exact critical exponent for the shortest path scaling function in percolation it is shown that the critical exponentg1 related to pair connectiveness and shortest path or chemical distance scaling recently studied by porto et al dokholyan et al and grassberger can be found exactly in two dimensions by using a crossing probability result of cardy with the outcomeg1d 25 24 this prediction is consistent with existing simulation results

frechet distance as a tool for diagnosing multivariate data abstract outliers can have dramatic effects on results from statistical analyses and on conclusions based on statistical analyses it is therefore important to detect outliers and influential points methods for detecting outliers in multivariate data may be based on assessments of changes in the estimate of the mean vector or changes in the estimate of the scatter matrix as data points are perturbed here we propose a method based on the frechet distance between the observed empirical distributions of the unperturbed and perturbed data as these distributions involve both the mean vector and the scatter matrix the resulting diagnostics are functions of both mean and scatter the proposed diagnostics are illustrated by two examples

nearest neighbor clutter removal for estimating features in spatial point processes abstract we consider the problem of detecting features in spatial point processes in the presence of substantial clutter one example is the detection of minefields using reconnaissance aircraft images that identify many objects that are not mines our solution uses kth nearest neighbor distances of points in the process to classify them as clutter or otherwise the observed kth nearest neighbor distances are modeled as a mixture distribution the parameters of which are estimated by a simple em algorithm this method allows for detection of generally shaped features that need not be path connected in the minefield example this method yields high detection and low false positive rates another application to outlining seismic faults is considered with some success the method works well in high dimensions the method can also be used to produce very high breakdown point robust estimators of a covariance matrix

on the universality of distribution of ranked cluster masses at critical percolation the distribution of masses of clusters smaller than the infinite cluster is evaluated at the percolation threshold the clusters are ranked according to their masses and the distribution p m ld r of the scaled masses m for any rank r shows a universal behaviour for different lattice sizes l d is the fractal dimension for different ranks however there is a universal distribution function only in the large rank limit i e p m ld r r y g mry ld y and are defined in the text where the universal scaling function g is found to be gaussian in nature

a multivariate bayesian model for embryonic growth most longitudinal growth curve models evaluate the evolution of each of the anthropometric measurements separately when applied to a reference population this exercise leads to univariate reference curves against which new individuals can be evaluated however growth should be evaluated in totality that is by evaluating all body characteristics jointly recently cole et al suggested the superimposition by translation and rotation sitar model which expresses individual growth curves by three subject specific parameters indicating their deviation from a flexible overall growth curve this model allows the characterization of normal growth in a flexible though compact manner in this paper we generalize the sitar model in a bayesian way to multiple dimensions the multivariate sitar model allows us to create multivariate reference regions which is advantageous for prediction the usefulness of the model is illustrated on longitudinal measurements of embryonic growth obtained in the first semester of pregnancy collected in the ongoing rotterdam predict study further we demonstrate how the model can be used to find determinants of embryonic growth copyright 2015 john wiley sons ltd

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

the masking breakdown point of multivariate outlier identification rules abstract in this article we consider simultaneous outlier identification rules for multivariate data generalizing the concept of so called outlier identifiers as presented by davies and gather for the case of univariate samples such multivariate outlier identifiers are based on estimators of location and covariance therefore it seems reasonable that characteristics of the estimators influence the behavior of outlier identifiers several authors mentioned that using estimators with low finite sample breakdown point is not recommended for identifying outliers to give a formal explanation we investigate how the finite sample breakdown points of estimators used in these identification rules influence the masking behavior of the rules

diagnostics in linear discriminant analysis abstract some new diagnostic measures in discriminant analysis are proposed they can be expressed in terms of the two fundamental influence statistics in discriminant analysis d i 2 and i a theorem on the asymptotic distributions of the fundamental statistics is derived based on the theorem the proposed measures can be shown to be asymptotically distributed as functions of independent chi squared and standard normal random variables critical values and expected quantiles of the measures can then be constructed hence influential observations are detected using q q plots and significance tests two measures have analogous forms in regression the theorem is also useful for getting the asymptotic distributions of existing measures that are functions of d i 2 and i a comparison of the diagnostics in linear discriminant analysis linear regression and linear logistic regression discriminant analysis is made although discriminant coefficients can be determined under a regression model regress

