on the projections of isotropic distributions the class of distributions on rm 1 c m oo which are the m dimensional marginal distributions of orthogonally invariant distributions on rm n is characterized this result is then used to provide a partial answer to the following question given a symmetric distribution on r and an integer n 2 under what conditions will there exist a random vector x e rn such that a x has the given distribution up to a positive scale factor for all a 0 a e rm

on extension results for n cyclically monotone operators in reflexive banach spaces abstract in this paper we provide some extension results for n cyclically monotone operators in reflexive banach spaces by making use of the fenchel duality in this way we give a positive answer to a question posed by bauschke and wang 2007 4

fitzpatrick functions and continuous linear monotone operators the notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex lower semicontinuous and proper function and any not necessarily symmetric continuous linear positive operator it was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing fitzpatrick functions in this paper we study fitzpatrick functions of continuous linear monotone operators defined on a hilbert space a novel characterization of skew operators is presented a result by brezis and haraux is reproved using the fitzpatrick function we investigate the fitzpatrick function of the sum of two operators and we show that a known upper bound is actually exact in finite dimensional and more general settings cyclic monotonicity properties are also analyzed and closed forms of the fitzpatrick functions of all orders are provided for all rotators in the euclidean plane

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

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

iterative coding for network coding we consider communication over a noisy network under randomized linear network coding possible error mechanisms include node or link failures byzantine behavior of nodes or an overestimate of the network min cut building on the work of kotter and kschischang we introduce a systematic oblivious random channel model within this model codewords contain a header this is the systematic part the header effectively records the coefficients of the linear encoding functions thus simplifying the decoding task under this constraint errors are modeled as random low rank perturbations of the transmitted codeword we compute the capacity of this channel and we define an error correction scheme based on random sparse graphs and a low complexity decoding algorithm by optimizing over the code degree profile we show that this construction achieves the channel capacity in complexity which is jointly quadratic in the number of coded information bits and sublogarithmic in the error probability

a comparison of constraint qualifications in infinite dimensional convex programming in this paper the relationships between various constraint qualifications for infinite dimensional convex programs are investigated using robinson s refinement of the duality result of rockafellar it is demonstrated that the constraint qualification proposed by rockafellar provides a systematic mechanism for comparing many constraint qualifications as well as establishing new results in different topological environments

on the stahel donoho estimator and depth weighted means of multivariate data the depth of multivariate data can be used to construct weighted means as robust estimators of location the use of projection depth leads to the stahel donoho estimator as a special case in contrast to maximal depth estimators the depth weighted means are shown to be asymptotically normal under appropriate conditions met by depth functions commonly used in the current literature we also confirm through a finite sample study that the stahel donoho estimator achieves a desirable balance between robustness and efficiency at gaussian models

asymptotic behaviour of s estimates of multivariate location parameters and dispersion matrices it is shown under appropriate conditions that rousseeuw s minimum volume estimator and other s estimators of multivariate location and dispersion parameters are consistent under certain differentiability conditions the estimates are asymptotically normally distributed with a norming factor of n 1 2

multivariate analysis by data depth descriptive statistics graphics and inference with discussion and a rejoinder by liu and singh a data depth can be used to measure the depth or outlyingness of a given multivariate sample with respect to its underlying distribution this leads to a natural center outward ordering of the sample points based on this ordering quantitative and graphical methods are introduced for analyzing multivariate distributional characteristics such as location scale bias skewness and kurtosis as well as for comparing inference methods all graphs are one dimensional curves in the plane and can be easily visualized and interpreted a sunburst plot is presented as a bivariate generalization of the box plot dd depth versus depth plots are proposed and examined as graphical inference tools some new diagnostic tools for checking multivariate normality are introduced one of them monitors the exact rate of growth of the maximum deviation from the mean while the others examine the ratio of the overall dispersion to the dispersion of a certain central region the affine invariance property of a data depth also leads to appropriate invariance properties for the proposed statistics and methods

