some multivariate tests of independence based on ranks of nearest neighbors abstractseveral parametric and nonparametric tests of independence between two random vectors are available in the literature but many of them perform poorly for high dimensional data and are not applicable when the dimension exceeds the sample size in this article we propose some tests based on ranks of nearest neighbors which can be conveniently used in high dimension low sample size situations several simulated and real data sets are analyzed to show the utility of the proposed tests codes for implementation of the proposed tests are available as supplementary materials

characterization of the shape stability for nonlinear elliptic problems abstract we characterize all geometric perturbations of an open set for which the solution of a nonlinear elliptic pde of p laplacian type with dirichlet boundary condition is stable in the l norm the necessary and sufficient conditions are jointly expressed by a geometric property associated to the p convergence if the dimension n of the space satisfies n 1 p n and if the number of the connected components of the complements of the moving domains are uniformly bounded a simple characterization of the uniform convergence can be derived in a purely geometric frame in terms of the hausdorff complementary convergence several examples are presented

measuring and testing dependence by correlation of distances distance correlation is a new measure of dependence between random vectors distance covariance and distance correlation are analogous to product moment covariance and correlation but unlike the classical definition of correlation distance correlation is zero only if the random vectors are independent the empirical distance dependence measures are based on certain euclidean distances between sample elements rather than sample moments yet have a compact representation analogous to the classical covariance and correlation asymptotic properties and applications in testing independence are discussed implementation of the test and monte carlo results are also presented

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

a nonparametric test of independence between two vectors abstract a new statistic qcirc n based on interdirections is proposed for testing whether two vector valued quantities are dependent the statistic which has an intuitive invariance property reduces to the quadrant statistic when the two quantities are each univariate under the null hypothesis of independence qcirc n has a limiting chi squared distribution when each vector is elliptically symmetric the new statistic is compared to the classical normal theory competitor wilks likelihood ratio criterion and a componentwise quadrant statistic using a novel model of dependence between the vectors pitman asymptotic relative efficiencies are s are computed the pitman are s indicate that qcirc n compares favorably to wilks likelihood ratio criterion when the vectors have heavy tailed elliptically symmetric distributions and is uniformly better than the componentwise quadrant statistic when the vectors are spherically symmetric a simulation study demonstrates that qcirc n performs better t

on visualization and aggregation of nearest neighbor classifiers nearest neighbor classification is one of the simplest and most popular methods for statistical pattern recognition a major issue in k nearest neighbor classification is how to find an optimal value of the neighborhood parameter k in practice this value is generally estimated by the method of cross validation however the ideal value of k in a classification problem not only depends on the entire data set but also on the specific observation to be classified instead of using any single value of k this paper studies results for a finite sequence of classifiers indexed by k along with the usual posterior probability estimates a new measure called the bayesian measure of strength is proposed and investigated in this paper as a measure of evidence for different classes the results of these classifiers and their corresponding estimated misclassification probabilities are visually displayed using shaded strips these plots provide an effective visualization of the evidence in favor of different classes when a given data point is to be classified we also propose a simple weighted averaging technique that aggregates the results of different nearest neighbor classifiers to arrive at the final decision based on the analysis of several benchmark data sets the proposed method is found to be better than using a single value of k

spectral stability of dirichlet second order uniformly elliptic operators we prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous dirichlet boundary conditions upon domain perturbation

multivariate nonparametric tests of independence new test statistics are proposed for testing whether two random vectors are independent gieser and randles as well as taskinen kankainen and oja have introduced and discussed multivariate extensions of the quadrant test of blomqvist this article serves as a sequel to this work and presents new multivariate extensions of kendall s tau and spearman s rho statistics two different approaches are discussed first interdirection proportions are used to estimate the cosines of angles between centered observation vectors and between differences of observation vectors second covariances between affine equivariant multivariate signs and ranks are used the test statistics arising from these two approaches appear to be asymptotically equivalent if each vector is elliptically symmetric the spatial sign versions are easy to compute for data in common dimensions and they provide practical robust alternatives to normal theory methods asymptotic theory is developed to approximate the finite sample null distr

