h sub spl infin control of discrete time uncertain systems this paper considers h sub spl infin control problems involving discrete time uncertain linear systems the uncertainty is supposed to belong to convex bounded domains and no additional assumptions are made as for instance matching conditions two h sub spl infin guaranteed cost control problems are solved the first one concerns the determination of a state feedback gain if one exists in such way the h sub spl infin norm of a certain transfer function remains bounded by a prespecified h sub spl infin level for all possible models the second one includes this bound as an additional variable to be minimized to achieve the smallest feasible limiting bound the results follow from the simple geometry of those problems which are shown to be convex in the particular parametric space under consideration an example illustrates the theory

on the observer design for descriptor systems for the descriptor system ex ax bu y cx du an observer of the order p rank e p sub 1 0

a multivariate two sample test based on the number of nearest neighbor type coincidences for independent d variate random samples x 1 cdots x n 1 i i d f x y 1 cdots y n 2 i i d g x where the densities f and g are assumed to be continuous a e consider the number t of all k nearest neighbor comparisons in which observations and their neighbors belong to the same sample we show that if f g a e the limiting normal distribution of t as min n 1 n 2 rightarrow infty n 1 n 1 n 2 rightarrow tau 0 tau 1 does not depend on f an omnibus procedure for testing the hypothesis h 0 f g a e is obtained by rejecting h 0 for large values of t the result applies to a general distance generated by a norm on mathbb r d for determining nearest neighbors and it generalizes to the multisample situation

nonparametric tests for homogeneity based on non bipartite matching given a sequence of observations has a change occurred in the underlying probability distribution with respect to observation order this problem of detecting change points arises in a variety of applications including health prognostics for mechanical systems syndromic disease surveillance in geographically dispersed populations anomaly detection in information networks and multivariate process control in general detecting change points in high dimensional settings is challenging and most change point methods for multidimensional problems rely upon distributional assumptions or the use of observation history to model probability distributions we present three new nonparametric statistical tests for heterogeneity based on the combinatorial properties of minimum non bipartite matching mnbm the key idea underlying each of these tests is that if a sequence of independent random observations undergoes a change in distribution either an abrupt shift or a gradual drift a mnbm based on inter point

bayesian aspects of some nonparametric problems we study the bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation we consider the asymptotic properties of bayes procedures for conjugate gaussian priors we show that so long as the prior puts nonzero measure on the very large parameter set of interest then the bayes estimators are not satisfactory more specifically we show that these estimators do not achieve the correct minimax rate over norm bounded sets in the parameter space thus all bayes estimators for proper gaussian priors have zero asymptotic efficiency in this minimax sense we then present a class of priors whose bayes procedures attain the optimal minimax rate of convergence these priors may be viewed as compound or hierarchical mixtures of suitable gaussian distributions

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

robust h control for linear discrete time systems with norm bounded time varying uncertainty the problem of robust h analysis and synthesis for linear discrete time systems with norm bounded time varying uncertainty is studied in this paper it will be shown that this problem is equivalent to the problem of h analysis and synthesis of an auxiliary system the necessary and sufficient conditions for the equivalency are proved thus the original problem can be solved by existing h control methods

on nonparametric confidence intervals an inequality is given for the expected length of a confidence interval given that a particular distribution generated the data and assuming that the confidence interval has a given coverage probability over a family of distributions as a corollary attempts to adapt to the regularity of the true density within derivative smoothness classes cannot improve the rate of convergence of the length of the confidence interval over minimax fixed length intervals and still maintain uniform coverage probability however adaptive confidence intervals can attain improved rates of convergence in some other classes of densities such as those satisfying a shape restriction

contour detection and hierarchical image segmentation this paper investigates two fundamental problems in computer vision contour detection and image segmentation we present state of the art algorithms for both of these tasks our contour detector combines multiple local cues into a globalization framework based on spectral clustering our segmentation algorithm consists of generic machinery for transforming the output of any contour detector into a hierarchical region tree in this manner we reduce the problem of image segmentation to that of contour detection extensive experimental evaluation demonstrates that both our contour detection and segmentation methods significantly outperform competing algorithms the automatically generated hierarchical segmentations can be interactively refined by user specified annotations computation at multiple image resolutions provides a means of coupling our system to recognition applications