classifier selection using the predicate depth typically one approaches a supervised machine learning problem by writing down an objective function and finding a hypothesis that minimizes it this is equivalent to finding the maximum a posteriori map hypothesis for a boltzmann distribution however map is not a robust statistic we present an alternative approach by defining a median of the distribution which we show is both more robust and has good generalization guarantees we present algorithms to approximate this median one contribution of this work is an efficient method for approximating the tukey median the tukey median which is often used for data visualization and outlier detection is a special case of the family of medians we define however computing it exactly is exponentially slow in the dimension our algorithm approximates such medians in polynomial time while making weaker assumptions than those required by previous work

on the relation between s estimators and m estimators of multivariate location and covariance we discuss the relation between s estimators and m estimators of multivariate location and covariance as in the case of the estimation of a multiple regression parameter s estimators are shown to satisfy first order conditions of m estimators we show that the influence function if x s f of s functionals exists and is the same as that of corresponding m functionals also we show that s estimators have a limiting normal distribution which is similar to the limiting normal distribution which is similar to the limiting normal distribution of m estimators finally we compare asymptotic variances and breakdown point of both types of estimators

projection based depth functions and associated medians order n uniform consistency depth regions and contours induced from projection based depth functions are investigated structural properties of depth regions and contours and general continuity and convergence results of sample depth regions are obtained affine equivariant multivariate medians induced from projection based depth functions are probed the limiting distributions as well as the strong and order n consistency of the sample projection medians are established the finite sample performance of projection medians is compared with that of a leading depth induced median the tukey halfspace median induced from the tukey halfspace depth function it turns out that with appropriate choices of univariate location and scale estimators the projection medians have a very high finite sample breakdown point and relative efficiency much higher than those of the halfspace median based on the results obtained it is found that projection depth functions and projection medians behave very well overall compared with their competitors and consequently are good alternatives to statistical depth functions and affine equivariant multivariate location estimators respectively

permutation tests for equality of distributions in high dimensional settings motivated by applications in high dimensional settings we suggest a test of the hypothesis h 0 that two sampled distributions are identical it is assumed that two independent datasets are drawn from the respective populations which may be very general in particular the distributions may be multivariate or infinite dimensional in the latter case representing for example the distributions of random functions from one euclidean space to another our test uses a measure of distance between data this measure should be symmetric but need not satisfy the triangle inequality so it is not essential that it be a metric the test is based on ranking the pooled dataset with respect to the distance and relative to any fixed data value and repeating this operation for each fixed datum a permutation argument enables a critical point to be chosen such that the test has concisely known significance level conditional on the set of all pairwise distances

a note on tyler s modification of the mad for the stahel donoho estimator in 1994 tyler proposed a modification of the mad in order to achieve the maximal possible finite sample breakdown point for the stahel donoho estimator based upon the median and the modified mad a proof is not given however and as far as we know has not been published yet moreover there seems to be an error concerning the exact definition of the modified mad which is needed to achieve this goal for the sake of theoretical completeness and in order to correct this error we calculate the finite sample breakdown point of modifications of the mad and point out which of these modifications lead to a stahel donoho estimator with maximal finite sample breakdown point

fitzpatrick functions cyclic monotonicity and rockafellar s antiderivative several deeper results on maximal monotone operators have recently found simpler proofs using fitzpatrick functions in this paper we study a sequence of fitzpatrick functions associated with a monotone operator the first term of this sequence coincides with the original fitzpatrick function and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties it is shown that for any maximal cyclically monotone operator the pointwise supremum of the sequence of fitzpatrick functions is closely related to rockafellar s antiderivative several examples are explicitly computed for the purpose of illustration in contrast to rockafellar s result a maximal 3 cyclically monotone operator need not be maximal monotone a simplified proof of asplund s observation that the rotation in the euclidean plane by n is n cyclically monotone but not n 1 cyclically monotone is provided the fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail we conclude with a new proof of moreau s result concerning the convexity of the set of proximal mappings

