outlier detection with the kernelized spatial depth function statistical depth functions provide from the deepest point a center outward ordering of multidimensional data in this sense depth functions can measure the extremeness or outlyingness of a data point with respect to a given data set hence they can detect outliers observations that appear extreme relative to the rest of the observations of the various statistical depths the spatial depth is especially appealing because of its computational efficiency and mathematical tractability in this article we propose a novel statistical depth the kernelized spatial depth ksd which generalizes the spatial depth via positive definite kernels by choosing a proper kernel the ksd can capture the local structure of a data set while the spatial depth fails we demonstrate this by the half moon data and the ring shaped data based on the ksd we propose a novel outlier detection algorithm by which an observation with a depth value less than a threshold is declared as an outlier the proposed algorithm is simple in structure the threshold is the only one parameter for a given kernel it applies to a one class learning setting in which normal observations are given as the training data as well as to a missing label scenario where the training set consists of a mixture of normal observations and outliers with unknown labels we give upper bounds on the false alarm probability of a depth based detector these upper bounds can be used to determine the threshold we perform extensive experiments on synthetic data and data sets from real applications the proposed outlier detector is compared with existing methods the ksd outlier detector demonstrates a competitive performance

a simple two sample test in high dimensions based on l2 norm testing the equality of two means is a fundamental inference problem for high dimensional data the hotelling s t2 test either performs poorly or becomes inapplicable several modifications have b

an affine invariant bivariate version of the sign test abstract the generalized median of h oja yields a notion of bivariate quantile and in turn an affine invariant bivariate analogue of the sign test its properties include a simple null covariance formula facilitating a permutation or sign change test in the case of bivariate symmetry normal efficiency coinciding with that of the oja median and bounded influence hence strong robustness author

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

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

exact nonparametric inference when doses are measured with random errors studies that estimate the effects of exposure to a possibly harmful agent often compare exposed subjects who received varied doses with matched controls who received zero dose if the doses are measured with error then one may wish to use the fallible doses to estimate a linear relationship between the unobserved true dose and the observed response if one is willing to assume that the dose errors for exposed subjects are symmetrically distributed about 0 that is the dose errors are pure errors and not say systematic underreporting of exposure then the presence of zero dose controls is all that is needed to obtain exact distribution free confidence intervals and tests and consistent point estimates the method is simpler for matched pairs than for matched sets with two or more matched subjects and it is illustrated using two studies one of each kind with matched pairs as in this first example the method uses wilcoxon s signed rank test as the basis for inference when there are several zero dos

analysis of fronts in reaction dispersal processes the existence of traveling wave front solutions with a minimum speed selected for reaction dispersal processes is studied we obtain a general existence condition in terms of the waiting time and dispersal distance probability distribution functions and we detail this result for situations of ecological interest in particular when particles disperse according to jumps of short length and any waiting time probability distribution function we show that the minimum speed selection for traveling wave fronts is not always possible so the waiting time and the dispersal distance distributions cannot be arbitrarily chosen

random walk in chemical space of cantor dust as a paradigm of superdiffusion we point out that the chemical space of a totally disconnected cantor dust k n ensuremath subset e n is a compact metric space c n with the spectral dimension d s d ensuremath ell ngd where d and d ensuremath ell n are the fractal and chemical dimensions of k n respectively hence we can define a random walk in the chemical space as a markovian gaussian process the mapping of a random walk in c n into k n ensuremath subset e n defines the quenched l evy flight on the cantor dust with a single step duration independent of the step length the equations describing the superdiffusion and diffusion reaction front propagation ruled by the local quenched l evy flight on k n ensuremath subset e n are derived the use of these equations to model superdiffusive phenomena observed in some physical systems in which propagators decay faster than algebraically is discussed

optimal tests for multivariate location based on interdirections and pseudo mahalanobis ranks we propose a family of tests based on randies 1989 concept of inter directions and the ranks of pseudo mahalanobis distances computed with respect to a multivariate m estimator of scatter due to tyler 1987 for the multivariate one sample problem under elliptical symmetry these tests which generalize the univariate signed rank tests are affine invariant depending on the score function considered van der waerden laplace they allow for locally asymptotically maximin tests at selected densities multivariate normal multivariate double exponential local powers and asymptotic relative efficiencies are derived with respect to hotelling s test randies 1989 multivariate sign test peters and randies 1990 wilcoxon type test and with respect to the oja median tests we moreover extend to the multivariate setting two famous univariate results the traditional chemoff savage 1958 property showing that hotelling s traditional procedure is uniformly dominated in the pitman sense by the van der waerden version of our tests and the celebrated hodges lehmann 1956 864 result providing for any fixed space dimension k the lower bound for the asymptotic relative efficiency of wilcoxon type tests with respect to hotelling s these asymptotic results are confirmed by a monte carlo investigation and application to a real data set