h control for descriptor systems a matrix inequalities approach abstract this paper considers the h control problem for descriptor systems that possibly have impulsive modes and or j axis zeros first we propose matrix inequalities that give a generalized stability condition and an h norm condition for descriptor systems using these matrix inequalities we show that the solvability of a set of matrix inequalities is necessary and sufficient to the existence of a proper controller that satisfies a prescribed h norm condition as well as stabilizing the closed loop system and eliminating all impulsive modes these inequalities are equivalent to certain linear matrix inequalities to which we can get solutions whenever they exist using efficient polynomial time algorithms

h sub infinity analysis and synthesis of discrete time systems with time varying uncertainty the problems of h sub infinity analysis and synthesis of discrete time systems with block diagonal real time varying uncertainty are considered it is shown that these problems can be converted into scaled h sub infinity analysis and synthesis problems the problems of quadratic stability analysis and quadratic stabilization of these types of systems are dealt with as a special case the results on synthesis are established for general linear dynamic output feedback control

an algebraic riccati equation approach to h optimization abstract this paper considers the general so called four block h optimal control problem with the assumption that system states are available for feedback it is shown that infimization of the h norm of the closed loop transfer function over all linear constant i e nondynamic stabilizing state feedback laws can be completely characterized via an algebraic riccati equation it is further shown that the optimal norm is not improved by allowing feedback to be dynamic thus the general state feedback h optimal control problem can be solved by iteratively solving one are and the controller can be chosen to be static gain

regularization of descriptor systems by output feedback conditions are given under which a descriptor or generalized state space system can be regularized by output feedback it is shown that under these conditions proportional and derivative output feedback controls can be constructed such that the closed loop system is regular and has index at most one this property ensures the solvability of the resulting system of dynamic algebraic equations a reduced form is given that allows the system properties as well as the feedback to be determined the construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way

the multivariate l1 median and associated data depth this paper gives three related results i a new simple fast monotonically converging algorithm for deriving the l1 median of a data cloud in d a problem that can be traced to fermat and has fascinated applied mathematicians for over three centuries ii a new general definition for depth functions as functions of multivariate medians so that different definitions of medians will correspondingly give rise to different dept functions and iii a simple closed form formula of the l1 depth function for a given data cloud in d

on a notion of simplicial depth abstract for a distribution f on rp and a point x in rp the simplicial depth d x which is the probability that x be inside a random simplex whose vertices are p 1 independent observations from f is introduced d x can be viewed as a measure of depth of the point x relative to f and its empirical version gives rise to a natural ordering of the data points from the center outward this ordering provides an approach for detecting outliers in a multivariate data cloud and leads to the introduction of affine equivariant multivariate generalizations of the univariate sample median and l statistics this sample median is shown to be consistent for the center of any angularly symmetric distribution

a constrained risk inequality with applications to nonparametric functional estimation a general constrained minimum risk inequality is derived given two densities f and f 0 we find a lower bound for the risk at the point given an upper bound for the risk at the point 0 the inequality sheds new light on superefficient estimators in the normal location problem and also on an adaptive estimation problem arising in nonparametric functional estimation

the interval testing procedure a general framework for inference in functional data analysis we introduce in this work the interval testing procedure itp a novel inferential technique for functional data the procedure can be used to test different functional hypotheses e g distributional equality between two or more functional populations equality of mean function of a functional population to a reference itp involves three steps i the representation of data on a possibly high dimensional functional basis ii the test of each possible set of consecutive basis coefficients iii the computation of the adjusted p values associated to each basis component by means of a new strategy here proposed we define a new type of error control the interval wise control of the family wise error rate particularly suited for functional data we show that itp is provided with such a control a simulation study comparing itp with other testing procedures is reported itp is then applied to the analysis of hemodynamical features involved with cerebral aneurysm pathology itp is implemented in the fdatest r package

an algebraic approach to h output feedback control problems abstract we derive an output feedback controller which stabilizes a system and satisfies a prescribed h norm bound of the closed loop transfer function the proposed design method is available for any system i e there are no restrictions on d12 and d21 this approach utilizes only algebraic operations thus proofs are simple and clear