finite sample breakdown points of projection based multivariate location and scatter statistics finite sample breakdown points are obtained for two classes of projection based multivariate location and scatter statistics the stahel donoho statistics and the maronna yohai statistics the definition of these multivariate statistics are dependent on the value of location and scale statistics for all univariate projections of the data and consequently their properties depend on the nature of the corresponding univariate location and scale statistics used on the projected data the finite sample breakdown points of the multivariate statistics though are not directly related to those of the corresponding univariate location and scale statistics a uniform finite sample breakdown point concept is needed the median and the median absolute deviation about the median m a d are one possible choice for the univariate location and scale statistics respectively for sparse data sets in high dimensions though it is not recommended that the m a d be used as the univariate scale statistic for the projected data since its uniform finite sample breakdown point is shown to be much less than optimum a simple modification to the m a d however is shown to alleviate this problem for various reasons one may wish to consider univariate location and scale statistics other than the median and the m a d respectively a very broad and natural class of univariate location and scale statistics to consider for the projected data is the simultaneous m estimates of location and scale new results on their breakdown properties are given in this paper implicit formulas for the breakdown points of monotonic simultaneous m estimates of location and scale are known and they tend to imply rather low breakdown points for smooth choices of the defining weight functions it is shown here that this phenomenon does not occur for a large class of nonmonotonic simultaneous m estimates of location and scale furthermore explicit rather than implicit expressions for the uniform finite sample breakdown points are given for these nonmonotonic m estimates

robust estimate of excitations in mechanical systems using m estimators theoretical background and numerical applications abstract model based methods are often used along with least squares to estimate or to identify in equivalent but in more engineering terms dynamic forces parameters and malfunctions in mechanical systems starting from experimental vibrations the effectiveness of these methods broadly proven and documented by several cases of study can be reduced if the model of the system is not accurate or if the experimental data are corrupted by noise especially if the mean value of the noise is not null or if biases are present a possible solution is the use of robust estimation techniques instead of traditional least squares in the ambit of model based identification the author proposes the application of the m estimators and discusses the problems related to their application to excitation identification in mechanical systems in this paper the necessary theory is presented in detail introducing several concepts of statistics in order to properly introduce the concept of robust estimation and the required algorithms based on iterative re weighted least squares are described then the different types of m estimators proposed in literature are introduced their performances with regard to mechanical applications are evaluated by means of a theoretical analysis and a couple of simple numerical examples a single input single output and a multiple inputs multiple outputs systems moreover the problem of the scale parameter which is not discussed in literature for complex numbers as the vibrations are conveniently represented is analyzed and a solution is proposed using a concept related to the data depth

a new condition for maximal monotonicity via representative functions abstract in this paper we give a weaker sufficient condition for the maximal monotonicity of the operator s a t a where s x x t y y are two maximal monotone operators a x y is a linear continuous mapping and x y are reflexive banach spaces we prove that our condition is weaker than the generalized interior point conditions given so far in the literature this condition is formulated using the representative functions of the operators involved in particular we rediscover some sufficient conditions given in the past using the so called fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator respectively

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

asymptotics of reweighted estimators of multivariate location and scatter we investigate the asymptotic behavior ofa weighted sample mean and covariance where the weights are determined by the mahalanobis distances with respect to initial robust estimators we derive an explicit expansion for the weighted estimators from this expansion it can be seen that reweighting does not improve the rate ofconvergence ofthe initial estimators we also show that ifone uses smooth s estimators to determine the weights the weighted estimators are asymptotically normal finally we will compare the efficiency and local robustness of the reweighted s estimators with two other improvements of s estimators estimators and constrained m estimators

maximal monotonicity for the precomposition with a linear operator we give the weakest constraint qualification known to us that ensures the maximal monotonicity of the operator a circ t circ a when a is a linear continuous mapping between two reflexive banach spaces and t is a maximal monotone operator as a special case we get the weakest constraint qualification that guarantees the maximal monotonicity of the sum of two maximal monotone operators on a reflexive banach space then we give a weak constraint qualification assuring the bre zis haraux type approximation of the range of the subdifferential of the precomposition to a of a proper convex lower semicontinuous function in nonreflexive banach spaces extending and correcting in a special case an older result due to riahi