interdirection tests for simple repeated measures designs abstract interdirection tests are proposed for a simple repeated measures design the test statistics proposed are applications of the one sample interdirection sign test and interdirection signed rank test to a repeated measurement setting the interdirection sign test has a small sample distribution free property and includes the two sided univariate sign test and blumen s bivariate sign test as special cases the interdirection signed rank test includes the two sided univariate wilcoxon signed rank test as a special case the proposed statistics are shown to have a limiting x p 1 2 null distribution when the underlying distribution is elliptically symmetric in addition the asymptotic distributions of the proposed statistics under certain contiguous alternatives are obtained for elliptically symmetric distributions with a particular density function form pitman asymptotic relative efficiencies and monte carlo studies show the proposed interdirection tests to be robust as compared to several competito

multivariate location estimation using extension of r estimates through u statistics type approach we consider a class of u statistics type estimates for multivariate location the estimates extend some r estimates to multivariate data in particular the class of estimates includes the multivariate median considered by gini and galvani 1929 and haldane 1948 and a multivariate extension of the well known hodges lehmann 1963 estimate we explore large sample behavior of these estimates by deriving a bahadur type representation for them in the process of developing these asymptotic results we observe some interesting phenomena that closely resemble the famous shrinkage phenomenon observed by stein 1956 in high dimensions interestingly the phenomena that we observe here occur even in dimension d 2

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

online estimation of the geometric median in hilbert spaces nonasymptotic confidence balls estimation procedures based on recursive algorithms are interesting and powerful techniques that are able to deal rapidly with very large samples of high dimensional data the collected data may be contaminated by noise so that robust location indicators such as the geometric median may be preferred to the mean in this context an estimator of the geometric median based on a fast and efficient averaged nonlinear stochastic gradient algorithm has been developed by bernoulli19 2013 18 43 this work aims at studying more precisely the nonasymptotic behavior of this nonlinear algorithm by giving nonasymptotic confidence balls in general separable hilbert spaces this new result is based on the derivation of improved l 2 rates of convergence as well as an exponential inequality for the nearly martingale terms of the recursive nonlinear robbins monro algorithm

on an adaptive transformation retransformation estimate of multivariate location an affine equivariant estimate of multivariate location based on an adaptive transformation and retransformation approach is studied the work is primarily motivated by earlier work on different versions of the multivariate median and their properties we explore an issue related to efficiency and equivariance that was originally raised by bickel and subsequently investigated by brown and hettmansperger our estimate has better asymptotic performance than the vector of co ordinatewise medians when the variables are substantially correlated the finite sample performance of the estimate is investigated by using monte carlo simulations some examples are presented to demonstrate the effect of the adaptive transformation retransformation strategy in the construction of multivariate location estimates for real data

probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion the proliferation and migration dichotomy of the tumor cell invasion is examined within a two component continuous time random walk ctrw model the balance equations for the cancer cells of two phenotypes with random switching between cell proliferation and migration are derived the transport of tumor cells is formulated in terms of the ctrw with an arbitrary waiting time distribution law while proliferation is modeled by a logistic growth the overall rate of tumor cell invasion for normal diffusion and subdiffusion is determined

innovated higher criticism for detecting sparse signals in correlated noise higher criticism is a method for detecting signals that are both sparse and weak although first proposed in cases where the noise variables are independent higher criticism also has reasonable performance in settings where those variables are correlated in this paper we show that by exploiting the nature of the correlation performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer indeed it turns out that the case of independent noise is the most difficult of all from a statistical viewpoint and that more accurate signal detection for a given level of signal sparsity and strength can be obtained when correlation is present we characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary the boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is toeplitz

estimation of error variance via ridge regression summarywe propose a novel estimator of error variance and establish its asymptotic properties based on ridge regression and random matrix theory the proposed estimator is valid under both low and high dimensional models and performs well not only in nonsparse cases but also in sparse ones the finite sample performance of the proposed method is assessed through an intensive numerical study which indicates that the method is promising compared with its competitors in many interesting scenarios

