inference on impulse response functions in structural var models skepticism toward traditional identifying assumptions based on exclusion restrictions has led to a surge in the use of structural var models in which structural shocks are identified by restricting the sign of the responses of selected macroeconomic aggregates to these shocks researchers commonly report the vector of pointwise posterior medians of the impulse responses as a measure of central tendency of the estimated response functions along with pointwise 68 percent posterior error bands it can be shown that this approach cannot be used to characterize the central tendency of the structural impulse response functions we propose an alternative method of summarizing the evidence from sign identified var models designed to enhance their practical usefulness our objective is to characterize the most likely admissible model s within the set of structural var models that satisfy the sign restrictions we show how the set of most likely structural response functions can be computed from the posterior mode of the joint distribution of admissible models both in the fully identified and in the partially identified case and we propose a highest posterior density credible set that characterizes the joint uncertainty about this set our approach can also be used to resolve the long standing problem of how to conduct joint inference on sets of structural impulse response functions in exactly identified var models we illustrate the differences between our approach and the traditional approach for the analysis of the effects of monetary policy shocks and of the effects of oil demand and oil supply shocks

on certain bivariate sign tests and medians abstract brown and hettmansperger proposed an affine invariant bivariate analogue of the sign test the os test based on the generalized median of oja on the other hand oja and nyblom introduced a family of locally most powerful affine invariant sign tests in the case of elliptic distributions the locally most powerful blumen s test and the os test are shown to be asymptotically equivalent formulas for calculating asymptotic relative efficiencies of the os test and the oja bivariate median are given it is shown that if the contours of a distribution are of a similar shape the relative efficiencies of the os test and blumen s test depend on the distribution only through the shape of the contours for the power family of contours x 1 p x 2 p c p 0 numerical calculations show that the efficiency of the os test relative to blumen s test attains its minimum 1 as p 2 spherical elliptic case and increases to infinity as p 0 in the bivariate laplace case with independent marginals p

dd classifier nonparametric classification procedure based on dd plot using the dd plot depth vs depth plot we introduce a new nonparametric classification algorithm and call it dd classifier the algorithm is completely nonparametric and it requires no prior knowledge of the underlying distributions or the form of the separating curve thus it can be applied to a wide range of classification problems the algorithm is completely data driven and its classification outcome can be easily visualized in a two dimensional plot regardless of the dimension of the data moreover it has the advantage of bypassing the estimation of underlying parameters such as means and scales which is often required by the existing classification procedures we study the asymptotic properties of the dd classifier and its misclassification rate specifically we show that dd classifier is asymptotically equivalent to the bayes rule under suitable conditions and it can achieve bayes error for a family broader than elliptical distributions the performance of the classifier is also examined u

an elliptic analogue of the multiple gamma function a hierarchy of functions including the elliptic gamma function is introduced it can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a q analogue of the multiple gamma function some properties of the functions are considered

hybrid confidence regions based on data depth summary we consider the general problem of constructing confidence regions for possibly multi dimensional parameters when we have available more than one approach for the construction these approaches may be motivated by different model assumptions different levels of approximation different settings of tuning parameters or different monte carlo algorithms their effectiveness is often governed by different sets of conditions which are difficult to vindicate in practice we propose two procedures for constructing hybrid confidence regions which endeavour to integrate all such individual approaches the procedures employ the concept of data depth to calibrate the confidence region in two different ways the first rendering its coverage error minimax and the second rendering its coverage error conservative the resulting region reconciles in many important aspects the discrepancies between the various approaches and is robust against misspecification of their governing conditions theoretical and empirical properties of our procedures are investigated in comparison with those of the constituent individual approaches

nested support vector machines one class and cost sensitive support vector machines svms are state of the art machine learning methods for estimating density level sets and solving weighted classification problems respectively however the solutions of these svms do not necessarily produce set estimates that are nested as the parameters controlling the density level or cost asymmetry are continuously varied such nesting not only reflects the true sets being estimated but is also desirable for applications requiring the simultaneous estimation of multiple sets including clustering anomaly detection and ranking we propose new quadratic programs whose solutions give rise to nested versions of one class and cost sensitive svms furthermore like conventional svms the solution paths in our construction are piecewise linear in the control parameters although here the number of breakpoints is directly controlled by the user we also describe decomposition algorithms to solve the quadratic programs these methods are compared to conventional non nested svms on synthetic and benchmark data sets and are shown to exhibit more stable rankings and decreased sensitivity to parameter settings