influence function and maximum bias of projection depth based estimators location estimators induced from depth functions increasingly have been pursued and studied in the literature among them are those induced from projection depth functions these projection depth based estimators have favorable properties among their competitors in particular they possess the best possible finite sample breakdown point robustness however robustness of estimators cannot be revealed by the finite sample breakdown point alone the influence function gross error sensitivity maximum bias and contamination sensitivity are also important aspects of robustness in this article we study these other robustness aspects of two types of projection depth based estimators projection medians and projection depth weighted means the latter includes the stahel donoho estimator as a special case exact maximum bias the influence function and contamination and gross error sensitivity are derived and studied for both types of estimators sharp upper bounds for the maximum bias and the influence functions are established comparisons based on these robustness criteria reveal that the projection depth based estimators enjoy desirable local as well as global robustness and are very competitive among their competitors

robust multivariate regression we introduce a robust method for multivariate regression based on robust estimation of the joint location and scatter matrix of the explanatory and response variables as a robust estimator of location and scatter we use the minimum covariance determinant mcd estimator of rousseeuw based on simulations we investigate the finite sample performance and robustness of the estimator to increase the efficiency we propose a reweighted estimator selected from several possible reweighting schemes the resulting multivariate regression does not need much computation time and is applied to real datasets we show that the multivariate regression estimator has the appropriate equivariance properties has a bounded influence function and inherits the breakdown value of the mcd estimator these theoretical robustness properties confirm the good finite sample results obtained from the simulations

on a notion of data depth based on random simplices for a distribution f on mathbb r p and a point x in mathbb r p the simplical depth d x is introduced which is the probability that the point x is contained inside a random simplex whose vertices are p 1 independent observations from f mathematically and heuristically it is argued that d x indeed can be viewed as a measure of depth of the point x with respect to f an empirical version of d cdot gives rise to a natural ordering of the data points from the center outward the ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and l statistics this generalized sample median and l statistics are affine equivariant

depth weighted scatter estimators general depth weighted scatter estimators are introduced and investigated for general depth functions we find out that these affine equivariant scatter estimators are fisher consistent and unbiased for a wide range of multivariate distributions and show that the sample scatter estimators are strong and n consistent and asymptotically normal and the influence functions of the estimators exist and are bounded in general we then concentrate on a specific case of the general depth weighted scatter estimators the projection depth weighted scatter estimators which include as a special case the well known stahel donoho scatter estimator whose limiting distribution has long been open until this paper large sample behavior including consistency and asymptotic normality and efficiency and finite sample behavior including breakdown point and relative efficiency of the sample projection depth weighted scatter estimators are thoroughly investigated the influence function and the maximum bias of the projection depth weighted scatter estimators are derived and examined unlike typical high breakdown competitors the projection depth weighted scatter estimators can integrate high breakdown point and high efficiency while enjoying a bounded influence function and a moderate maximum bias curve comparisons with leading estimators on asymptotic relative efficiency and gross error sensitivity reveal that the projection depth weighted scatter estimators behave very well overall and consequently represent very favorable choices of affine equivariant multivariate scatter estimators

coding for errors and erasures in random network coding the problem of error control in random linear network coding is considered a ldquononcoherentrdquo or ldquochannel obliviousrdquo model is assumed where neither transmitter nor receiver is assumed to have knowledge of the channel transfer characteristic motivated by the property that linear network coding is vector space preserving information transmission is modeled as the injection into the network of a basis for a vector space v and the collection by the receiver of a basis for a vector space u a metric on the projective geometry associated with the packet space is introduced and it is shown that a minimum distance decoder for this metric achieves correct decoding if the dimension of the space v capu is sufficiently large if the dimension of each codeword is restricted to a fixed integer the code forms a subset of a finite field grassmannian or equivalently a subset of the vertices of the corresponding grassmann graph sphere packing and sphere covering bounds as well as a generalization of the singleton bound are provided for such codes finally a reed solomon like code construction related to gabidulin s construction of maximum rank distance codes is described and a sudan style ldquolist 1rdquo minimum distance decoding algorithm is provided