a critical exponent for shortest path scaling in continuum percolation we carry out monte carlo experiments to study the scaling behavior of shortest path lengths in continuum percolation these studies suggest that the critical exponent governing this scaling is the same for both continuum and lattice percolation we use splitting a technique that has not yet been fully exploited in the physics literature to increase the speed of our simulations this technique can also be applied to other models where clusters are grown sequentially

multiple imputation and analysis for high dimensional incomplete proteomics data multivariable analysis of proteomics data using standard statistical models is hindered by the presence of incomplete data we faced this issue in a nested case control study of 135 incident cases of myocardial infarction and 135 pair matched controls from the framingham heart study offspring cohort plasma protein markers k 861 were measured on the case control pairs n 135 and the majority of proteins had missing expression values for a subset of samples in the setting of many more variables than observations k n we explored and documented the feasibility of multiple imputation approaches along with subsequent analysis of the imputed data sets initially we selected proteins with complete expression data k 261 and randomly masked some values as the basis of simulation to tune the imputation and analysis process we randomly shuffled proteins into several bins performed multiple imputation within each bin and followed up with stepwise selection using conditional logistic regression within each bin this process was repeated hundreds of times we determined the optimal method of multiple imputation number of proteins per bin and number of random shuffles using several performance statistics we then applied this method to 544 proteins with incomplete expression data 40 missing values from which we identified a panel of seven proteins that were jointly associated with myocardial infarction

statistical quality control of variety trial data summary i propose detection criteria for identifying an abnormal or erroneous data vector provided by a single variety trial in a longer series of variety trials the test criteria are based on the linear effects estimated separately for each studied trial using global variance components estimated from the whole series of variety trials the criteria comprise three mutually independent test statistics the first one is a quadratic form in the estimated fixed effects the second one is a quadratic form in the estimated realized linear random effects not including the residual effects and the third one is a quadratic form in the estimated realized residual effects under the null hypothesis defining a valid data vector the three quadratics have independent x2 distributions under natural alternative hypotheses they have noncentral x2 distributions decomposing the total variation of the data vector studied into quadratic forms due to different types of the realized linear effects intuitively justifies the resulting test criteria the decomposition may also be used to show that the resulting tests are likelihood ratio tests i further present computational procedures that allow us to dispense with the need for prior estimation of the linear effects

fast very robust methods for the detection of multiple outliers abstract a few repeats of a simple forward search from a random starting point are shown to provide sufficiently robust parameter estimates to reveal masked multiple outliers the stability of the patterns obtained is exhibited by the stalactite plot the robust estimators used are least median of squares for regression and the minimum volume ellipsoid for multivariate outliers the forward search also has potential as an algorithm for calculation of these parameter estimates for large problems parallel computing provides appreciable reduction in computational time

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

the use and interpretation of residuals based on robust estimation abstract residual plots and diagnostic techniques are important tools for examining the fit of a regression model in the case of least squares fits plots of residuals provide a visual assessment of the adequacy of various aspects of the fitted model an important question is whether plots of robust residuals can be interpreted in the same manner as their least squares counterparts this article addresses this problem for two popular classes of robust estimates m estimates and gm estimates of the mallows and schweppe types first order properties of the residuals and fitted values are derived under correct and misspecified models these properties are insightful on the general interpretability of robust residual plots and on their ability to detect curvature in misspecified models the results of a simulation study consisting of tests for randomness and curvature in residual plots supports these theoretical properties standardization of robust residuals is also presented

trimming and likelihood robust location and dispersion estimation in the elliptical model robust estimators of location and dispersion are often used in the elliptical model to obtain an uncontaminated and highly representative subsample by trimming the data outside an ellipsoid based in the associated mahalanobis distance here we analyze some one or k step maximum likelihood estimators computed on a subsample obtained with such a procedure we introduce different models which arise naturally from the ways in which the discarded data can be treated leading to truncated or censored likelihoods as well as to a likelihood based on an only outliers gross errors model results on existence uniqueness robustness and asymptotic properties of the proposed estimators are included a remarkable fact is that the proposed estimators generally keep the breakdown point of the initial robust estimators but they could improve the rate of convergence of the initial estimator because our estimators always converge at rate n 1 2 independently of the rate of convergence of the initial estimator