incomplete generalized l statistics given data x 1 x n and a kernel h with m arguments serfling introduced the class of generalized l statistics gl statistics which is defined by taking linear combinations of the ordered h x i1 x im where i 1 i m ranges over all n n m distinct m tuples of 1 n in this paper we derive a class of incomplete generalized l statistics igl statistics by taking linear combinations of the ordered elements from a subset of h x i1 x im with size n n a special case is the class of incomplete u statistics introduced by blom under very general conditions the igl statistic is asymptotically equivalent to the gl statistic as soon as n n n as n which makes the igl much more computationally feasible we also discuss various ways of selecting the subset of h x i x im several examples are discussed in particular some new estimates of the scale parameter in nonparametric regression are introduced it is shown that these estimates are asymptotically equivalent to an igl statistic some extensions for example functionals other than l and multivariate kernels are also addressed

twisting with a flip the art of pestunization we construct mathcal n 2 supersymmetric yang mills theory on 4d manifolds with a killing vector field with isolated fixed points it turns out that for every fixed point one can allocate either instanton or anti instanton contributions to the partition function and that this is compatible with supersymmetry the equivariant donaldson witten theory is a special case of our construction we present a unified treatment of pestun s calculation on s 4 and equivariant donaldson witten theory by generalizing the notion of self duality on manifolds with a vector field we conjecture the full partition function for a mathcal n 2 theory on any 4d manifold with a killing vector using this new notion of self duality to localize a supersymmetric theory is what we call pestunization

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

multivariate functional halfspace depth this article defines and studies a depth for multivariate functional data by the multivariate nature and by including a weight function it acknowledges important characteristics of functional data namely differences in the amount of local amplitude shape and phase variation we study both population and finite sample versions the multivariate sample of curves may include warping functions derivatives and integrals of the original curves for a better overall representation of the functional data via the depth we present a simulation study and data examples that confirm the good performance of this depth function supplementary materials for this article are available online

estimation of extreme depth based quantile regions consider the extreme quantile region induced by the halfspace depth function hd of the form q fx 2 rd hd x p g such that pq p for a given very small p 0 this region can hardly be estimated through a fully nonparametric procedure since the sample halfspace depth is 0 outside the convex hull of the data using extreme value theory we construct a natural semiparametric estimator of this quantile region and prove a refined consistency result a simulation study clearly demonstrates the good performance of our estimator we use the procedure for risk management by applying it to stock market returns

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

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

a decomposition of total variation depth for understanding functional outliers there has been extensive work on data depth based methods for robust multivariate data analysis recent developments have moved to infinite dimensional objects such as functional data in this wor

exploring the hydrological robustness of model parameter values with alpha shapes estimation of parameter values in hydrological models has gradually moved from subjective trial and error methods into objective estimation methods translation of nature s complexity to bit opera

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

gluing nekrasov partition functions in this paper we summarise the localisation calculation of 5d super yang mills on simply connected toric sasaki einstein se manifolds we show how various aspects of the computation including the equivariant index the asymptotic behaviour and the factorisation property are governed by the combinatorial data of the toric geometry we prove that the perturbative partition function on a simply connected se manifold corresponding to an n gon toric diagram factorises to n copies of perturbative part zero instanton sector of the nekrasov partition function this leads us to conjecture a prescription for the computation of the complete partition function by gluing n copies of the full nekrasov partition functions this work is a generalisation of some earlier computation carried out on yp q manifolds whose moment map cone has a quadrangle base and our result is valid for manifolds whose moment map cones have pentagon base hexagon base etc the algorithm we used for dealing with general cones may also be of independent interest

multivariate quantiles and multiple output regression quantiles from l1 optimization to halfspace depth a new multivariate concept of quantile based on a directional version of koenker and bassett s traditional regression quantiles is introduced for multivariate location and multiple output regression problems in their empirical version those quantiles can be computed efficiently via linear programming techniques consistency bahadur representation and asymptotic normality results are established most importantly the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of tukey this relation does not only allow for efficient depth contour computations by means of parametric linear programming but also for transferring from the quantile to the depth universe such asymptotic results as bahadur representations finally linear programming duality opens the way to promising developments in depth related multivariate rank based inference

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

nonparametric imputation by data depth abstractwe present single imputation method for missing values which borrows the idea of data depth a measure of centrality defined for an arbitrary point of a space with respect to a probability d

concentration of the empirical level sets of tukey s halfspace depth tukey s halfspace depth has attracted much interest in data analysis because it is a natural way of measuring the notion of depth relative to a cloud of points or more generally to a probability measure given an i i d sample we investigate the concentration of upper level sets of the tukey depth relative to that sample around their population version we show that under some mild assumptions on the underlying probability measure concentration occurs at a parametric rate and we deduce moment inequalities at that same rate in a computational prospective we study the concentration of a discretized version of the empirical upper level sets