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

multidimensional trimming based on projection depth as estimators of location parameters univariate trimmed means are well known for their robustness and efficiency they can serve as robust alternatives to the sample mean while possessing high efficiencies at normal as well as heavy tailed models this paper introduces multidimensional trimmed means based on projection depth induced regions robustness of these depth trimmed means is investigated in terms of the influence function and finite sample breakdown point the influence function captures the local robustness whereas the breakdown point measures the global robustness of estimators it is found that the projection depth trimmed means are highly robust locally as well as globally asymptotics of the depth trimmed means are investigated via those of the directional radius of the depth induced regions the strong consistency asymptotic representation and limiting distribution of the depth trimmed means are obtained relative to the mean and other leading competitors the depth trimmed means are highly efficient at normal or symmetric models and overwhelmingly more efficient when these models are contaminated simulation studies confirm the validity of the asymptotic efficiency results at finite samples

elliptical graphical modelling we propose elliptical graphical models based on conditional uncorrelatedness as a gen eralization of gaussian graphical models by letting the population distribution be elliptical instead of normal allowing the tting of data with arbitrarily heavy tails we study the class of propor tionally ane equivariant scatter estimators and show how they can be used to perform elliptical graphical modelling leading to a new class of partial correlation estimators and analogues of the classical deviance test general expressions for the asymptotic variance of partial correlation esti mators unconstrained and under decomposable models are given and the asymptotic chi square approximation of the pseudo deviance test statistic is proved the feasibility of our approach is demonstrated by a simulation study using among others tyler s scatter estimator which is distribution free within the elliptical model our approach provides a robustication of gauss ian graphical modelling the latter is likelihood based and known to be very sensitive to model misspecication and outlying observations 1 introduction and notation the statistical theory of undirected graphical models for continuous variables is usually based on the assumption of multivariate normality in practice data may deviate from the normal model in various ways outliers and heavy tails pose a problem of particular gravity they frequently occur and the normal likelihood methods such as the sample covariance matrix are very susceptible to them our objective is to deal with heavy tailed data and to safeguard graphical modelling against the impact of faulty outliers we restrict our attention to the case where we have only continuous variables and only undirected edges joint multivariate normality is often assumed in this situation and the statistical methodology is called gaussian graphical modelling we propose the class of elliptical distributions as a more general model and call our approach elliptical graphical modelling the lack of robustness of gaussian graphical modelling has been noted by several authors four proposals of robust approaches to gaussian graphical modelling are known to us becker 2005 and gottard and pacillo 2010 suggest replacing the sample covariance matrix by the reweighted minimum covariance determinant estimator miyamura and kano 2006 propose an alternative m type estimation and finegold drton arxiv 1009 3669 consider robustied versions of the graphical lasso by friedman et al 2008 this article delivers a systematic treatment of the plug in approach used in the rst two ref erences we show that the sample covariance matrix may be replaced by any ane equivariant

packing and covering properties of rank metric codes this paper investigates packing and covering properties of codes with the rank metric first we investigate packing properties of rank metric codes then we study sphere covering properties of rank metric codes derive bounds on their parameters and investigate their asymptotic covering properties

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

communication over finite field matrix channels this paper is motivated by the problem of error control in network coding when errors are introduced in a random fashion rather than chosen by an adversary an additive multiplicative matrix channel is considered as a model for random network coding the model assumes that n packets of length m are transmitted over the network and up to t erroneous packets are randomly chosen and injected into the network upper and lower bounds on capacity are obtained for any channel parameters and asymptotic expressions are provided in the limit of large field or matrix size a simple coding scheme is presented that achieves capacity in both limiting cases the scheme has decoding complexity o n 2 m and a probability of error that decreases exponentially both in the packet length and in the field size in bits extensions of these results for coherent network coding are also presented