differential galois theory of linear difference equations we present a galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations with this we are able to reprove holder s theorem that the gamma function satisfies no polynomial differential equation and are able to give general results that imply for example that no differential relationship holds among solutions of certain classes of q hypergeometric equations

investigating the geometry of a p dimensional data set abstract this article concerns itself with exploratory data analysis in hyperspace and discusses a method of data reduction intended to allow insight into the geometry of a p dimensional data set where p 2 the essential idea is to examine various views through the data set using a suitably chosen line the baseline for defining a viewing direction for a given data point its distance from a suitable point on the baseline called the vertex and the angular separation between the baseline and the line connecting the vertex to the data point are taken as coordinates of an ordinary polar coordinate plot in two dimensions the bulk of the article discusses empirical evidence for the utility of such a plot which can be referred to as a coneplot because points are essentially identified with the intersection in p dimensions of a hypersphere and a ray of a hypercone

pair connectedness and shortest path scaling in critical percolation we present high statistics data on the distribution of shortest path lengths between two near by points on the same cluster at the percolation threshold our data are based on a new and very efficient algorithm for d 2 they clearly disprove a recent conjecture by m porto et al 1998 phys rev e 58 r5205 our data also provide upper bounds on the probability that two near by points are on different infinite clusters

socp relaxation bounds for the optimal subset selection problem applied to robust linear regression abstract this paper deals with the problem of finding the globally optimal subset of h elements from a larger set of n elements in d space dimensions so as to minimize a quadratic criterion with an special emphasis on applications to computing the least trimmed squares estimator ltse for robust regression the computation of the ltse is a challenging subset selection problem involving a nonlinear program with continuous and binary variables linked in a highly nonlinear fashion the selection of a globally optimal subset using the branch and bound bb algorithm is limited to problems in very low dimension typically d 5 as the complexity of the problem increases exponentially with d we introduce a bold pruning strategy in the bb algorithm that results in a significant reduction in computing time at the price of a negligeable accuracy lost the novelty of our algorithm is that the bounds at nodes of the bb tree come from pseudo convexifications derived using a linearization technique with approximate bounds for the nonlinear terms the approximate bounds are computed solving an auxiliary semidefinite optimization problem we show through a computational study that our algorithm performs well in a wide set of the most difficult instances of the ltse problem

robpca a new approach to robust principal component analysis we introduce a new method for robust principal component analysis pca classical pca is based on the empirical covariance matrix of the data and hence is highly sensitive to outlying observations two robust approaches have been developed to date the first approach is based on the eigenvectors of a robust scatter matrix such as the minimum covariance determinant or an s estimator and is limited to relatively low dimensional data the second approach is based on projection pursuit and can handle high dimensional data here we propose the robpca approach which combines projection pursuit ideas with robust scatter matrix estimation robpca yields more accurate estimates at noncontaminated datasets and more robust estimates at contaminated data robpca can be computed rapidly and is able to detect exact fit situations as a by product robpca produces a diagnostic plot that displays and classifies the outliers we apply the algorithm to several datasets from chemometrics and engineering

estimating bayes factors via posterior simulation with the laplace metropolis estimator abstract the key quantity needed for bayesian hypothesis testing and model selection is the integrated or marginal likelihood of a model we describe a way to use posterior simulation output to estimate integrated likelihoods we describe the basic laplace metropolis estimator for models without random effects for models with random effects we introduce the compound laplace metropolis estimator we apply this estimator to data from the world fertility survey and show it to give accurate results batching of simulation output is used to assess the uncertainty involved in using the compound laplace metropolis estimator the method allows us to test for the effects of independent variables in a random effects model and also to test for the presence of the random effects