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

the limit shape of convex hull peeling we prove that the convex peeling of a random point set in dimension d approximates motion by the 1 d 1 power of gaussian curvature we use viscosity solution theory to interpret the limiting partial differential equation pde we use the martingale method to solve the cell problem associated to convex peeling our proof follows the program of armstrong and cardaliaguet for homogenization of geometric motions but with completely different ingredients

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

on the limiting distributions of multivariate depth based rank sum statistics and related tests a depth based rank sum statistic for multivariate data introduced by liu and singh j amer statist assoc 88 1993 252 260 as an extension of the wilcoxon rank sum statistic for univariate data has been used in multivariate rank tests in quality control and in experimental studies those applications however are based on a conjectured limiting distribution provided by liu and singh j amer statist assoc 88 1993 252 260 the present paper proves the conjecture under general regularity conditions and therefore validates various applications of the rank sum statistic in the literature the paper also shows that the corresponding rank sum tests can be more powerful than hotelling s t 2 test and some commonly used multivariate rank tests in detecting location scale changes in multivariate distributions

on the concept of depth for functional data the statistical analysis of functional data is a growing need in many research areas in particular a robust methodology is important to study curves which are the output of many experiments in applied statistics as a starting point for this robust analysis we propose analyze and apply a new definition of depth for functional observations based on the graphic representation of the curves given a collection of functions it establishes the centrality of an observation and provides a natural center outward ordering of the sample curves robust statistics such as the median function or a trimmed mean function can be defined from this depth definition its finite dimensional version provides a new depth for multivariate data that is computationally feasible and useful for studying high dimensional observations thus this new depth is also suitable for complex observations such as microarray data images and those arising in some recent marketing and financial studies natural properties of these

first order analytic difference equations and integrable quantum systems we present a new solution method for a class of first order analytic difference equations the method yields explicit minimal solutions that are essentially unique special difference equations give rise to minimal solutions that may be viewed as generalized gamma functions of hyperbolic trigonometric and elliptic type euler s gamma function being of rational type we study these generalized gamma functions in considerable detail the scattering and weight functions u and w functions associated to various integrable quantum systems can be expressed in terms of our generalized gamma functions we obtain detailed information on these u and w functions exploiting the difference equations they satisfy

median based classifiers for high dimensional data conventional distance based classifiers use standard euclidean distance and so can suffer from excessive volatility if vector components have heavy tailed distributions this difficulty can be alleviated by replacing the l2 distance by its l1 counterpart for example the l1 version of the popular centroid classifier would allocate a new data value to the population to whose centroid it was closest in l1 terms however this approach can lead to inconsistency because the centroid is defined using l2 rather than l1 distance in particular by mixing l1 and l2 approaches we produce a classifier that can seriously misidentify data in cases where the means and medians of marginal distributions take different values these difficulties motivate replacing centroids by medians however in the very high dimensional settings commonly encountered today this can be problematic if we attempt to work with a conventional spatial median therefore we suggest using componentwise medians to construct a robust class

maximal inequalities for degenerate u processes with applications to optimization estimators maximal inequalities for degenerate u processes of order k k geq 1 are established the results rest on a moment inequality due to bonami for k th order forms and on extensions of chaining and symmetrization inequalities from the theory of empirical processes rates of uniform convergence are obtained the maximal inequalities can be used to determine the limiting distribution of estimators that optimize criterion functions having u process structure as an application a semiparametric regression estimator that maximizes a u process of order 3 is shown to be sqrt n consistent and asymptotically normally distributed

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

a rank based multivariate cusum procedure we consider statistical process control when measurements are multivariate a cumulative sum cusum procedure is suggested in detecting a shift in the mean vector of the measurements which is based on the cross sectional antiranks of the measurements at each time point the measurements are ordered and their antiranks which are the indices of the order statistics are recorded when the process is in control and the joint distribution of the multivariate measurements satisfies some regularity conditions the antirank vector at each time point has a given distribution this distribution changes to some other distribution when the process is out of control and the components of the shift in the mean vector of the process are not all the same this cusum can therefore detect shifts in all directions except the one in which the components of the shift in the mean vector are all the same but not 0 the shift with equal components however can be easily detected by another univariate cusum the former cusu

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

control charts for multivariate processes abstract this article uses the concept of data depth to introduce several new control charts for monitoring processes of multivariate quality measurements for any dimension of the measurements these charts are in the form of two dimensional graphs that can be visualized and interpreted just as easily as the well known univariate x x and cusum charts moreover they have several significant advantages first they can detect simultaneously the location shift and scale increase of the process unlike the existing methods which can detect only the location shift second their construction is completely nonparametric in particular it does not require the assumption of normality for the quality distribution which is needed in standard approaches such as the 2 and hotelling s t 2 charts thus these new charts generalize the principle of control charts to multivariate settings and apply to a much broader class of quality distributions