a multivariate signed rank test for the one sample location problem abstract an affine invariant signed rank test is proposed for the one sample multivariate location problem the test suggested is a modification of randles s multivariate sign test based on interdirections which extends blumen s bivariate procedure to the multidimensional setting comparisons are made between the proposed statistic and several competitors via pitman asymptotic relative efficiencies and monte carlo results the signed rank statistic appears to be robust it performs better than its competitors when the distribution is light tailed and virtually as well as hotelling s t 2 under multivariate normality for heavy tailed distributions the signed rank statistic performs better than hotelling s t 2 but not as well as randles s statistic

affine invariant rank methods in the bivariate location model abstract the generalized multivariate median of h oja is used to define a multivariate notion of quantile or rank and to define a measure of scatter of multivariate linear models the latter when applied to the one and two sample bivariate location models yields affine invariant analogs of the wilcoxon rank sum and signed rank tests and of the corresponding estimates additional keywords dispersion measures rank statics author

multidelayed random walks theory and application to the neolithic transition in europe we present a model in which particles or individuals of a biological population disperse with a rest time between consecutive motions or migrations which may take several possible values from a discrete set particles or individuals may also react or reproduce we derive a new equation for the effective rest time stackrel ifmmode tilde else fi t of the random walk application to the neolithic transition in europe makes it possible to derive more realistic theoretical values for its wavefront speed than those following from the single delayed framework presented previously j fort and v m endez phys rev lett 82 867 1999 the new results are consistent with the archaeological observations of this important historical process

generalized pseudo riemannian geometry generalized tensor analysis in the sense of colombeau s construction is employed to introduce a nonlinear distributional pseudo riemannian geometry in particular after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo riemannian metric generalized connection and generalized curvature tensor we prove a fundamental lemma of pseudo riemannian geometry in this setting and define the notion of geodesics of a generalized metric finally we present applications of the resulting theory to general relativity

tests for high dimensional data based on means spatial signs and spatial ranks tests based on sample mean vectors and sample spatial signs have been studied in the recent literature for high dimensional data with the dimension larger than the sample size for suitable sequences of alternatives we show that the powers of the mean based tests and the tests based on spatial signs and ranks tend to be same as the data dimension grows to infinity for any sample size when the coordinate variables satisfy appropriate mixing conditions further their limiting powers do not depend on the heaviness of the tails of the distributions this is in striking contrast to the asymptotic results obtained in the classical multivariate setup on the other hand we show that in the presence of stronger dependence among the coordinate variables the spatial sign and rank based tests for high dimensional data can be asymptotically more powerful than the mean based tests if in addition to the data dimension the sample size also grows to infinity the sizes of some mean based tests for high dimensional data studied in the recent literature are observed to be significantly different from their nominal levels this is due to the inadequacy of the asymptotic approximations used for the distributions of those test statistics however our asymptotic approximations for the tests based on spatial signs and ranks are observed to work well when the tests are applied on a variety of simulated and real datasets

sign tests in multidimension inference based on the geometry of the data cloud abstract multivariate sign tests attracted several statisticians in the past and it is evident from recent nonparametric literature that they still continue to draw attention one of the most important features of the univariate sign test is that it does not involve much technical assumptions or complicacy and this makes it quite popular among statistics users in this article we have come up with a new method for constructing multivariate sign tests that have reasonable statistical properties and can be used conveniently to solve one sample location problems our principal strategy here is to make a wise utilization of certain geometric structures in the constellation of data points for making inference about the location of their distribution as we proceed with the development of a fairly broad and general methodology we indicate its relationship with previous work done by others and sometimes attempt to unify some of the earlier ideas in particular we pick up some well known tests for uniform dis

on a geometric notion of quantiles for multivariate data abstract an extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered the approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles both of which have been extensively studied in the literature these geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other l estimates of multivariate location and they lead to a directional notion of central and extreme points in a multidimensional setup such quantiles can be defined as meaningful and natural objects even in infinite dimensional hilbert and banach spaces and they yield an effective generalization of quantile regression in multiresponse linear model problems desirable equivariance properties are shown to hold for these multivariate quantiles and issues related to their computation for data in finite dimensional spaces are discussed n 1 2 consistenc