the influence function and maximum bias of tukey s median tukey s median is among one of the earliest known high breakdown point multivariate location statistics aside from its breakdown point though little else appears to be known about its robustness properties in this paper we investigate other aspects of tukey s median and in particular we derive and study its influence function and its maximum contamination bias function when judged by these other robustness criteria tukey s median again proves to be highly robust

structural properties and convergence results for contours of sample statistical depth functions statistical depth functions have become increasingly used in nonparametric inference for multivariate data here the contours of such functions are studied structural properties of the regions enclosed by contours such as affine equivariance nestedness connectedness and compactness and almost sure convergence results for sample depth contours are established also specialized results are established for some popular depth functions including halfspace depth and for the case of elliptical distributions finally some needed foundational results on almost sure convergence of sample depth functions are provided

finite field matrix channels for network coding in 2010 silva et al studied certain classes of finite field matrix channels in order to model random linear network coding where exactly t random errors are introduced in this paper we consider a generalization of these matrix channels where the number of errors is not required to be constant indeed the number of errors may follow any distribution we show that a capacity achieving input distribution can always be taken to have a very restricted form the distribution should be uniform given the rank of the input matrix this result complements and is inspired by a paper of nobrega et al which establishes a similar result for a class of matrix channels that model network coding with link erasures our result shows that the capacity of our channels can be expressed as maximization over probability distributions on the set of possible ranks of input matrices a set of linear rather than exponential size

on fractile transformation of covariates in regression the need for comparing two regression functions arises frequently in statistical applications comparison of the usual regression functions is not very meaningful in situations where the distributions and the ranges of the covariates are different for the populations for instance in econometric studies the prices of commodities and people s incomes observed at different time points may not be on comparable scales due to inflation and other economic factors in this article we describe a method of standardizing the covariates and estimating the transformed regression function which then become comparable we develop smooth estimates of the fractile regression function and study its statistical properties analytically as well as numerically we also provide a few real examples that illustrate the difficulty in comparing the usual regression functions and motivate the need for the fractile transformation our analysis of the real examples leads to new and useful statistical conclusions that are missed b

min max bias robust regression abstract this paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over epsilon contamination neighborhoods for the joint distribution of the response and carriers two classes of estimates are treated 1 m estimates with bounded function rho applied to the scaled residuals using a very general class of scale estimates and 2 bounded influence function type generalized m estimates estimates in the first class are obtained as the solution of a minimization problem while estimates in the second class are specified by an estimating equation the first class of m estimates is sufficiently general to include both huber proposal 2 simultaneous estimates of regression coefficients and residuals scale and rousseeuw yohai s estimates of regression it is shown than an s estimate based on a jump function type rho solves the min max bias problem for the class of m estimates with very general scale this estimate is obtained by the minimization of the alpha quantile of the squared residuals where alpha epsilon depends on the fraction of contamination epsilon when epsilon approaches limit of 5 alpha epsilon approaches limit of 5 and the min max estimator approaches the least median of squared residuals estimator introduced by rousseeuw for the bounded influence class of gm estimates it is shown the a sign type nonlinearity yields the min max estimate this estimate coincides with the minimum gross error sensitivity gm estimate

a quality index based on data depth and multivariate rank tests let f and g be the distribution functions of two given populations on rp p 1 we introduce and study a parameter q q f g which measures the overall outlyingness of population g relative to population f the parameter q can be defined using any concept of data depth its value ranges from 0 to 1 and is 5 when f and g are identical we show that within the dass of elliptical distributions when g departs from f in location or g has a larger spread or both the value of q dwindles down from 5 hence q can be used to detect the loss of accuracy or precision of a manufacturing process and thus it should serve as an important measure in quality assurance this in fact is the reason why we refer to q as a quality index in this article in addition to studying the properties of q we provide an exact rank test for testing q 5 vs q 5 this can be viewed as a multivariate analog of wilcoxon s rank sum test the tests proposed here have power against location change and scale increase simultaneo