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

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

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

limit theorems for u processes a u process is a collection of u statistics concretely a u process over a family h of kernels h of m variables based on a probability measure p on s s is the collection u n m h p h in mathcal h of u statistics this chapter is devoted to the asymptotic theory of u processes we are interested in finding conditions on h and p ensuring that the law of large numbers or the central limit theorem or the law of the iterated logarithm hold for u n m h p uniformly in h h the theory of empirical processes deals with the same questions for the case m 1 and u process theory is patterned after it this is a relatively new subject at least in the generality presented here h being an arbitrary collection of kernels which are defined on a general measurable space there is therefore a need to indicate its usefulness to this effect a section on applications is added at the end of the chapter section 5 5 there we illustrate the use of each of the main theorems in this chapter by deriving properties of certain multidimensional generalizations of the median and in general of m estimators and by studying estimators of the cumulative hazard and distribution functions of a random variable based on truncated data

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

convergence of depth contours for multivariate datasets contours of depth often provide a good geometrical understanding of the structure of a multivariate dataset they are also useful in robust statistics in connection with generalized medians and data ordering if the data constitute a random sample from a spherical or elliptic distribution the depth contours are generally required to converge to spherical or elliptical shapes we consider contour constructions based on a notion of data depth and prove a uniform contour convergence theorem under verifiable conditions on the depth measure applications to several existing depth measures discussed in the literature are also considered

an approach to multivariate covariate dependent quantile contours with application to bivariate conditional growth charts multivariate quantile contours are useful in numerous applications and have been studied in different contexts however no easy solutions exist when dynamic and conditional quantile contours are needed without strong distributional assumptions in this article we propose a new form of bivariate quantile contours and a two stage estimation procedure to take time effect into account the proposed procedure relies on quantile regression for longitudinal data and is flexible to include potentially important covariates as necessary in addition we propose a visual model assessment tool and discuss a practical guideline for model selection the performance of the proposed methodology is demonstrated by a simulation study as well as an application to joint height weight screening of young children in the united states we construct bivariate growth charts by a nested sequence of age dependent and covariate varying quantile contours of height and weight and use it to locate an individual subject s percentile

the spatial distribution in infinite dimensional spaces and related quantiles and depths the spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data in this paper we investigate the concept of spatial distribution for data in infinite dimensional banach spaces many technical difficulties are encountered in such spaces that are primarily due to the non compactness of the closed unit ball in this work we prove some glivenko cantelli and donsker type results for the empirical spatial distribution process in infinite dimensional spaces the spatial quantiles in such spaces can be obtained by inverting the spatial distribution function a bahadur type asymptotic linear representation and the associated weak convergence results for the sample spatial quantiles in infinite dimensional spaces are derived a study of the asymptotic efficiency of the sample spatial median relative to the sample mean is carried out for some standard probability distributions in function spaces the spatial distribution can be used to define the spatial depth in infinite dimensional banach spaces and we study the asymptotic properties of the empirical spatial depth in such spaces we also demonstrate the spatial quantiles and the spatial depth using some real and simulated functional data

from continuous time random walks to the fractional fokker planck equation we generalize the continuous time random walk ctrw to include the effect of space dependent jump probabilities when the mean waiting time diverges we derive a fractional fokker planck equation ffpe this equation describes anomalous diffusion in an external force field and close to thermal equilibrium we discuss the domain of validity of the fractional kinetic equation for the force free case we compare between the ctrw solution and that of the ffpe

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

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

a distribution free multivariate phase i location control chart for subgrouped data from elliptical distributions in quality control a proper phase i analysis is essential to the success of phase ii monitoring a literature review reveals no distribution free phase i multivariate techniques in existence this research develops a phase i location control chart for multivariate elliptical processes the resulting in control reference sample can then be used to estimate the parameters for phase ii monitoring using monte carlo simulation the proposed method is compared with the hotelling s t2 phase i chart although hotelling s t2 chart is preferred when the data are multivariate normal the proposed method is shown to perform significantly better under nonnormality this article has supplementary material online

on depth and deep points a calculus for a general definition of depth in data analysis a differential like calculus is constructed in which the location case the framework of tukey s median plays a fundamental role similar to that of linear functions in the mathematical analysis as an application a lower bound for maximal regression depth is proved in the general multidimensional case as conjectured by rousseeuw and hubert and others this lower bound is demonstrated to have an impact on the breakdown point of the maximum depth estimator