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

efficient robust methods via monitoring for clustering and multivariate data analysis abstract monitoring the properties of single sample robust analyses of multivariate data as a function of breakdown point or efficiency leads to the adaptive choice of the best values of these parameters eliminating arbitrary decisions about their values and so increasing the quality of estimators monitoring the trimming proportion in robust cluster analysis likewise leads to improved estimators we illustrate these procedures on a sample of 424 cows with bovine phlegmon for clustering we use a method which includes constraints on the eigenvalues of the dispersion matrices so avoiding thread shaped clusters the car bike plot reveals the stability of clustering as the trimming level changes the pattern of clusters and outliers alters appreciably for low levels of trimming

identifying multiple outliers in multivariate data summary we propose a procedure for the detection of multiple outliers in multivariate data let xbe an n x p data matrix representing n observations onp variates we first order the n observations using an appropriately chosen robust measure of outlyingness then divide the data set into two initial subsets a basic subset which containsp 1 good observations and a nonbasic subset which contains the remaining n p 1 observations second we compute the relative distance from each point in the data set to the centre of the basic subset relative to the possibly singular covariance matrix of the basic subset third we rearrange the n observations in ascending order accordingly then divide the data set into two subsets a basic subset which contains the first p 2 observations and a non basic subset which contains the remaining n p 2 observations this process is repeated until an appropriately chosen stopping criterion is met the final non basic subset of observations is declared an outlying subset the procedure proposed is illustrated and compared with existing methods by using several data sets the procedure is simple computationally inexpensive suitable for automation computable with widely available software packages effective in dealing with masking and swamping problems and most importantly successful in identifying multivariate outliers

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

procedures for the identification of multiple outliers in linear models abstract we consider the problem of identifying and testing multiple outliers in linear models the available outlier identification methods often do not succeed in detecting multiple outliers because they are affected by the observations they are supposed to identify we introduce two test procedures for the detection of multiple outliers that appear to be less sensitive to this problem both procedures attempt to separate the data into a set of clean data points and a set of points that contain the potential outliers the potential outliers are then tested to see how extreme they are relative to the clean subset using an appropriately scaled version of the prediction error the procedures are illustrated and compared to various existing methods using several data sets known to contain multiple outliers also the performances of both procedures are investigated by a monte carlo study the data sets and the monte carlo indicate that both procedures are effective in the detection of multiple outliers

influence analysis based on the case sensitivity function the case sensitivity function approach to influence analysis is introduced as a natural smooth extension of influence curve methodology in which both the insights of geometry and the power of convex analysis are available in it perturbation is defined as movement between probability vectors defining weighted empirical distributions a euclidean geometry is proposed giving such perturbations both size and direction the notion of the salience of a perturbation is emphasized this approach has several benefits a general probability case weight analysis results answers to a number of outstanding questions follow directly rescaled versions of the three usual finite sample influence curve measures seen now to be required for comparability across different sized subsets of cases are readily available these new diagnostics directly measure the salience of the infinitesimal perturbations involved their essential unity both within and between subsets is evident geometrically finally it is shown how a relaxation strategy in which a high dimensional 0 ncm discrete problem is replaced by a low dimensional o n continuous problem can combine with convex optimization results to deliver better performance in challenging multiple case influence problems further developments are briefly indicated

robust diagnostic data analysis transformations in regression we introduce a very general forward method of data analysis that starts from a small robustly chosen subset of the data and shows the effect of adding observations by a forward search powerful diagnostic procedures result the observations are ordered by their agreement with the proposed transformation masking is overcome and the inferential effect of each observation is clear we apply the resulting method to the transformation of both univariate and multivariate data other applications of the forward search are mentioned

generalized radius processes for elliptically contoured distributions the use of mahalanobis distances has a long history in statistics given a sample of size n and general location and scatter estimators mn and n we can define generalized radii as if we wish to trim observations based on the estimators mn and n then it is natural to first remove the most remote ones i e those with the largest s with this in mind we define a process that maps the trimming proportion in 0 1 to the generalized radius of the observation that has just been removed by this level of trimming we analyze the asymptotic behavior of this process for elliptically contoured distributions we show that the limit law depends only on the elliptical family considered and how n serves to estimate the underlying scale factor through its determinant we carry out monte carlo simulations for finite sample sizes and outline an application for assessing fit to a fixed elliptical family and also for the case where a proportion of outlying observations is discarded

existence of parameterized picard vessiot extensions over fields with algebraically closed constants abstract the purpose of this short note is to establish the existence of parameterized picard vessiot extensions for systems of linear difference differential equations over difference differential fields with an algebraically closed field of constants