biased random walks and propagation failure the critical value of the reaction rate able to sustain the propagation of an invasive front is obtained for general non markovian biased random walks with reactions from the hamilton jacobi equation corresponding to the mean field equation we find that the critical reaction rate depends only on the mean waiting time and on the statistical properties of the jump length probability distribution function and is always underestimated by the diffusion approximation if the reaction rate is larger than the jump frequency invasion always succeeds even in the case of maximal bias numerical simulations support our analytical predictions

bahadur representation of m sb m estimates we take a unified approach to asymptotic properties of m m estimates based on i i d observations defined through the minimization of a real valued criterion function of one or more variables our results are applicable to a host of location and scale estimators found in the literature

diagonal likelihood ratio test for equality of mean vectors in high dimensional data we propose a likelihood ratio test framework for testing normal mean vectors in high dimensional data under two common scenarios the one sample test and the two sample test with equal covariance matrices we derive the test statistics under the assumption that the covariance matrices follow a diagonal matrix structure in comparison with the diagonal hotelling s tests our proposed test statistics display some interesting characteristics in particular they are a summation of the log transformed squared t statistics rather than a direct summation of those components more importantly to derive the asymptotic normality of our test statistics under the null and local alternative hypotheses we do not need the requirement that the covariance matrices follow a diagonal matrix structure as a consequence our proposed test methods are very flexible and readily applicable in practice simulation studies and a real data analysis are also carried out to demonstrate the advantages of our likelihood ratio test methods

tests for high dimensional regression coefficients with factorial designs we propose simultaneous tests for coefficients in high dimensional linear regression models with factorial designs the proposed tests are designed for the large p small n situations where the conventional f test is no longer applicable we derive the asymptotic distribution of the proposed test statistic under the high dimensional null hypothesis and various scenarios of the alternatives which allow power evaluations we also evaluate the power of the f test for models of moderate dimension the proposed tests are employed to analyze a microarray data on yorkshire gilts to find significant gene ontology terms which are significantly associated with the thyroid hormone after accounting for the designs of the experiment

a distribution free multivariate sign test based on interdirections abstract distribution free tests are investigated for the one sample multivariate location problem counts called interdirections which measure the angular distance between two observation vectors relative to the positions of the other observations are introduced these counts are invariant under nonsingular linear transformations and have a small sample distribution free property over a broad class of population models called distributions with elliptical directions which includes all elliptically symmetric populations and many skewed populations a sign test based on interdirections is described including as special cases the two sided univariate sign test and blumen s bivariate sign test the statistic is shown to have a limiting 2 p null distribution and because it is based on interdirections it is also seen to be invariant and to have a small sample distribution free property pitman asymptotic relative efficiencies and a monte carlo study show the test to perform well compared with hotell

anomalous diffusion with linear reaction dynamics from continuous time random walks to fractional reaction diffusion equations we have revisited the problem of anomalously diffusing species modeled at the mesoscopic level using continuous time random walks to include linear reaction dynamics if a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term if the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term we have also developed further extensions of the ctrw model to include more general reaction dynamics

front propagation in hyperbolic fractional reaction diffusion equations from the continuous time random walk scheme and assuming a l evy waiting time distribution typical of subdiffusive transport processes we study a hyperbolic reaction diffusion equation involving time fractional derivatives the linear speed selection of wave fronts in this equation is analyzed when the reaction diffusion dimensionless number and the fractional index satisfy a certain condition we find fronts exhibiting an unphysical behavior they travel faster in the subdiffusive than in the diffusive regime

a high dimensional nonparametric multivariate test for mean vector this work is concerned with testing the population mean vector of nonnormal high dimensional multivariate data several tests for high dimensional mean vector based on modifying the classical hotelling t2 test have been proposed in the literature despite their usefulness they tend to have unsatisfactory power performance for heavy tailed multivariate data which frequently arise in genomics and quantitative finance this article proposes a novel high dimensional nonparametric test for the population mean vector for a general class of multivariate distributions with the aid of new tools in modern probability theory we proved that the limiting null distribution of the proposed test is normal under mild conditions when p is substantially larger than n we further study the local power of the proposed test and compare its relative efficiency with a modified hotelling t2 test for high dimensional data an interesting finding is that the newly proposed test can have even more substantial power gain with l