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

applications of robust distances for regression the dd plot introduced by rousseeuw and van driessen 1999 is a plot of classical vs robust mahalanobis distances mdi vs rdi the dd plot can be used as a diagnostic for multivariate normality and elliptical symmetry and to assess the success of numerical transformations towards elliptical symmetry in the regression context many procedures can be adversely aected if strong nonlinearities are present in the predictors even if strong nonlinearities are present the robust distances can be used to help visualize important regression models such as generalized linear models

penalty based aggregation of multidimensional data abstract research in aggregation theory is nowadays still mostly focused on algorithms to summarize tuples consisting of observations in some real interval or of diverse general ordered structures of course in practice of information processing many other data types between these two extreme cases are worth inspecting this contribution deals with the aggregation of lists of data points in r d for arbitrary d 1 even though particular functions aiming to summarize multidimensional data have been discussed by researchers in data analysis computational statistics and geometry there is clearly a need to provide a comprehensive and unified model in which their properties like equivariances to geometric transformations internality and monotonicity may be studied at an appropriate level of generality the proposed penalty based approach serves as a common framework for all idempotent information aggregation methods including componentwise functions pairwise distance minimizers and data depth based medians it also allows for deriving many new practically useful tools

a direct verification argument for the hamilton jacobi equation continuum limit of nondominated sorting abstract nondominated sorting is a combinatorial algorithm that sorts points in euclidean space into layers according to a partial order it was recently shown that nondominated sorting of random points has a hamilton jacobi equation continuum limit the original proof given in calder et al 2014 relies on a continuum variational problem in this paper we give a new proof using a direct verification argument that completely avoids the variational interpretation we believe this may be generalized to apply to other stochastic homogenization problems for which there is no obvious underlying variational principle

the modular properties and the integral representations of the multiple elliptic gamma functions abstract we show the modular properties of the multiple elliptic gamma functions which are an extension of those of the theta function and the elliptic gamma function the modular property of the theta function is known as jacobi s transformation and that of the elliptic gamma function was provided by felder and varchenko in this paper we deal with the multiple sine functions since the modular properties of the multiple elliptic gamma functions result from the equivalence between two ways to represent the multiple sine functions as infinite products we also derive integral representations of the multiple sine functions and the multiple elliptic gamma functions we introduce correspondences between the multiple elliptic gamma functions and the multiple sine functions

orientational relaxation in a dispersive dynamic medium generalization of the kubo ivanov anderson jump diffusion model to include fractional environmental dynamics the ivanov anderson model and an earlier treatment by kubo envisages a decay of the orientational correlation by random but large amplitude molecular jumps as opposed to infinitesimal small jumps assumed in brownian diffusion recent computer simulation studies on water and viscous liquids have shown that large amplitude motions may indeed be more of a rule than exception existing theoretical studies on jump diffusion mostly assume an exponential poissonian waiting time distribution for jumps thereby again leading to an exponential decay here we extend the existing formalism of ivanov and anderson to include an algebraic waiting time distribution between two jumps as a result the first l 1 and second l 2 rank orientational time correlation functions show the same long time power law but their short time decay behavior is quite different the predicted cole cole plot of dielectric relaxation reproduces various features of non debye behavior observed experimentally we also developed a theory where both unrestricted small jumps and large angular jumps coexist simultaneously the small jumps are shown to have a large effect on the long time decay particularly in mitigating the effects of algebraic waiting time distribution and in giving rise to an exponential like decay with a time constant surprisingly less than the time constant that arises from small amplitude decay alone

the elliptic gamma function and sl 3 z z3 abstract the elliptic gamma function is a generalization of the euler gamma function and is associated to an elliptic curve its trigonometric and rational degenerations are the jackson q gamma function and the euler gamma function respectively the elliptic gamma function appears in baxter s formula for the free energy of the eight vertex model and in the hypergeometric solutions of the elliptic qkzb equations in this paper the properties of this function are studied in particular we show that elliptic gamma functions are generalizations of automorphic forms of g sl 3 z z 3 associated to a non trivial class in h3 g z

multiple elliptic gamma functions associated to cones abstract we define generalizations of the multiple elliptic gamma functions and the multiple sine functions associated to good rational cones we explain how good cones are related to collections of s l r z elements and prove that the generalized multiple sine and multiple elliptic gamma functions enjoy infinite product representations and modular properties determined by the cone this generalizes the modular properties of the elliptic gamma function studied by felder and varchenko and the results about the usual multiple sine and elliptic gamma functions found by narukawa

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

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems