robust kalman filter based on a generalized maximum likelihood type estimator a new robust kalman filter is proposed that detects and bounds the influence of outliers in a discrete linear system including those generated by thick tailed noise distributions such as impulsive noise besides outliers induced in the process and observation noises we consider in this paper a new type called structural outliers for a filter to be able to counter the effect of these outliers observation redundancy in the system is necessary we have therefore developed a robust filter in a batch mode regression form to process the observations and predictions together making it very effective in suppressing multiple outliers a key step in this filter is a new prewhitening method that incorporates a robust multivariate estimator of location and covariance the other main step is the use of a generalized maximum likelihood type gm estimator based on schweppe s proposal and the huber function which has a high statistical efficiency at the gaussian distribution and a positive breakdown point in regression the latter is defined as the largest fraction of contamination for which the estimator yields a finite maximum bias under contamination this gm estimator enables our filter to bound the influence of residual and position where the former measures the effects of observation and innovation outliers and the latter assesses that of structural outliers the estimator is solved via the iteratively reweighted least squares irls algorithm in which the residuals are standardized utilizing robust weights and scale estimates finally the state estimation error covariance matrix of the proposed gm kalman filter is derived from its influence function simulation results revealed that our filter compares favorably with the h filter in the presence of outliers

high breakdown point and high efficiency robust estimates for regression a class of robust estimates for the linear model is introduced these estimates called mm estimates have simultaneously the following properties i they are highly efficient when the errors have a normal distribution and ii their breakdown point is 0 5 the mm estimates are defined by a three stage procedure in the first stage an initial regression estimate is computed which is consistent robust and with high breakdown point but not necessarily efficient in the second stage an m estimate of the errors scale is computed using residuals based on the initial estimate finally in the third stage an m estimate of the regression parameters based on a proper redescending psi function is computed consistency and asymptotical normality of the mm estimates assuming random carriers are proved a convergent iterative numerical algorithm is given finally the asymptotic biases under contamination of optimal bounded influence estimates and mm estimates are compared

robust estimation in structured linear regression a structured linear regression model is one in which there are permanent dependencies among some p row vectors of the n x p design matrix to study structured linear regression we introduce a new class of robust estimators called d estimators which can be regarded as a generalization of the least median of squares and least trimmed squares estimators they minimize a dispersion function of the ordered absolute residuals up to the rank h we investigate their breakdown point and exact fit point as a function of h in structured linear regression it is found that the d and s estimators can achieve the highest possible breakdown point for h appropriately chosen it is shown that both the maximum breakdown point and the corresponding optimal value of h h op are sample dependent they hinge on the design but not on the response the relationship between the breakdown point and the design vanishes when h is strictly larger than h op however when h is smaller than h op the breakdown point depends in a complicated way on the design as well as on the response

a central limit theorem for multivariate generalized trimmed k means a central limit theorem for generalized trimmed k means is obtained in a very general framework that covers the multivariate setting general penalty functions and general k 1 several applications including the location estimator case k 1 for elliptical distributions and the construction of multivariate not necessarily connected tolerance zones are also given

the aronsson euler equation for absolutely minimizing lipschitz extensions with respect to carnot caratheodory metrics we derive the euler lagrange equation also known in this setting as the aronsson euler equation for absolute minimizers of the l variational problem formula math where c g is an open subset of a carnot group 0 u denotes the horizontal gradient of u r and the lipschitz class is defined in relation to the carnot caratheodory metric in particular we show that absolute minimizers are infinite harmonic in the viscosity sense as a corollary we obtain the uniqueness of absolute minimizers in a large class of groups this result extends previous work of jensen and of crandall evans and gariepy we also derive the aronsson euler equation for more regular absolutely minimizing lipschitz extensions corresponding to those carnot caratheodory metrics which are associated to free systems of vector fields

c α preserving operators on separable banach spaces abstract let t be a bounded linear operator from a separable banach space x to a banach space y a necessary and sufficient condition on t for the existence of a subspace z of x such that z is isomorphic to c and the restriction of t to z is an isomorphism is given the special case where x is the disc algebra is then considered and results similar to those previously obtained by the author for c k spaces are obtained for the disc algebra finally some additional results of the same type are proved for subspaces of c k with small annihilator

