distribution comes from the elli

the fisher bingham distribution on the sphere abstract the fisher distribution is the analogue on the sphere of the isotropic bivariate normal distribution in the plane the purpose of this paper is to propose and analyze a spherical analogue of the general bivariate normal distribution estimation hypothesis testing and confidence regions are also discussed author

scaled von mises fisher distributions and regression models for paleomagnetic directional data we propose a new distribution for analyzing paleomagnetic directional data that is a novel transformation of the von mises fisher distribution the new distribution has ellipse like symmetry as

computing the multiplicative group of residue class rings let k be a global field with maximal order 0k and let m0 be an ideal of 0k we present algorithms for the computation of the multiplicative group 0k m0 of the residue class ring 0k m0 and the discrete logarithm therein based on the explicit representation of the group of principal units we show how these algorithms can be combined with other methods in order to obtain more efficient algorithms they are applied to the computation of the ray class group clkm modulo m m0m where m denotes a formal product of real infinite places and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields

robust measures of location for directional data established techniques for robust m estimation of a location parameter are adapted for use in directional data in particular a periodic version of any of the commonly used functions can be used to define a comparable estimator of angular location this technique is illustrated with a numerical example and a small simulation study the proposed estimators appear to perform at efficiency levels similar to those of ordinary m estimators in the linear case

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

computing generators of the unit group of an integral abelian group ring abstract we describe an algorithm for obtaining generators of the unit group of the integral group ring z g of a finite abelian group g we used our implementation in magma of this algorithm to compute the unit groups of z g for g of order up to 110 in particular for those cases we obtained the index of the group of hoechsmann units in the full unit group at the end of the paper we describe an algorithm for the more general problem of finding generators of an arithmetic group corresponding to a diagonalisable algebraic group

robust bounded influence tests in general parametric models abstract we introduce robust tests for testing hypotheses in a general parametric model these are robust versions of the wald scores and likelihood ratio tests and are based on general m estimators their asymptotic properties and influence functions are derived it is shown that the stability of the level is obtained by bounding the self standardized sensitivity of the corresponding m estimator furthermore optimally bounded influence tests are derived for the wald and scores type tests applications to real and simulated data sets are given to illustrate the tests performance

regression models for an angular response summary this paper considers the problem of regressing an angular response variate on a set of linear explanatory variables a general class of models is proposed in which the mean direction and dispersion of a von mises variate are related to the explanatory variables by general link functions appropriate regression diagnostics estimation and testing procedures are developed for fitting the models the meaning of correlation between an angular and a linear variable is clarified and leads to a general notion of multiple correlation associated with the regression model the methods are applied to a set of data arising from a study of movements of intertidal gastropods

fractional differential equations with a constant delay the paper discusses stability and asymptotic properties of a fractional order differential equation involving both delayed as well as non delayed terms as the main results explicit necessary and sufficient conditions guaranteeing asymptotic stability of the zero solution are presented including asymptotic formulae for all solutions the studied equation represents a basic test equation for numerical analysis of delay differential equations of fractional type therefore the knowledge of optimal stability conditions is crucial among others for numerical stability investigations of such equations theoretical conclusions are supported by comments and comparisons distinguishing behaviour of a fractional order delay equation from its integer order pattern

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

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

a distribution free m estimator of multivariate scatter the existence and uniqueness of a limiting form of a huber type m estimator of multivariate scatter is established under certain conditions on the observed sample these conditions hold with probability one when sampling randomly from a continuous multivariate distribution the existence of the estimator is proven by showing that it is the limiting point of a specific algorithm hence the proof is constructive for continuous populations the estimator of multivariate scatter is shown to be strongly consistent and asymptotically normal an important property of the estimator is that its asymptotic distribution is distribution free with respect to the class of continuous elliptically distributed populations this distribution free property also holds for the finite sample size distribution when the location parameter is known in addition the estimator is the most robust estimator of the scatter matrix of an elliptical distribution in the sense of minimizing the maximum asymptotic variance