robust mean and covariance structure analysis through iteratively reweighted least squares robust schemes in regression are adapted to mean and covariance structure analysis providing an iteratively reweighted least squares approach to robust structural equation modeling each case is properly weighted according to its distance based on first and second order moments from the structural model a simple weighting function is adopted because of its flexibility with changing dimensions the weight matrix is obtained from an adaptive way of using residuals test statistic and standard error estimators are given based on iteratively reweighted least squares the method reduces to a standard distribution free methodology if all cases are equally weighted examples demonstrate the value of the robust procedure

galois groups for integrable and projectively integrable linear difference equations we consider first order linear difference systems over c x c x with respect to a difference operator that is either a shift x x 1 x x 1 q dilation x qx x qx with q c q c not a root of unity or mahler operator x xq x xq with q z 2q z 2 such a system is integrable if its solutions also satisfy a linear differential system it is projectively integrable if it becomes integrable after moding out by scalars we apply recent results of schafke and singer to characterize which groups can occur as galois groups of integrable or projectively integrable linear difference systems in particular such groups must be solvable finally we give hypertranscendence criteria

the length of the shorth soit n 0 1 la longueur de l fraction la plus courte de l echantillon ordonne x 1 n x 2 n x n n on utilise le concept de derivees compactes des fonctionnelles statistiques pour etudier le comportement asymptotique de n lorsque n

probability distribution of the shortest path on the percolation cluster its backbone and skeleton we consider the mean distribution functions ensuremath phi r mathcal l ensuremath phi mathrm b r mathcal l and ensuremath phi mathrm s r mathcal l giving the probability that two sites on the incipient percolation cluster on its backbone and on its skeleton respectively connected by a shortest path of length l are separated by an euclidean distance r following a scaling argument due to de gennes for self avoiding walks we derive analytical expressions for the exponents g 1 d f d mathrm min ensuremath d and g 1 mathrm b g 1 mathrm s 3d mathrm min ensuremath d which determine the scaling behavior of the distribution functions in the limit x ensuremath equiv r mathcal l stackrel ifmmode tilde else fi ensuremath nu ensuremath ll 1 i e ensuremath phi r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 ensuremath phi mathrm b r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 mathrm b and ensuremath phi mathrm s r mathcal l ensuremath propto mathcal l ensuremath stackrel ifmmode tilde else fi ensuremath nu d x g 1 mathrm s with stackrel ifmmode tilde else fi ensuremath nu ensuremath equiv 1 d mathrm min where d f and d mathrm min are the fractal dimensions of the percolation cluster and the shortest path respectively the theoretical predictions for g 1 g 1 mathrm b and g 1 mathrm s are in very good agreement with our numerical results

unmasking multivariate outliers and leverage points abstract detecting outliers in a multivariate point cloud is not trivial especially when there are several outliers the classical identification method does not always find them because it is based on the sample mean and covariance matrix which are themselves affected by the outliers that is how the outliers get masked to avoid the masking effect we propose to compute distances based on very robust estimates of location and covariance these robust distances are better suited to expose the outliers in the case of regression data the classical least squares approach masks outliers in a similar way also here the outliers may be unmasked by using a highly robust regression method finally a new display is proposed in which the robust regression residuals are plotted versus the robust distances this plot classifies the data into regular observations vertical outliers good leverage points and bad leverage points several examples are discussed

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

high breakdown rank regression abstract a weighted rank estimate is proposed that has 50 breakdown and is asymptotically normal at rate n based on this theory inferential procedures including asymptotic confidence and tests and diagnostic procedures such as studentized residuals are developed the influence function of the estimate is derived and shown to be continuous and bounded everywhere in x y space examples show that robustness against outlying high leverage clusters may approach that of the least median of squares while retaining more stability against inliers the estimator uses weights that correct for both factor and response spaces a monte carlo study shows that the estimate is more efficient than the generalized rank estimates which are generalized r estimates with weights that only correct for factor space when weights are constant the estimate reduces to the regular wilcoxon rank estimate