dirichlet problems on varying domains abstract the aim of the paper is to characterise sequences of domains for which solutions to an elliptic equation with dirichlet boundary conditions converge to a solution of the corresponding problem on a limit domain necessary and sufficient conditions are discussed for strong and uniform convergence for the corresponding resolvent operators examples are given to illustrate that most results are optimal

singularity of eigenfunctions at the junction of shrinking tubes part ii abstract consider two domains connected by a thin tube it can be shown that generically the mass of a given eigenfunction of the dirichlet laplacian concentrates in only one of them the restriction to the other domain when suitably normalized develops a singularity at the junction of the tube as the channel section tends to zero our main result states that under a nondegeneracy condition the normalized limiting profile has a singularity of order n 1 where n is the space dimension we give a precise description of the asymptotic behavior of eigenfunctions at the singular junction which provides us with some important information about its sign near the tunnel entrance more precisely the solution is shown to be one sign in a neighborhood of the singular junction in other words we prove that the nodal set does not enter inside the channel

distance to ck hypersurfaces abstract it is shown by elementary means that a ck hypersurface m of positive reach in r n 1 has the property that the signed distance function to it is ck k 1 this extends and complements work of federer gilbarg and trudinger and serrin

the singularly perturbed domain and the characterization for the eigenfunctions with neumann boundary condition on considere un domaine mobile 0 parametre qui degenere partiellement quand 0 et on donne une caracterisation elaboree des comportements des fonctions propres du laplacien avec la condition limite de neumann

estimating the prediction function and the number of unseen species in sampling with replacement abstract a sample of n units is taken from a population consisting of an unknown number of species we are interested in estimating the number of species and the prediction function for future sampling the prediction function is defined as the expected number of new species that will be found if an additional sample of size tn is taken for any positive real number t in this paper we point out that an estimator suggested by efron and thisted lacks some essential properties of the true prediction function for example the property of alternating copositivity as a result it cannot be used for large values of t we propose an alternative estimator that possesses the essential properties and is easily obtained we illustrate our estimator with two numerical examples and a simulation study

a new coefficient of correlation abstract is it possible to define a coefficient of correlation which is a as simple as the classical coefficients like pearson s correlation or spearman s correlation and yet b consistently es

graph theoretic measures of multivariate association and prediction interpoint distance based graphs can be used to define measures of association that extend kendall s notion of a generalized correlation coefficient the authors present particular statistics that provide distribution free tests of independence sensitive to alternatives involving non monotonic relationships moreover since ordering plays no essential role the ideas that fully applicable in a multivariate setting the authors also define an asymmetric coefficient measuring the extent to which a vector x can be used to make single valued predictions of a vector y the authors discuss various techniques for proving that such statistics are asymptotically normal as an example of the effectiveness of their approach the authors present an application to the examination of residuals from multiple regression 18 references 2 figures 1 table

nonparametric estimation of species richness two procedures the jackknife and the bootstrap are discussed as methods for estimating the number of species by the sampling of quadrats explicit formulas for both procedures are presented and evaluated under a model with a random distribution of individuals the jackknife and bootstrap are shown to reduce the bias although they underestimate the actual number of species if there is a large number of rare species and the number of quadrats sampled is small when a small number of quadrats is sampled the jackknife is shown to give better estimates when the number of quadrats is large the jackknife tends to overestimate the number of species and the bootstrap performs better

sharp convergence rate of eigenvalues in a domain with a shrinking tube abstract in this paper we consider a class of singularly perturbed domains obtained by attaching a cylindrical tube to a fixed bounded region and letting its section shrink to zero we use an almgren type monotonicity formula to evaluate the sharp convergence rate of perturbed simple eigenvalues via courant fischer min max characterization and blow up analysis for scaled eigenfunctions

a consistent multivariate test of association based on ranks of distances summary we consider the problem of detecting associations between random vectors of any dimension few tests of independence exist that are consistent against all dependent alternatives we propose a powerful test that is applicable in all dimensions and consistent against all alternatives the test has a simple form is easy to implement and has good power

the control of the false discovery rate in multiple testing under dependency benjamini and hochberg suggest that the false discovery rate may be the appropriate error rate to control in many applied multiple testing problems a simple procedure was given there as an fdr controlling procedure for independent test statistics and was shown to be much more powerful than comparable procedures which control the traditional familywise error rate we prove that this same procedure also controls the false discovery rate when the test statistics have positive regression dependency on each of the test statistics corresponding to the true null hypotheses this condition for positive dependency is general enough to cover many problems of practical interest including the comparisons of many treatments with a single control multivariate normal test statistics with positive correlation matrix and multivariate t furthermore the test statistics may be discrete and the tested hypotheses composite without posing special difficulties for all other forms of dependency a simple conservative modification of the procedure controls the false discovery rate thus the range of problems for which a procedure with proven fdr control can be offered is greatly increased 1 1 simultaneous hypotheses testing the control of the increased type i error when testing simultaneously a family of hypotheses is a central issue in the area of multiple comparisons rarely are we interested only in whether all hypotheses are jointly true or not which is the test of the intersection null hypothesis in most applications we infer about the individual hypotheses realizing that some of the tested hypotheses are usually true we hope not all and some are not we wish to decide which ones are not true indicating statistical discoveries an important such problem is that of multiple endpoints in a clinical trial a new treatment is compared with an existing one in terms of a large number of potential benefits endpoints

canonical correlation analysis when the data are curves summary it is not immediately straightforward to extend canonical correlation analysis to the context of functional data analysis where the data are themselves curves or functions the obvious approach breaks down and it is necessary to use a method involving smoothing in some way such a method is introduced and discussed with reference to a data set on human gait the breakdown of the unsmoothed method is illustrated in a practical context and is demonstrated theoretically a consistency theorem for the smoothed method is proved in an increasing number of problems in a wide range of fields the data observed are not the univariate or multivariate observations of classical statistics but are func tions observed continuously ramsay and dalzell 1991 give the name functional data analysis to the analysis of data of this kind in most cases the observations will be functions of time or a closely related variable but there are clearly applications where the functions are surfaces observed over two or three dimensional space the motivating example for the present paper is a set of data on children s gait collected by the motion analysis laboratory at children s hospital san diego california see olshen et al 1989 for full details for each of a number of children several angles made by the child s joints knee hip etc are observed during the child s gait cycle one aim of the study is to gain understanding of the gait cycle of a normal child to make comparisons with children suffering from walking diffi culties these data motivated rice and silverman 1991 to discuss the extension of principal component analysis to the functional setting and to explain how smoothing can be incorporated into the analysis in a natural way in a substantial complemen tary paper ramsay and dalzell 1991 discuss different approaches to regression and principal component analysis in functional data analysis illustrating their work by a meteorological example the insights of functional data analysis may be helpful in chemometrics for instance a standard chemometric problem is the analysis of spectra observed in chromatography and these are to all intents and purposes functional observations another obvious area of potential relevance is the analysis of growth curve data the focus in the present paper will be on canonical correlation analysis cca in the gait example one might ask how variability in the knee angle cycle is related to

classification based on hybridization of parametric and nonparametric classifiers parametric methods of classification assume specific parametric models for competing population densities e g gaussian population densities can lead to linear and quadratic discriminant analysis and they work well when these model assumptions are valid violation in one or more of these parametric model assumptions often leads to a poor classifier on the other hand nonparametric classifiers e g nearest neighbor and kernel based classifiers are more flexible and free from parametric model assumptions but the statistical instability of these classifiers may lead to poor performance when we have small numbers of training sample observations nonparametric methods however do not use any parametric structure of population densities therefore even when one has some additional information about population densities that important information is not used to modify the nonparametric classification rule this paper makes an attempt to overcome these limitations of parametric and nonparametric approaches and combines their strengths to develop some hybrid classification methods we use some simulated examples and benchmark data sets to examine the performance of these hybrid discriminant analysis tools asymptotic results on their misclassification rates have been derived under appropriate regularity conditions