diagnostic testing and evaluation of maximum likelihood models abstract the paper develops a unified theory of likelihood specification testing based on m estimators of auxiliary parameters the theory is sufficiently general to encompass a wide class of specification tests including moment based tests pearson type goodness of fit tests the information matrix test and the cox test the paper also presents a framework based on frechet differentiation for determining the effects of misspecification on the almost sure limits of parameter estimates and specification test statistics

a general theory for jackknife variance estimation the delete 1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles this well known deficiency can be rectified by using a more general jackknife with d the number of observations deleted depending on a smoothness measure of the point estimator our general theory explains why jackknife works or fails it also shows that i for sufficiently smooth estimators the jackknife variance estimators with bounded d are consistent and asymptotically unbiased and ii for nonsmooth estimators d has to go to infinity at a rate explicitly determined by a smoothness measure to ensure consistency and asymptotic unbiasedness improved results are obtained for several classes of estimators in particular for the sample p quantiles the jackknife variance estimators with d satisfying n 1 2 d rightarrow 0 and n d rightarrow infty are consistent and asymptotically unbiased

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

robust covariance matrix estimation in heterogeneous low rank context this paper addresses the problem of robust covariance matrix cm estimation in the context of a disturbance composed of a low rank lr heterogeneous clutter plus an additive white gaussian noise the lr clutter is modeled by a spherically invariant random vector with assumed high clutter to noise ratio in such a context adaptive process should require less training samples than classical methods to reach equivalent performance as in a full rank clutter configuration the main issue is that classical robust estimators of the cm cannot be computed in the undersampled configuration to overcome this issue the current approach is based on regularization methods nevertheless most of these solutions are enforcing the estimate to be well conditioned while in our context it should be lr structured this paper therefore addresses this issue and derives an algorithm to compute the maximum likelihood estimator of the cm for the considered disturbance model several relaxations and robust generalizations of the result are discussed performance is finally illustrated on numerical simulations and on a space time adaptive processing for airborne radar application

point estimation of the central orientation of random rotations data as three dimensional rotations have application in computer science kinematics and materials sciences among other areas estimating the central orientation from a sample of such data is an important problem which is complicated by the fact that several different approaches exist for this motivated by various geometrical and decision theoretical considerations however little is known about how such estimators compare especially on common distributions for location models with random rotations we examine four location estimators three of which are commonly found in different literatures and the fourth estimator a projected median is newly introduced our study unifies existing literature and provides a detailed numerical investigation of location estimators for three commonly used rotation distributions in statistics and materials science while the data generating model influences the best choice of an estimator the proposed projected median emerges as an overall good performer which can

m estimators of location for gaussian and related processes with slowly decaying serial correlations abstract we investigate the behavior of m estimators of the location parameter for stochastic processes with long range dependence the processes considered are gaussian or one dimensional transformations of gaussian processes it turns out that up to a constant all m estimators are asymptotically equivalent to the arithmetic mean for gaussian processes this constant is always equal to one independently of the function in view of the case of iid observations the results are surprising they are related to earlier work by gastwirth and rubin some simulations illustrate the results

joint covariance estimation with mutual linear structure we consider the problem of joint estimation of structured covariance matrices assuming the structure is unknown estimation is achieved using heterogeneous training sets namely given groups of measurements coming from centered populations with different covariances our aim is to determine the mutual structure of these covariance matrices and estimate them supposing that the covariances span a low dimensional affine subspace in the space of symmetric matrices we develop a new efficient algorithm discovering the structure and using it to improve the estimation our technique is based on the application of principal component analysis in the matrix space we also derive an upper performance bound of the proposed algorithm in the gaussian scenario and compare it with the cramer rao lower bound numerical simulations are presented to illustrate the performance benefits of the proposed method

statistical analysis for the angular central gaussian distribution on the sphere summary the angular central gaussian distribution is an alternative to the bingham distribution for modeling antipodal symmetric directional data in this paper the statistical theory for the angular central gaussian model is presented some topics treated are maximum likelihood estimation of the parameters testing for uniformity and circularity and principal components analysis comparisons to methods based upon the sample second moments are made via an example

robustness of estimators for directional data on introduit une robustesse de biais standardisee autorisant des considerations de robustesse uniformes pour des espaces parametriques bornes