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

cramer von mises distributions and permutation tests summary in the k sample problem or one way analysis of variance a multiple response permutation procedure based on within group sums of absolute rank differences is shown to have a certain large sample limit distribution this distribution is the convolution of k 1 copies of the usual cramer von mises distribution and so is a cramer von mises analogue of chi squared quantiles and a simple chi squared approximation are outlined

on bayesian supremum norm contraction rates building on ideas from castillo and nickl 6 a method is pro vided to study nonparametric bayesian posterior convergence rates when strong measures of distances such as the sup norm are considered in particular we show that likelihood methods can achieve optimal minimax sup norm rates in density estimation on the unit interval the introduced methodology is used to prove that commonly used families of prior distri butions on densities namely log density priors and dyadic random density histograms can indeed achieve optimal sup norm rates of convergence new results are also derived in the gaussian white noise model as a further il lustration of the presented techniques ams 2000 subject classications primary 62g20 secondary 62g05 62g07

adaptive bayesian estimation using a gaussian random field with inverse gamma bandwidth we consider nonparametric bayesian estimation inference using a rescaled smooth gaussian field as a prior for a multidimensional function the rescaling is achieved using a gamma variable and the procedure can be viewed as choosing an inverse gamma bandwidth the procedure is studied from a frequentist perspective in three statistical settings involving replicated observations density estimation regression and classification we prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax optimal up to a logarithmic factor whatever the regularity level of the data generating distribution thus the hierachical bayesian procedure with a fixed prior is shown to be fully adaptive

adaptive bayesian inference on the mean of an infinite dimensional normal distribution we consider the problem of estimating the mean of an infinite break dimensional normal distribution from the bayesian perspective under the assumption that the unknown true mean satisfies a smoothness condition we first derive the convergence rate of the posterior distribution for a prior that is the infinite product of certain normal distributions and compare with the minimax rate of convergence for point estimators although the posterior distribution can achieve the optimal rate of convergence the required prior depends on a smoothness parameter q when this parameter q is unknown besides the estimation of the mean we encounter the problem of selecting a model in a bayesian approach this uncertainty in the model selection can be handled simply by further putting a prior on the index of the model we show that if q takes values only in a discrete set the resulting hierarchical prior leads to the same convergence rate of the posterior as if we had a single model a slightly weaker result is presented when q is unrestricted an adaptive point estimator based on the posterior distribution is also constructed primary subjects 62g20 secondary subjects 62c10 62g05 keywords adaptive bayes procedure convergence rate minimax risk posterior distribution model selection

a wilcoxon mann whitney type test for infinite dimensional data the wilcoxon mann whitney test is a robust competitor of the t test in the univariate setting for finite dimensional multivariate non gaussian data several extensions of the wilcoxon mann whitney test have been shown to outperform hotelling s t 2 test in this paper we study a wilcoxon mann whitney type test based on spatial ranks in infinite dimensional spaces we investigate its asymptotic properties and compare it with several existing tests the proposed test is shown to be robust with respect to outliers and to have better power than some competitors for certain distributions with heavy tails we study its performance using real and simulated data

ensemble subsampling for imbalanced multivariate two sample tests some existing nonparametric two sample tests for equality of multivariate distributions perform unsatisfactorily when the two sample sizes are unbalanced in particular the power of these tests tends to diminish with increasingly unbalanced sample sizes in this article we propose a new testing procedure to solve this problem the proposed test based on the nearest neighbor method by schilling employs a novel ensemble subsampling scheme to remedy this issue more specifically the test statistic is a weighted average of a collection of statistics each associated with a randomly selected subsample of the data we derive the asymptotic distribution of the test statistic under the null hypothesis and show that the new test is consistent against all alternatives when the ratio of the sample sizes either goes to a finite limit or tends to infinity via simulated data examples we demonstrate that the new test has increasing power with increasing sample size ratio when the size of the smaller sample is fixe

a kernel two sample test we propose a framework for analyzing and comparing distributions which we use to construct statistical tests to determine if two samples are drawn from different distributions our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel hilbert space rkhs and is called the maximum mean discrepancy mmd we present two distribution free tests based on large deviation bounds for the mmd and a third test based on the asymptotic distribution of this statistic the mmd can be computed in quadratic time although efficient linear time approximations are available our statistic is an instance of an integral probability metric and various classical metrics on distributions are obtained when alternative function classes are used in place of an rkhs we apply our two sample tests to a variety of problems including attribute matching for databases using the hungarian marriage method where they perform strongly excellent performance is also obtained when comparing distributions over graphs for which these are the first such tests