leverage and breakdown in l 1 regression abstract in this article the notion of leverage of a design point when fitting a linear regression model is interpreted geometrically in the case of least squares fitting the leverage indicators based on the diagonal of the hat matrix are widely applied by interpreting these hat matrix indicators geometrically leverage can be generalized to groups of design points as well as to other methods of fitting the article introduces a leverage indicator that is appropriate for l 1 regression and discusses some aspects of this new diagnostic it is shown that in the case of l 1 regression the leverage indicators have a precise interpretation they tell us about the breakdown and or the exactness of fit as an application the article considers the maximal possible breakdown value of l 1 regression and the choice of designs that achieve this maximum

preserving operators on spaces let a c k x be a bounded linear operator where if is a compact hausdorff space and x is a separable banach space sufficient conditions are given for a to be an isomorphism into when restricted to a subspace y of c k such that y is isometrically isomorphic to a space c a of continuous functions on the space of ordinal numbers less than or equal to the countable ordinal a 1 for a compact hausdorff space k c k will denote the banach space of continuous real valued functions on k with the supremum norm the greek letters a s and y will be reserved for countable ordinal numbers we consider the set 0 a y 0 x 0 for each countable ordinal a we will define a family pa x b of open subsets g and associated measures ju g g b let p0 x b g jttg g is an open set in k ixg g b and g g x if a s 1 then pa x b g nc g jiig g p0 x b and there is a disjoint sequence g of open subsets of g and associated measures xg such that g ftg g ps x b ju gn jug and ung c g if a is a limit ordinal then x b g xg g p0 x b there is a disjoint sequence of open subsets gn of g with associated measures xg such that fic nc ung c g and g jug g pa x b where an a the notation a a means that a is an increasing sequence of ordinals with a sup an a simple example illustrating the sets pa x b is contained in 4 received by the editors march 20 1979 and in revised form july 30 1981 1980 mathematics subject classification primary 46e15 research on this paper was partially supported by the national science foundation grant number mcs 77 01690 1982 american mathematical society o0o2 9939 82 oo0o o726 so5 oo

derivation of the aronsson equation for c1 hamiltonians it is proved herein that any absolute minimizer u for a suitable hamiltonian h c r r u is a viscosity solution of the aronsson equation hp du u x h du u x x 0 in u the primary advance is to weaken the assumption that h c used by previous authors to the natural condition that h c

an efficient derivation of the aronsson equation for 1 p the equation which characterizes minima of the functional u u du p dx subject to fixed values of u on u is p u 0 here p is the well known p laplacian when p the corresponding functional is u du 2 l u a new feature arises in that minima are no longer unique unless u is allowed to vary leading to the idea of absolute minimizers aronsson showed that then the appropriate equation is u 0 that is u is infinity harmonic as explained below jensen showed that infinity harmonic functions understood in the viscosity sense are precisely the absolute minimizers here we advance results of barron jensen and wang concerning more general functionals u f x u du l u by giving a simplified derivation of the corresponding necessary condition under weaker hypotheses

operators on c k spaces preserving copies of schreier spaces it is proved that an operator t c k x k compact metrizable x a separable banach space for which the e szlenk index of t bx is greater than or equal to 1 is an isomorphism on a subspace of c k isomorphic to x the schreier space of order as a corollary one obtains that a complemented subspace of c k with szlenk index equal to 1 contains a subspace isomorphic to x

calculating the breakdown point of sparse linear models in robust statistics the concept of breakdown point was introduced to quantify the robustness of an estimator in a linear regression model computing the breakdown point is useful in tuning some robust regression estimators e g the least trimmed squares estimator computing the breakdown point for a structured linear model i e one with dependencies among some p rows of the n p design matrix x can be very demanding this article presents an algorithm for calculating the maximum breakdown point for sparse linear models which are a special type of structured linear model whose design matrix has many zero entries the algorithm decomposes a sparse design matrix into smaller submatrixes on which the computation is performed thereby leading to substantial savings in computation an assembly process along with a few numerical examples illustrate the application of the algorithm and demonstrate its computational benefits

l variational problems and aronsson equations in this paper we prove that viscosity solutions of aronsson equations are absolute minimizers in certain l variational problems

another way to say harmonic it is known that solutions of u n i j 1 u xi u xj u xjxj 0 that is the harmonic functions are exactly those functions having a comparison property with respect to the family of translates of the radial solutions g x a x we establish a more difficult linear result a function in r n is harmonic if it has the comparison property with respect to sums of n translates of the radial harmonic functions g x a x 2 n for n 2 and g x b ln x for n 2 an attempt to generalize these results for u 0 p and u 0 p 2 to the general p laplacian leads to the fascinating discovery that certain sums of translates of radial p superharmonic functions are again p superharmonic mystery remains the class of p superharmonic functions so constructed for p 2 does not suffice to characterize p subharmonic functions