robust and consistent variable selection in high dimensional generalized linear models generalized linear models are popular for modelling a large variety of data we consider variable selection through penalized methods by focusing on resistance issues in the presence of outlying data and other deviations from assumptions we highlight the weaknesses of widely used penalized m estimators propose a robust penalized quasilikelihood estimator and show that it enjoys oracle properties in high dimensions and is stable in a neighbourhood of the model we illustrate its finite sample performance on simulated and real data

robust principal component analysis for power transformed compositional data geochemical surveys collect sediment or rock samples measure the concentration of chemical elements and report these typically either in weight percent or in parts per million ppm there are usually a large number of elements measured and the distributions are often skewed containing many potential outliers we present a new robust principal component analysis pca method for geochemical survey data that involves first transforming the compositional data onto a manifold using a relative power transformation a flexible set of moment assumptions are made which take the special geometry of the manifold into account the kent distribution moment structure arises as a special case when the chosen manifold is the hypersphere we derive simple moment and robust estimators ro of the parameters which are also applicable in high dimensional settings the resulting pca based on these estimators is done in the tangent space and is related to the power transformation method used in correspondence analysis t

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

doubly robust type estimation for covariate adjustment in latent variable modeling due to the difficulty in achieving a random assignment a quasi experimental or observational study design is frequently used in the behavioral and social sciences if a nonrandom assignment depends on the covariates multiple group structural equation modeling that includes the regression function of the dependent variables on the covariates that determine the assignment can provide reasonable estimates under the condition of correct specification of the regression function however it is usually difficult to specify the correct regression function because the dimensions of the dependent variables and covariates are typically large therefore the propensity score adjustment methods have been proposed since they do not require the specification of the regression function and have been applied to several applied studies however these methods produce biased estimates if the assignment mechanism is incorrectly specified in order to make a more robust inference it would be more useful to develop an estimation method that integrates the regression approach with the propensity score methodology in this study we propose a doubly robust type estimation method for marginal multiple group structural equation modeling this method provides a consistent estimator if either the regression function or the assignment mechanism is correctly specified a simulation study indicates that the proposed estimation method is more robust than the existing methods

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

use of the von mises distribution to analyse continuous proportions summary the von mises distribution in p dimensions is suggested for the analysis of data which are proportions of a continuum such as time or volume the one way analysis ofvariance technique developed by watson 1956 is extended to a multiway layout and the methods are illustrated on data giving proportions of time spent in various activities other examples are briefly mentioned

on the rotational invariant l1 norm pca abstract principal component analysis pca is a powerful tool for dimensionality reduction unfortunately it is sensitive to outliers so that various robust pca variants were proposed in the literature one of the most frequently applied methods for high dimensional data reduction is the rotational invariant l 1 norm pca of ding and coworkers so far no convergence proof for this algorithm was available the main topic of this paper is to fill this gap we reinterpret this robust approach as a conditional gradient algorithm and show moreover that it coincides with a gradient descent algorithm on grassmann manifolds based on the latter point of view we prove global convergence of the whole series of iterates to a critical point using the kurdyka ojasiewicz property of the objective function where we have to pay special attention to so called anchor points where the function is not differentiable

analytic study on linear neutral fractional differential equations this paper is devoted to investigating linear neutral fractional differential equations with constant coefficients the existence uniqueness and iterative formula of the solution are obtained meanwhile dependence of the solution on initial value and the general solution represented by a fundamental solution matrix are discussed several examples are given to illustrate the applications of our results some conclusions in the literature are extended greatly

modeling sparse longitudinal data on riemannian manifolds modern data collection often entails longitudinal repeated measurements that assume values on a riemannian manifold analyzing such longitudinal riemannian data is challenging because of both the sparsity of the observations and the nonlinear manifold constraint addressing this challenge we propose an intrinsic functional principal component analysis for longitudinal riemannian data information is pooled across subjects by estimating the mean curve with local frechet regression and smoothing the covariance structure of the linearized data on tangent spaces around the mean dimension reduction and imputation of the manifold valued trajectories are achieved by utilizing the leading principal components and applying best linear unbiased prediction we show that the proposed mean and covariance function estimates achieve state of the art convergence rates for illustration we study the development of brain connectivity in a longitudinal cohort of alzheimer s disease and normal participants by modeling the connectivity on the manifold of symmetric positive definite matrices with the affine invariant metric in a second illustration for irregularly recorded longitudinal emotion compositional data for unemployed workers we show that the proposed method leads to nicely interpretable eigenfunctions and principal component scores data used in preparation of this article were obtained from the alzheimer s disease neuroimaging initiative database