an adaptation theory for nonparametric confidence intervals a nonparametric adaptation theory is developed for the construction of confidence intervals for linear functionals a between class modulus of continuity captures the expected length of adaptive confidence intervals sharp lower bounds are given for the expected length and an ordered modulus of continuity is used to construct adaptive confidence procedures which are within a constant factor of the lower bounds in addition minimax theory over nonconvex parameter spaces is developed

from depth to local depth a focus on centrality aiming at analyzing multimodal or nonconvexly supported distributions through data depth we introduce a local extension of depth our construction is obtained by conditioning the distribution to appropriate depth based neighborhoods and has the advantages among others of maintaining affine invariance and applying to all depths in a generic way most importantly unlike their competitors which for extreme localization rather measure probability mass the resulting local depths focus on centrality and remain of a genuine depth nature at any locality level we derive their main properties establish consistency of their sample versions and study their behavior under extreme localization we present two applications of the proposed local depth for classification and for symmetry testing and we extend our construction to the regression depth context throughout we illustrate the results on several datasets both artificial and real univariate and multivariate supplementary materials for this artic

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

rates of contraction for posterior distributions in lr metrics 1 r the frequentist behavior of nonparametric bayes estimates more specifically rates of contraction of the posterior distributions to shrinking lr norm neighborhoods 1 r of the unknown parameter are studied a theorem for nonparametric density estimation is proved under general approximation theoretic assumptions on the prior the result is applied to a variety of common examples including gaussian process wavelet series normal mixture and histogram priors the rates of contraction are minimax optimal for 1 r 2 but deteriorate as r increases beyond 2 in the case of gaussian nonparametric regression a gaussian prior is devised for which the posterior contracts at the optimal rate in all lr norms 1 r

data driven nonparametric existence and association problems we investigate two closely related nonparametric hypothesis testing problems in the first problem i e the existence problem we test whether a testing data stream is generated by one of a set of composite distributions in the second problem i e the association problem we test which one of the multiple distributions generates a testing data stream we assume that some distributions in the set are unknown and instead only training sequences generated by the corresponding distributions are available for both problems we construct the generalized likelihood tests and characterize the error exponents of the maximum error probabilities for the existence problem we show that the error exponent is mainly captured by the chernoff information between the set of composite distributions and alternative distributions for the association problem we show that the error exponent is captured by the minimum chernoff information between each pair of distributions as well as the kullback leibler divergences between the approximated distributions via training sequences and the true distributions we also show that the ratio between the lengths of training and testing sequences plays an important role in determining the error decay rate

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

supremum norm posterior contraction and credible sets for nonparametric multivariate regression in the setting of nonparametric multivariate regression with unknown error variance we study asymptotic properties of a bayesian method for estimating a regression function f and its mixed partial derivatives we use a random series of tensor product of b splines with normal basis coefficients as a prior for f and the error variance is either estimated using the empirical bayes approach or is endowed with a suitable prior in a hierarchical bayes approach we establish pointwise l2 and supremum norm posterior contraction rates for f and its mixed partial derivatives and show that they coincide with the minimax rates our results cover even the anisotropic situation where the true regression function may have different smoothness in different directions using the convergence bounds we show that pointwise l2 and supremum norm credible sets for f and its mixed partial derivatives have guaranteed frequentist coverage with optimal size new results on tensor products of b splines are also obtained in the course

bounded real criteria for linear time delay systems considers the problem of finding the h sub spl infin norm of a linear time varying system having a delayed state and with an output that comprises a linear combination of the current state as well as of the delayed state both delay independent and delay dependent bounded real criteria are derived which provide sufficient conditions for the system to possess an h sub spl infin norm which is less than or equal to a prescribed bound the stationary case for time invariant systems is also tackled and in this situation the proposed bounded real criteria are given in terms of linear matrix inequalities two examples are presented which compare the guaranteed bounds that are achieved by our results with the actual h sub spl infin norm of the system