robust clustering with applications in computer vision a clustering algorithm based on the minimum volume ellipsoid mve robust estimator is proposed the mve estimator identifies the least volume region containing h percent of the data points the clustering algorithm iteratively partitions the space into clusters without prior information about their number at each iteration the mve estimator is applied several times with values of h decreasing from 0 5 a cluster is hypothesized for each ellipsoid the shapes of these clusters are compared with shapes corresponding to a known unimodal distribution by the kolmogorov smirnov test the best fitting cluster is then removed from the space and a new iteration starts constrained random sampling keeps the computation low the clustering algorithm was successfully applied to several computer vision problems formulated in the feature space paradigm multithresholding of gray level images analysis of the hough space and range image segmentation

crossover from weak to strong disorder regime in the duration of epidemics abstract we study the susceptible infected recovered sir model in complex networks considering that not all individuals in the population interact in the same way this heterogeneity between contacts is modeled by a continuous disorder in our model the disorder represents the contact time or the closeness between individuals we find that the duration time of an epidemic has a crossover with the system size from a power law regime to a logarithmic regime depending on the transmissibility related to the strength of the disorder using percolation theory we find that the duration of the epidemic scales as the average length of the branches of the infection our theoretical findings supported by simulations explains the crossover between the two regimes

calculating differential galois groups of parametrized differential equations with applications to hypertranscendence the main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations via the parameterized and differential galois theories to achieve this we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential galois groups for differential equations whose coefficients are rational functions berman and singer presented an algorithm calculating the differential galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators we use their algorithm as a part of our algorithm as a result we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter

on b robust instrumental variable estimation of the linear model with panel data the aim of this paper is to demonstrate how to acquire robust consistent estimates of the linear model when the fundamental orthogonality condition is not fulfilled with this end in view we develop two estimation procedures two stage generalized m 2sgm and robust generalized method of moments rgmm both estimators are b robust i e their associated influence function is bounded consistent and asymptotic normally distributed our simulation results indicate that the relatively efficient rgmm estimator in regressions with heteroskedastic and or autocorrelated errors provides accurate parameter estimates of a panel data model with all variables subject to measurement errors even if a substantial portion of the data is contaminated with aberrant observations the traditional estimation techniques such as 2sls and gmm break down when outliers corrupt the data

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

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

breakdown in nonlinear regression abstract the breakdown point is considered an important measure of the robustness of a linear regression estimator this article addresses the concept of breakdown in nonlinear regression because it is not invariant to nonlinear reparameterization the usual definition of the breakdown point in linear regression is inadequate for nonlinear regression the original definition of breakdown due to hampel is more suitable for nonlinear problems but may indicate breakdown when the fitted values change very little we introduce breakdown functions which measure breakdown of the fitted values using the breakdown functions we introduce a new definition of the breakdown point for the linear regression model our definition of the breakdown point coincides with the usual definition for linear regression as well as with hampel s definition for most nonlinear regression functions we show that the breakdown point of the least squares estimator is 1 n we prove that for a large class of unbounded regression func

location of several outliers in multiple regression data using elemental sets the outlying tendency of any case in a multiple regression of p predictors may be estimated by drawing all subsets of size p from the remaining cases and fitting the model each such subset yields an elemental residual for the case in question and a suitable summary statistic of them can be used as an estimate of the case s outlying tendency we propose two such summary statistics an unweighted median which is of bounded influence and a weighted median which is more efficient but less robust the computational load of the procedure is reduced by using random samples in place of the full set of subsets of size p as a byproduct the method yields useful information on the influence or leverage of cases and the mutual masking of high leverage points

practical bayesian density estimation using mixtures of normals abstract mixtures of normals provide a flexible model for estimating densities in a bayesian framework there are some difficulties with this model however first standard reference priors yield improper posteriors second the posterior for the number of components in the mixture is not well defined if the reference prior is used third posterior simulation does not provide a direct estimate of the posterior for the number of components we present some practical methods for coping with these problems finally we give some results on the consistency of the method when the maximum number of components is allowed to grow with the sample size

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

finding an unknown number of multivariate outliers we use the forward search to provide robust mahalanobis distances to detect the presence of outliers in a sample of multivariate normal data theoretical results on order statistics and on estimation in truncated samples provide the distribution of our test statistic we also introduce several new robust distances with associated distributional results comparisons of our procedure with tests using other robust mahalanobis distances show the good size and high power of our procedure we also provide a unification of results on correction factors for estimation from truncated samples

outlier detection in functional observations with applications to profile monitoring the presence of outliers has serious adverse effects on the modeling and forecasting of functional data therefore outlier detection aiming at identifying abnormal functional curves from a dataset is quite important this article proposes a new testing procedure based on functional principal component analysis under mild conditions the null distribution of the test statistic is shown to be asymptotically pivotal with a well known asymptotic distribution simulation results demonstrate good finite sample performance of the asymptotic test and detection procedure finally by illustrating the connection between profile monitoring in statistical process control and outlier detection in functional data we apply the proposed approach to a real data example from a manufacturing process and show that it performs quite well in detecting outlying profiles supplementary material for this article is posted online on the journal web site

a unified approach to exploratory factor analysis with missing data nonnormal data and in the presence of outliers factor analysis is regularly used for analyzing survey data missing data data with outliers and consequently nonnormal data are very common for data obtained through questionnaires based on covariance matrix estimates for such nonstandard samples a unified approach for factor analysis is developed by generalizing the approach of maximum likelihood under constraints statistical properties of the estimates for factor loadings and error variances are obtained a rescaled bartlett corrected statistic is proposed for evaluating the number of factors equivariance and invariance of parameter estimates and their standard errors for canonical varimax and normalized varimax rotations are discussed numerical results illustrate the sensitivity of classical methods and advantages of the proposed procedures

differential transcendence of solutions of difference riccati equations and tietze s treatment abstract there is the paper by h tietze published in 1905 on differential transcendence of solutions of difference riccati equations in this paper we clarify the essence of tietze s treatment and make it purely algebraic as an application the q airy equation is studied

shortest path fractal dimension for percolation in two and three dimensions we carry out a high precision monte carlo study of the shortest path fractal dimension d mathrm min for percolation in two and three dimensions using the leath alexandrowicz method which grows a cluster from an active seed site a variety of quantities are sampled as a function of the chemical distance including the number of activated sites a measure of the radius and the survival probability by finite size scaling we determine d mathrm min 1 130 phantom rule 0 16em 0ex 77 2 and 1 3756 6 in two and three dimensions respectively the result in two dimensions rules out the recently conjectured value d mathrm min 217 192 deng et al phys rev e 81 020102 r 2010

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

weighted wilcoxon type smoothly clipped absolute deviation method summary shrinkage type variable selection procedures have recently seen increasing applications in biomedical research however their performance can be adversely influenced by outliers in either the response or the covariate space this article proposes a weighted wilcoxon type smoothly clipped absolute deviation ww scad method which deals with robust variable selection and robust estimation simultaneously the new procedure can be conveniently implemented with the statistical software r we establish that the ww scad correctly identifies the set of zero coefficients with probability approaching one and estimates the nonzero coefficients with the rate n 1 2 moreover with appropriately chosen weights the ww scad is robust with respect to outliers in both the x and y directions the important special case with constant weights yields an oracle type estimator with high efficiency in the presence of heavier tailed random errors the robustness of the ww scad is partly justified by its asymptotic performance under local shrinking contamination we propose a bayesian information criterion type tuning parameter selector for the ww scad the performance of the ww scad is demonstrated via simulations and by an application to a study that investigates the effects of personal characteristics and dietary factors on plasma beta carotene level

robust multivariate tolerance regions influence function and monte carlo study in this article we define a class of multivariate tolerance regions that turn out to be more resistant than the classical ones to outliers the tolerance factors are numerically evaluated under the central model and the sensitivity to deviations from the normal distribution for moderate samples is studied through a monte carlo study moreover the influence function of the coverage probability allows us to compare the sensitivity of different proposals to anomalous data finally real data examples are discussed

the detection of influential subsets in linear regression by using an influence matrix this paper presents a new method to identify influential subsets in linear regression problems the procedure uses the eigenstructure of an influence matrix which is defined as the matrix of uncentred covariances of the effect on the whole data set of deleting each observation normalized to include the univariate cook statistics on the diagonal it is shown that the eigenstructure of the influence matrix is useful to identify influential subsets and a procedure for detecting influential sets is proposed the method is illustrated with two examples

robust and high breakdown fits of polynomial models a common strategy for fitting a polynomial regression model is to use a combination of residual plots and significance tests on the regression coefficients in this article we consider whether this strategy can be used effectively for fits based on two classes of robust estimatesnamely m estimates and gm estimates we show that the asymptotic relative efficiency of mallows type high breakdown gm estimates can be quite low for detecting curvature in polynomial models furthermore the gm residual plots may also be of little help in detecting curvature an example and a simulation study illustrate the discussion

computation of high breakdown nonlinear regression parameters most algorithms for estimating high breakdown regression estimators in linear regression rely on finding the least squares fit to many p point elemental sets where p is the dimension of the x matrix such an approach is computationally infeasible in nonlinear regression this article presents a new algorithm for computing high breakdown estimates in nonlinear regression that requires only a small number of least squares fits to p points the algorithm is used to compute rousseeuw s least median of squares lms estimate and yohai s mm estimate in both simulations and examples it is also used to compute bootstrapped and monte carlo standard error estimates for mm estimates which are compared with asymptotic standard errors ase s using the progress algorithm for a two parameter nonlinear model with sample size 30 would require finding the least squares fit to 435 two point subsets of the data in the settings considered in this article the proposed algorithm performs just as well with 25 as with 435

the accuracy of elemental set approximations for regression abstract the elemental set algorithm involves performing many fits to a data set each fit made to a subsample of size just large enough to estimate the parameters in the model elemental sets have been proposed as a computational device to approximate estimators in the areas of high breakdown regression and multivariate location scale estimation where exact optimization of the criterion function is computationally intractable although elemental set algorithms are used widely and for a variety of problems the quality of the approximation they give has not been studied this article shows that they provide excellent approximations for the least median of squares least trimmed squares and ordinary least squares criteria it is suggested that the approach likely will be equally effective in the other problem areas in which exact optimization of a criterion is difficult or impossible

multiple influential point detection in high dimensional regression spaces influence diagnosis should be routinely conducted when one aims to construct a regression model despite its importance the problem of influence quantification is severely under investigated in a high dimensional setting mainly due to the difficulty of establishing a coherent theoretical framework and the lack of easily implementable procedures although some progress has been made in recent years existing approaches are ineffective in detecting multiple influential points especially due to the notorious masking and swamping effects to address this challenge we propose a new group deletion procedure referred to as mip by introducing two novel quantities named max and min statistics these two statistics have complimentary properties in that the max statistic is effective for overcoming the masking effect while the min statistic is useful for overcoming the swamping effect combining their strengths we further propose an efficient algorithm that can detect influential points with prespecified guarantees for wider applications we focus on developing the new proposal for the multiple response regression model encompassing the univariate response linear model as a special case the proposed influential point detection procedure is simple to implement efficient to run and enjoys attractive theoretical properties its effectiveness is verified empirically via extensive simulation study and data analysis

the metrically trimmed mean as a robust estimator of location the metrically trimmed mean is defined as the average of observations remaining after a fixed number of outlying observations have been removed a metric the distance from the median is used to determine which points are outlying the influence curve and the asymptotic normality of the metrically trimmed mean are derived using von mises expansions the relative merits of the median the trimmed mean and the metrically trimmed mean are discussed in neighborhoods of nonparametric models with natural parameters it is observed that the metrically trimmed mean works well for the center of symmetry of a symmetric distribution function with asymmetric contamination a multivariate extension of the metrically trimmed mean is discussed

the impact of outliers and leverage points for technical efficiency measurement using high breakdown procedures given that most data used for production studies have not been accumulated for such purposes it is important that the quantitative tools for messy data which can affect the accuracy of computed technical efficiency measures be found in this study high breakdown robust methods are used in conjunction with a robust distance measure defined relative to the minimum volume ellipsoid estimator the standardized robust residuals from the high breakdown estimators and the robust distance measures are used to statistically and graphically depict both multivariate outliers and leverage points once these points are found their relationship to those observations that exhibit strong technically efficient or inefficient behavior scale inefficiency and or unusual production characteristics is analyzed for three linerboard manufacturing facilities additionally the impact of the outliers and leverage points on the estimated least squares coefficients which are used by the corrected ordinary least squares methodology to compute the full frontier technical efficiency measures is explored finally a sensitivity analysis of the impact of outliers and leverage points on the computed linear programming based technical efficiency measures is presented