high order asymptotic expansions for robust tests summary this paper provides high order asymptotic expansions for the m ratio and the signedroot m ratio robust test statistics which allow one to compute accurate approximations to significance or critical values using the edgeworth approximation the bartlett correction the variance correction or the saddlepoint approximation specific results are obtained for the linear regression model with the huber m estimator a monte carlo study illustrates the numerical accuracy of these approximations with respect to the usual first order approximations

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

differentiability of statistical functionals and consistency of the jackknife in statistical applications the unknown parameter of interest can frequently be defined as a functional 0 t f where f is an unknown population statistical inferences about 0 are usually made based on the statistic t fn where fn is the empirical distribution assessing t fn as an estimator of 0 or making large sample inferences usually requires a consistent estimator of the asymptotic variance of t fn asymptotic behaviour of the jackknife variance estimator is closely related to the smoothness of the functional t this paper studies the smoothness of t through the differentiability of t and establishes some general results for the consistency of the jackknife variance estimators the results are applied to some examples in which the statistics t fn are l m estimators and some test statistics

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

hermite and smith normal form algorithms over dedekind domains we show how the usual algorithms valid over euclidean domains such as the hermite normal form the modular hermite normal form and the smith normal form can be extended to dedekind rings in a sequel to this paper we will explain the use of these algorithms for computing in relative extensions of number fields

comprehensive definitions of breakdown points for independent and dependent observations summary we provide a new definition of breakdown in finite samples with an extension to asymptotic breakdown previous definitions centre on defining a critical region for either the parameter or the objective function if for a particular outlier configuration the critical region is entered breakdown is said to occur in contrast with the traditional approach we leave the definition of the critical region implicit our proposal encompasses previous definitions of breakdown in linear and non linear regression settings in some cases it leads to a different and more intuitive notion of breakdown than other procedures that are available an important advantage of our new definition is that it also applies to models for dependent observations where current definitions of breakdown typically fail we illustrate our suggestion by using examples from linear and non linear regression and time series

regression for compositional data by using distributions defined on the hypersphere summary compositional data can be transformed to directional data by the square root transformation and then modelled by using distributions defined on the hypersphere one advantage of this approach is that zero components are catered for naturally in the models the kent distribution for directional data is a good candidate model because it has a sufficiently general covariance structure we propose a new regression model which models the mean direction of the kent distribution as a function of a vector of covariates our estimators can be regarded as asymptotic maximum likelihood estimators we show that these estimators perform well and are suitable for typical compositional data sets including those with some zero components

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

tyler shape depth summaryin many problems from multivariate analysis the parameter of interest is a shape matrix a normalized version of the corresponding scatter or dispersion matrix in this article we propose a notion of depth for shape matrices that involves data points only through their directions from the centre of the distribution we refer to this concept as tyler shape depth since the resulting estimator of shape namely the deepest shape matrix is the median based counterpart of the m estimator of shape due to tyler 1987 besides estimation shape depth like its tyler antecedent also allows hypothesis testing on shape its main benefit however lies in the ranking of the shape matrices it provides the practical relevance of which is illustrated by applications to principal component analysis and shape based outlier detection we study the invariance quasi concavity and continuity properties of tyler shape depth the topological and boundedness properties of the corresponding depth regions and the existence of a deepest shape matrix and we prove fisher consistency in the elliptical case finally we derive a glivenko cantelli type result and establish almost sure consistency of the deepest shape matrix estimator

projection estimates of multivariate location in this paper we study the maximum asymptotic bias of the projection estimate for multivariate location based on univariate estimates of location and dispersion in particular we study the projection estimate that uses the median and median absolute deviation about the median mad as univariate location and dispersion estimates respectively this estimator may be considered a natural affine equivariant multivariate median for spherical distributions the maximum bias of this estimate depends only on the marginal distributions and not on the dimension and is approximately twice the maximum bias of the univariate median we also show that for multivariate normal distributions its maximum bias compares favorably with those of the donoho stahel minimum volume ellipsoid and minimum covariance determinant estimates in all these cases the maximum bias increases with the dimension p

a practical affine equivariant multivariate median summary a robust affine equivariant estimator of location for multivariate data is proposed which becomes the univariate median for data of dimension one the estimator is robust in the sense that it has a bounded influence function a positive breakdown value and has high efficiency compared to the sample mean for heavy tailed distributions perhaps its greatest strength is that unlike other affine equivariant multivariate medians it is easily computed for data in any practical dimension

convergence and fluctuations of regularized tyler estimators this paper studies the behavior of regularized tyler estimators rtes of scatter matrices the advantages of these estimators are twofold first they guarantee by construction a good conditioning of the estimate and second being derivatives of robust tyler estimators they inherit their robustness properties notably their resilience to outliers nevertheless one major problem that poses the use of rtes is represented by the question of setting the regularization parameter while a high value of is likely to push all the eigenvalues away from zero it comes at the cost of a larger bias with respect to the population covariance matrix a deep understanding of the statistics of rtes is essential to come up with appropriate choices for the regularization parameter this is not an easy task and requires working under asymptotic regimes wherein the number of observations n and or their size n increase together first asymptotic results have recently been obtained when n and n are large and commensurable interestingly no results concerning the regime of n going to infinity with n fixed exist this motivates our work in particular we prove in this paper that the rtes converge to a deterministic matrix when n with n fixed which is expressed as a function of the theoretical covariance matrix we also derive the fluctuations of the rtes around this limit and establish that these fluctuations converge in distribution to a multivariate gaussian distribution with parameters depending on the population covariance and the regularization coefficient

new criteria on finite time stability of fractional order hopfield neural networks with time delays in this article the finite time stability fts of fractional order hopfield neural networks with time delays fhnntds is studied a widely used inequality in investigating the stability of the fractional order neural networks is fractional order gronwall inequality related to the mittag leffler function which cannot be directly used to study the stability of the factional order neural networks with time delays in the existing works related to fractional order gronwall inequality with time delays the order 0 was divided into two cases 0 0 5 and 0 5 in this article a new fractional order gronwall integral inequality with time delay and the unified form for all the fractional order 0 is developed which can be widely applied to investigate fts of various fractional order systems with time delays based on this new inequality a new criterion for the fts of fhnntds is derived compared with the existing criteria in which fractional order 0 1 was divided into two cases 0 0 5 and 0 5 1 the obtained results in this article are presented in the unified form of fractional order 0 1 and convenient to verify more importantly the criteria in this article are less conservative than some existing ones finally two numerical examples are given to demonstrate the validity of the proposed results

blockwise bootstrapped empirical process for stationary sequences we apply the bootstrap for general stationary observations proposed by kunsch to the empirical process for p dimensional random vectors it is known that the empirical process in the multivariate case converges weakly to a certain gaussian process we show that the bootstrapped empirical process converges weakly to the same gaussian process almost surely assuming that the block length l for constructing bootstrap replicates satisfies l n o n 1 2 varepsilon 0 varepsilon frac 1 2 and l n rightarrow infty an example where the multivariate setup arises are the robust gm estimates in an autoregressive model we prove the asymptotic validity of the bootstrap approximation by showing that the functional associated with the gm estimates is frechet differentiable

some tests for common principal component subspaces in several groups summary an approximate test based on sample eigenprojections is obtained for testing the hypothesis that the subspaces spanned by the first m principal components of several different covariance matrices are identical this procedure can fairly easily be extended to an analogous test of hypothesis concerning correlation matrices extensions to a robust principal components analysis are also considered in recent years principal components analysis has been generalized from the typical single population setting to that of several populations this paper focuses on an extension that may be useful in those situations in which one needs a variance preserving simultaneous reduction of dimensionality in several groups this sort of reduction will be possible when the subspace spanned by the most important principal component vectors of a particular group is the same as that of any other group formally suppose the same p variables are being measured on objects in g different groups with the covariance