a probabilistic nearest neighbour method for statistical pattern recognition summary nearest neighbour algorithms are among the most popular methods used in statistical pattern recognition the models are conceptually simple and empirical studies have shown that their performance is highly competitive against other techniques however the lack of a formal framework for choosing the size of the neighbourhood k is problematic furthermore the method can only make discrete predictions by reporting the relative frequency of the classes in the neighbourhood of the prediction point we present a probabilistic framework for the k nearest neighbour method that largely overcomes these difficulties uncertainty is accommodated via a prior distribution on k as well as in the strength of the interaction between neighbours these prior distributions propagate uncertainty through to proper probabilistic predictions that have continuous support on 0 1 the method makes no assumptions about the distribution of the predictor variables the method is also fully automatic with no user set parameters and empirically it proves to be highly accurate on many bench mark data sets

estimating the number of species a review how many kinds are there suppose that a population is partitioned into c classes in many situations interest focuses not on estimation of the relative sizes of the classes but on estimation of c itself for example biologists and ecologists may be interested in estimating the number of species in a population of plants or animals numismatists may be concemed with estimating the number of dies used to produce an ancient coin issue and linguists may be interested in estimating the size of an author s vocabulary in this article we review the problem of statistical estimation of c many approaches have been proposed some purely data analytic and others based in sampling theory in the latter case numerous variations have been considered the population may be finite or infinite if finite samples may be taken with replacement multinomial sampling or without replacement hypergeometric sampling or by bernoulli sampling if infinite sampling may be multinomial or bernoulli or the sample may be th

on the sharp effect of attaching a thin handle on the spectral rate of convergence abstract consider two domains connected by a thin tube it can be shown that the resolvent of the dirichlet laplacian is continuous with respect to the channel section parameter this in particular implies the continuity of isolated simple eigenvalues and the corresponding eigenfunctions with respect to domain perturbation under an explicit nondegeneracy condition we improve this information providing a sharp control of the rate of convergence of the eigenvalues and eigenfunctions in the perturbed domain to the relative eigenvalue and eigenfunction in the limit domain as an application we prove that again under an explicit nondegeneracy condition the case of resonant domains features polynomial splitting of the two eigenvalues and a clear bifurcation of eigenfunctions

consistent nonparametric tests of independence three simple and explicit procedures for testing the independence of two multi dimensional random variables are described two of the associated test statistics l1 log likelihood are defined when the empirical distribution of the variables is restricted to finite partitions a third test statistic is defined as a kernel based independence measure two kinds of tests are provided distribution free strong consistent tests are derived on the basis of large deviation bounds on the test statistics these tests make almost surely no type i or type ii error after a random sample size asymptotically level tests are obtained from the limiting distribution of the test statistics for the latter tests the type i error converges to a fixed non zero value and the type ii error drops to zero for increasing sample size all tests reject the null hypothesis of independence if the test statistics become large the performance of the tests is evaluated experimentally on benchmark data

optimal prediction of the number of unseen species abstract estimating the number of unseen species is an important problem in many scientific endeavors its most popular formulation introduced by fisher et al fisher ra corbet as williams cb 1943 j animal ecol 12 1 42 58 uses n samples to predict the number u of hitherto unseen species that would be observed if t n t n new samples were collected of considerable interest is the largest ratio t between the number of new and existing samples for which u can be accurately predicted in seminal works good and toulmin good i toulmin g 1956 biometrika 43 102 45 63 constructed an intriguing estimator that predicts u for all t 1 t 1 subsequently efron and thisted efron b thisted r 1976 biometrika 63 3 435 447 proposed a modification that empirically predicts u even for some t 1 t 1 but without provable guarantees we derive a class of estimators that provably predict u all of the way up to t log n t log n we also show that this range is the best possible and that the estimator s mean square error is near optimal for any t our approach yields a provable guarantee for the efron thisted estimator and in addition a variant with stronger theoretical and experimental performance than existing methodologies on a variety of synthetic and real datasets the estimators are simple linear computationally efficient and scalable to massive datasets their performance guarantees hold uniformly for all distributions and apply to all four standard sampling models commonly used across various scientific disciplines multinomial poisson hypergeometric and bernoulli product