nonparametric detection of anomalous data streams a nonparametric anomalous hypothesis testing problem is investigated in which there are totally n observed sequences out of which s anomalous sequences are to be detected each typical sequence consists of m independent and identically distributed i i d samples drawn from a distribution p whereas each anomalous sequence consists of m i i d samples drawn from a distribution q that is distinct from p the distributions p and q are assumed to be unknown in advance distribution free tests are constructed by using the maximum mean discrepancy as the metric which is based on mean embeddings of distributions into a reproducing kernel hilbert space the probability of error is bounded as a function of the sample size m the number s of anomalous sequences and the number n of sequences it is shown that with s known the constructed test is exponentially consistent if m is greater than a constant factor of log n for any p and q whereas with s unknown m should have an order strictly greater than log n furthermore it is shown that no test can be consistent for arbitrary p and q if m is less than a constant factor of log n thus the order level optimality of the proposed test is established numerical results are provided to demonstrate that the proposed tests outperform or perform as well as tests based on other competitive approaches under various cases

the geometric median on riemannian manifolds with application to robust atlas estimation one of the primary goals of computational anatomy is the statistical analysis of anatomical variability in large populations of images the study of anatomical shape is inherently related to the construction of transformations of the underlying coordinate space which map one anatomy to another it is now well established that representing the geometry of shapes or images in euclidian spaces undermines our ability to represent natural variability in populations in our previous work we have extended classical statistical analysis techniques such as averaging principal components analysis and regression to riemannian manifolds which are more appropriate representations for describing anatomical variability in this paper we extend the notion of robust estimation a well established and powerful tool in traditional statistical analysis of euclidian data to manifold valued representations of anatomical variability in particular we extend the geometric median a classic robust estimator of centrality for data in euclidean spaces we formulate the geometric median of data on a riemannian manifold as the minimizer of the sum of geodesic distances to the data points we prove existence and uniqueness of the geometric median on manifolds with non positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds generalizing the weiszfeld procedure for finding the geometric median of euclidean data we present an algorithm for computing the geometric median on an arbitrary manifold we show that this algorithm converges to the unique solution when it exists in this paper we exemplify the robustness of the estimation technique by applying the procedure to various manifolds commonly used in the analysis of medical images using this approach we also present a robust brain atlas estimation technique based on the geometric median in the space of deformable images

spatial medians depth functions and multivariate jensen s inequality for any given partial order in a d dimensional euclidean space under mild regularity assumptions we show that the intersection of closed generalized intervals containing more than 1 2 of the probability mass is a non empty compact interval this property is shared with common intervals on real line where the intersection is the median set of the underlying probability distribution so obtained multivariate medians with respect to a partial order can be observed as special cases of centers of distribution in the sense of type d depth functions introduced by y zuo and r serfling em ann stat bf 28 2000 461 482 we show that the halfspace depth function can be realized via compact convex sets or for example closed balls in place of halfspaces and discuss structural properties of halfspace and related depth functions and their centers among other things we prove that in general the maximal guaranteed depth is frac 1 d 1 as an application of these results we provide a jensen s type inequality for functions of several variables with medians in place of expectations which is an extension of the previous work by m merkle em stat prob letters bf 71 2005 277 281

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

on regularizing descriptor systems by output feedback in this paper linear time invariant descriptor variable systems are considered which are either nonregular or if they are regular they have impulsive modes it is shown that if a certain necessary and sufficient condition is satisfied then both regularity can be established and impulsive modes can be eliminated by output feedback we have also considered the problem of achieving structural properties of the regularized system

adaptive bernstein von mises theorems in gaussian white noise we investigate bernstein von mises theorems for adaptive nonparametric bayesian procedures in the canonical gaussian white noise model we consider both a hilbert space and multiscale setting with applications in l 2 and l infty respectively this provides a theoretical justification for plug in procedures for example the use of certain credible sets for sufficiently smooth linear functionals we use this general approach to construct optimal frequentist confidence sets based on the posterior distribution we also provide simulations to numerically illustrate our approach and obtain a visual representation of the geometries involved

fastmmd ensemble of circular discrepancy for efficient two sample test the maximum mean discrepancy mmd is a recently proposed test statistic for the two sample test its quadratic time complexity however greatly hampers its availability to large scale applications to accelerate the mmd calculation in this study we propose an efficient method called fastmmd the core idea of fastmmd is to equivalently transform the mmd with shift invariant kernels into the amplitude expectation of a linear combination of sinusoid components based on bochner s theorem and fourier transform rahimi recht 2007 taking advantage of sampling the fourier transform fastmmd decreases the time complexity for mmd calculation from to where n and d are the size and dimension of the sample set respectively here l is the number of basis functions for approximating kernels that determines the approximation accuracy for kernels that are spherically invariant the computation can be further accelerated to by using the fastfood technique le sarlos smola 2013 the uniform convergence of our method has also been theoretically proved in both unbiased and biased estimates we also provide a geometric explanation for our method ensemble of circular discrepancy which helps us understand the insight of mmd and we hope will lead to more extensive metrics for assessing the two sample test task experimental results substantiate that the accuracy of fastmmd is similar to that of mmd and with faster computation and lower variance than existing mmd approximation methods

multivariate spacings based on data depth i construction of nonparametric multivariate tolerance regions this paper introduces and studies multivariate spacings the spacings are developed using the order statistics derived from data depth specifically the spacing between two consecutive order statistics is the region which bridges the two order statistics in the sense that the region contains all the points whose depth values fall between the depth values of the two consecutive order statistics these multivariate spacings can be viewed as a data driven realization of the so called statistically equivalent blocks these spacings assume a form of center outward layers of shells rings in the two dimensional case where the shapes of the shells follow closely the underlying probabilistic geometry the properties and applications of these spacings are studied in particular the spacings are used to construct tolerance regions the construction of tolerance regions is nonparametric and completely data driven and the resulting tolerance region reflects the true geometry of the underlying distribution this is different from most existing approaches which require that the shape of the tolerance region be specified in advance the proposed tolerance regions are shown to meet the prescribed specifications in terms of content and expectation they are also asymptotically minimal under elliptical distributions finally a simulation and comparison study on the proposed tolerance regions is presented

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

rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density in this paper we investigate the asymptotic properties of nonparametric bayesian mixtures of betas for estimating a smooth density on 0 1 we consider a parametrization of beta distributions in terms of mean and scale parameters and construct a mixture of these betas in the mean parameter while putting a prior on this scaling parameter we prove that such bayesian nonparametric models have good frequentist asymptotic properties we determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a holder class with regularity for all positive leading to a minimax adaptive estimating procedure of the density we also believe that the approximating results obtained on these mixtures of beta densities can be of interest in a frequentist framework

generalised weighted cramer von mises distance estimators summary generalised weighted cramer von mises distance estimators in an arbitrary model with a k dimensional parameter vector are investigated the distance function is defined as a function g of model based residuals for a specified target model f and the empirical cumulative distribution function f over the real line invoking a weight function w it is shown that the estimator is fisher consistent asymptotically multivariate normal and nearly efficient with desirable robustness properties if the true model is equal to the target model the residual function g does not affect the limiting distribution the weight function w controls the asymptotic distribution and the robustness of the estimator three different classes of the weight functions are introduced for different outlier patterns these weight functions produce estimators asymptotically as efficient as the maximum likelihood estimators at the true model an alternative way of calculating the estimators is considered simulation results indicate that asymptotic results are useful for moderate sample sizes and that the estimators are stable at the neighbourhood of the target model

comments on bounded real criteria for linear time delay systems xiefu jiang et al point out some some misleading statements in the above mentioned paper shaked et al 1998 and relation results are given also wen shyong yu points out serious mistakes in the same paper and also untenable main results

thomas bayes walk on manifolds convergence of the bayes posterior measure is considered in canonical statistical settings where observations sit on a geometrical object such as a compact manifold or more generally on a compact metric space verifying some conditions a natural geometric prior based on randomly rescaled solutions of the heat equation is considered upper and lower bound posterior contraction rates are derived

nonlinear processing and analysis of angular signals physical quantities referring to angles like vector direction color hue etc exhibit an inherently periodic nature due to this periodicity digital filters and edge operators proposed for data on the line cannot be applied on such data we introduce filters for angular signals circular mean circular median circular a trimmed mean circular modified trimmed mean particular emphasis is given to the circular median filter for which some interesting properties are derived we also use estimators of circular dispersion to introduce edge detectors for angular signals three variations for the extension of quasirange to circular data are proposed and expressions for their output pdf are derived these circular quasiranges have good and user controlled properties as edge detectors in noisy angular signals the performance of the proposed edge operators is evaluated on angular edges using certain quantitative criteria finally a series of experiments featuring one dimensional 1 d angular signals and hue images is used to illustrate the operation of the new filters and edge detectors

bayesian estimation of the spectral density of a time series this article describes a bayesian approach to estimating the spectral density of a stationary time series a nonparametric prior on the spectral density is described through bernstein polynomials because the actual likelihood is very complicated a pseudoposterior distribution is obtained by updating the prior using the whittle likelihood a markov chain monte carlo algorithm for sampling from this posterior distribution is described that is used for computing the posterior mean variance and other statistics a consistency result is established for this pseudoposterior distribution that holds for a short memory gaussian time series and under some conditions on the prior to prove this asymptotic result a general consistency theorem of schwartz is extended for a triangular array of independent nonidentically distributed observations this extension is also of independent interest a simulation study is conducted to compare the proposed method with some existing methods the method is illustrated with

morphological operators on the unit circle images encoding angular information are common in image analysis examples include the hue band of color images or images encoding directional texture information applying mathematical morphology to image data distributed on the unit circle is not immediately possible as the unit circle is not a lattice three approaches to solving this problem are presented first difference based operators are studied e g gradient top hat second a definition of grouped circular data is suggested and pseudo morphological operators which operate only on grouped data are introduced finally processing using pixel labeling is presented leading to the development of a cyclic opening operator applications for treating the hue band of color images and for finding perturbations in wood texture are given

adaptive bayesian multivariate density estimation with dirichlet mixtures summary we show that rate adaptive multivariate density estimation can be performed using bayesian methods based on dirichlet mixtures of normal kernels with a prior distribution on the kernel s covariance matrix parameter we derive sufficient conditions on the prior specification that guarantee convergence to a true density at a rate that is minimax optimal for the smoothness class to which the true density belongs no prior knowledge of smoothness is assumed the sufficient conditions are shown to hold for the dirichlet location mixture of normals prior with a gaussian base measure and an inverse wishart prior on the covariance matrix parameter locally h older

on adaptive posterior concentration rates tion of holder balls and that moreover achieve our lower bound we analyse the consequences in terms of asymptotic behaviour of poste rior credible balls as well as frequentist minimax adaptive estimation our results are appended with an upper bound for the contraction rate under an arbitrary loss in a generic regular experiment the up per bound is attained for certain sieve priors and enables to extend our results to density estimation

a distribution free two sample run test applicable to high dimensional data we propose a multivariate generalization of the univariate two sample run test based on the shortest hamiltonian path the proposed test is distribution free in finite samples while most existing two sample tests perform poorly or are even inapplicable to high dimensional data our test can be conveniently used in high dimension low sample size situations we investigate its power when the sample size remains fixed and the dimension of the data grows to infinity simulated and real datasets demonstrate our method s superiority over existing nonparametric two sample tests

median filtering in constant time the median filter is one of the basic building blocks in many image processing situations however its use has long been hampered by its algorithmic complexity o tau of in the kernel radius with the trend toward larger images and proportionally larger filter kernels the need for a more efficient median filtering algorithm becomes pressing in this correspondence a new simple yet much faster algorithm exhibiting o 1 runtime complexity is described and analyzed it is compared and benchmarked against previous algorithms extensions to higher dimensional or higher precision data and an approximation to a circular kernel are presented as well

balanced confidence regions based on tukey s depth and the bootstrap summary we propose and study the bootstrap confidence regions for multivariate parameters based on tukey s depth the bootstrap is based on the normalized or studentized statistic formed from an independent and identically distributed random sample obtained from some unknown distribution in rq the bootstrap points are deleted on the basis of tukey s depth until the desired confidence level is reached the proposed confidence regions are shown to be second order balanced in the context discussed by beran we also study the asymptotic consistency of tukey s depth based bootstrap confidence regions the applicability of the method proposed is demonstrated in a simulation study

nonparametric detection of geometric structures over networks nonparametric detection of the possible existence of an anomalous structure over a network is investigated nodes corresponding to the anomalous structure if one exists receive samples generated by a distribution q which is different from a distribution p generating samples for other nodes if an anomalous structure does not exist all nodes receive samples generated by p it is assumed that the distributions p and q are arbitrary and unknown the goal is to design statistically consistent tests with probability of errors converging to zero as the network size becomes asymptotically large kernel based tests are proposed based on maximum mean discrepancy which measures the distance between mean embeddings of distributions into a reproducing kernel hilbert space detection of an anomalous interval over a line network is first studied sufficient conditions on minimum and maximum sizes of candidate anomalous intervals are characterized in order to guarantee that the proposed test is consistent it is also shown that certain necessary conditions must hold in order to guarantee that any test is universally consistent comparison of sufficient and necessary conditions yields that the proposed test is order level optimal and nearly optimal respectively in terms of minimum and maximum sizes of candidate anomalous intervals generalization of the results to other networks is further developed numerical results are provided to demonstrate the performance of the proposed tests

a transfer function approach to the problems of discrete time systems h sub infinity optimal linear control and filtering a solution is derived to the problems of h sub infinity optimal linear state regulation and filtering the solution method for both problems is based on a transfer function approach which applies standard spectral factorization return difference relations are given which are extensions of the relations associated with the linear quadratic problem

technical communique stabilization of discrete time singular systems a matrix inequalities approach this paper is concerned with the state feedback stabilization problem of discrete time singular systems the singular system under consideration is not assumed to be regular the problem we address is to design state feedback controllers such that the resulting closed loop system is regular causal and stable conditions for the existence of solutions to this problem are obtained expressed in terms of certain matrix inequalities when these conditions are satisfied the explicit formula of desired state feedback controllers is also given without resorting to decomposing the system matrices

on the discrete time bounded real lemma with application in the characterization of static state feedback h controllers abstract this paper deals with the discrete time bounded real lemma and its application in the characterization of all static state feedback h controllers for discrete time systems we first present two versions of the discrete time bounded real lemma the results apply to non minimal stable state space realizations and are given in terms of either the positive semi definite strong solution or the positive semi definite stabilizing solution of a discrete algebraic riccati equation based on the discrete time bounded real lemma we obtain a characterization of all static state feedback h controllers for discrete time systems as an application of the parameterization of all static state feedback h controllers we develop a solution to the discrete time h control problem via static output feedback also a numerical procedure is proposed to determine an output feedback gain

notions of limiting p values based on data depth and bootstrap abstract in this article we propose some new notions of limiting p values for hypothesis testing the limiting p value lp here not only provides the usual attractive interpretation of a p value as the strength in support of the null hypothesis coming from the observed data but also has several advantages first it allows us to resample directly from the empirical distribution in the bootstrap implementations rather than from the estimated population distribution satisfying the null constraints second it serves as a test statistic and as a p value simultaneously and thus enables us to obtain test results directly without having to construct an explicit test statistic and then establish or approximate its sampling distribution these are the two steps generally required in a standard testing procedure using bootstrap and the concept of data depth we have provided lp s for a broad class of testing problems where the parameters of interest can be either finite or infinite dimensional some compute

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

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

regression depth commentaries rejoinder in this article we introduce a notion of depth in the regression setting it provides the rank of any line plane rather than ranks of observations or residuals in simple regression we can compute the depth of any line by a fast algorithm for any bivariate dataset z n of size n there exists a line with depth at least n 3 the largest depth in z n can be used as a measure of linearity versus convexity in both simple and multiple regression we introduce the deepest regression method which generalizes the univariate median and is equivariant for monotone transformations of the response throughout the errors may be skewed and heteroscedastic we also consider depth based regression quantiles they estimate the quantiles of y given x as do the koenker bassett regression quantiles but with the advantage of being robust to leverage outliers we explore the analogies between depth in regression and in location where tukey s halfspace depth is a special case of our general definition also liu s simplicial depth can be extended to the regression framework

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

breakdown properties of location estimates based on halfspace depth and projected outlyingness we describe multivariate generalizations of the median trimmed mean and w estimates the estimates are based on a geometric construction related to projection pursuit they are both affine equivariant coordinate free and have high breakdown point the generalization of the median has a breakdown point of at least 1 d 1 in dimension d and the breakdown point can be as high as 1 3 under symmetry in contrast various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than 1 d 1 in dimension d

convergence rates of posterior distributions for non i i d observations we consider the asymptotic behavior of posterior distributions and bayes estimators based on observations which are required to be neither independent nor identically distributed we give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion we then specialize our results to independent nonidentically distributed observations markov processes stationary gaussian time series and the white noise model we apply our general results to several examples of infinite dimensional statistical models including nonparametric regression with normal errors binary regression poisson regression an interval censoring model whittle estimation of the spectral density of a time series and a nonlinear autoregressive model