inhomogeneous infinity laplace equation abstract we present the theory of the viscosity solutions of the inhomogeneous infinity laplace equation x i u x j u x i x j 2 u f in domains in r n we show existence and uniqueness of a viscosity solution of the dirichlet problem under the intrinsic condition f does not change its sign we also discover a characteristic property which we call the comparison with standard functions property of the viscosity sub and super solutions of the equation with constant right hand side applying these results and properties we prove the stability of the inhomogeneous infinity laplace equation with nonvanishing right hand side which states the uniform convergence of the viscosity solutions of the perturbed equations to that of the original inhomogeneous equation when both the right hand side and boundary data are perturbed in the end we prove the stability of the well known homogeneous infinity laplace equation x i u x j u x i x j 2 u 0 which states the viscosity solutions of the perturbed equations converge uniformly to the unique viscosity solution of the homogeneous equation when its right hand side and boundary data are perturbed simultaneously

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

generalized s estimators abstract in this article we introduce a new type of positive breakdown regression method called a generalized s estimator or gs estimator based on the minimization of a generalized m estimator of residual scale we compare the class of gs estimators with the usual s estimators including least median of squares it turns out that gs estimators attain a much higher efficiency than s estimators at the cost of a slightly increased worst case bias we investigate the breakdown point the maxbias curve and the influence function of gs estimators we also give an algorithm for computing gs estimators and apply it to real and simulated 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

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

aspects of robust linear regression section 1 of the paper contains a general discussion of robustness in section 2 the influence function of the hampel rousseeuw least median of squares estimator is derived linearly invariant weak metrics are constructed in section 3 it is shown in section 4 that s estimators satisfy an exact holder condition of order 1 2 at models with normal errors in section 5 the breakdown points of the hampel krasker dispersion and regression functionals are shown to be 0 the exact breakdown point of the krasker welsch dispersion functional is obtained as well as bounds for the corresponding regression functional section 6 contains the construction of a linearly equivariant high breakdown and locally lipschitz dispersion functional for any design distribution in section 7 it is shown that there is no inherent contradiction between efficiency and a high breakdown point section 8 contains a linearly equivariant high breakdown regression functional which is lipschitz continuous at models with normal errors

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

uniqueness and nonuniqueness of viscosity solutions to aronsson s equation for a bounded domain omega subset mathbb r n and g in c delta omega assume that h in c 2 mathbb r n is convex and coercive and that p in mathbb r n h p rm min mathbb r n h has no interior points then we establish the uniqueness of viscosity solutions to the dirichlet problem of aronsson s equation begin aligned mathcal a u h p nabla u otimes h p nabla u nabla 2 u 0 rm in omega u g rm on delta omega end aligned for h h p x depending on x we illustrate the connection between the uniqueness and nonuniqueness of viscosity solutions to aronsson s equation and that of the hamilton jacobi equation h nabla u x c

an axiomatic approach to scalar data interpolation on surfaces we discuss possible algorithms for interpolating data given on a set of curves in a surface of 3 we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolutely minimizing lipschitz extension model amle is singled out and studied in more detail we study the correctness of our numerical approach and we show experiments illustrating the interpolation of data on some simple test surfaces like the sphere and the torus

l variational problems and weak kam theory in the first part of this paper we extend several results in crandall evans and gariepy 11 and crandall and evans 10 to absolute minimizers of more general l functionals in the second part we present some interesting connections between l variational problems and weak kam theory as an application we will advance the main result in fathi and siconolfi 17 i e the existence of a c1 subsolution of the hamilton jacobi equation moreover we will propose a possible approximation of the projected aubrey set by a variational approach that was first used by evans in 14 2007 wiley periodicals inc

high breakdown point estimates of regression by means of the minimization of an efficient scale abstract a new class of robust estimates estimates is introduced the estimates have simultaneously the following properties a they are qualitatively robust b their breakdown point is 5 and c they are highly efficient for regression models with normal errors they are defined by minimizing a new scale estimate applied to the residuals asymptotically a estimate is equivalent to an m estimate with a function given by a weighted average of two functions one corresponding to a very robust estimate and the other to a highly efficient estimate the weights are adaptive and depend on the underlying error distribution we prove consistency and asymptotic normality and give a convergent iterative computing algorithm finally we compare the biases produced by gross error contamination in the estimates and optimal bounded influence estimates