tests alternative to higher criticism for high dimensional means under sparsity and column wise dependence we consider two alternative tests to the higher criticism test of donoho and jin ann statist 32 2004 962 994 for high dimensional means under the sparsity of the nonzero means for sub gaussian distributed data with unknown column wise dependence the two alternative test statistics are constructed by first thresholding l 1 and l 2 statistics based on the sample means respectively followed by maximizing over a range of thresholding levels to make the tests adaptive to the unknown signal strength and sparsity the two alternative tests can attain the same detection boundary of the higher criticism test in ann statist 32 2004 962 994 which was established for uncorrelated gaussian data it is demonstrated that the maximal l 2 thresholding test is at least as powerful as the maximal l 1 thresholding test and both the maximal l 2 and l 1 thresholding tests are at least as powerful as the higher criticism test

pricing on electricity market based on coupled continuous time random walk concept abstract in this paper we propose a model of electricity market based on the forward rate dynamics described by a diffusion with jumps as a generalization of the classical diffusion approach we consider jump components resulting from a coupled continuous time random walk ctrw with jump lengths proportional to the corresponding inter jump time intervals in the framework of the model we derive a formula for the euro price of a standard european call option showing applicability of ctrw processes for pricing of financial instruments the result obtained by an advance theory of semimartingales is an essential extension of the pricing formula derived in the classical diffusion model of the forward rate dynamics it indicates an influence of both the continuous and the jump parts of the forward rate process on the option price

statistical uses of the spatial median the spatial median called the mediancentre in an earlier paper by j c gower is defined as the bivariate location measure to minimize the sum of absolute distances to observations its asymptotic efficiency relative to the sample mean vector with normal data is shown to exceed the usual univariate 2 ir 0 637 its estimating equations have an angular aspect and are used to develop angle tests which are analogues of sign tests in one dimension

multivariate sign based high dimensional tests for the two sample location problem abstractthis article concerns tests for the two sample location problem when data dimension is larger than the sample size existing multivariate sign based procedures are not robust against high dimensionality producing tests with type i error rates far away from nominal levels this is mainly due to the biases from estimating location parameters we propose a novel test to overcome this issue by using the leave one out idea the proposed test statistic is scalar invariant and thus is particularly useful when different components have different scales in high dimensional data asymptotic properties of the test statistic are studied compared with other existing approaches simulation studies show that the proposed method behaves well in terms of sizes and power supplementary materials for this article are available online

asymptotic data analysis on manifolds given an m dimensional compact submanifold m of euclidean space r s the concept of mean location of a distribution related to mean or expected vector is generalized to more general r s valued functionals including median location which is derived from the spatial median the asymptotic statistical inference for general functionals of distributions on such submanifolds is elaborated convergence properties are studied in relation to the behavior of the underlying distributions with respect to the cutlocus an application is given in the context of independent but not identically distributed samples in particular to a multisample setup

description of diffusive and propagative behavior on fractals the known properties of diffusion on fractals are reviewed in order to give a general outlook of these dynamic processes after that we propose a description developed in the context of the intrinsic metric of fractals which leads us to a differential equation able to describe diffusion in real fractals in the asymptotic regime we show that our approach has a stronger physical justification than previous works on this field the most important result we present is the introduction of a dependence on time and space for the conductivity in fractals which is deduced by scaling arguments and supported by computer simulations finally the diffusion equation is used to introduce the possibility of reaction diffusion processes on fractals and analyze their properties specifically an analytic expression for the speed of the corresponding travelling fronts which can be of great interest for application purposes is derived

comparing large covariance matrices under weak conditions on the dependence structure and its application to gene clustering comparing large covariance matrices has important applications in modern genomics where scientists are often interested in understanding whether relationships e g dependencies or co regulations among a large number of genes vary between different biological states we propose a computationally fast procedure for testing the equality of two large covariance matrices when the dimensions of the covariance matrices are much larger than the sample sizes a distinguishing feature of the new procedure is that it imposes no structural assumptions on the unknown covariance matrices hence the test is robust with respect to various complex dependence structures that frequently arise in genomics we prove that the proposed procedure is asymptotically valid under weak moment conditions as an interesting application we derive a new gene clustering algorithm which shares the same nice property of avoiding restrictive structural assumptions for high dimensional genomics data using an asthma gene expression dataset we illustrate how the new test helps compare the covariance matrices of the genes across different gene sets pathways between the disease group and the control group and how the gene clustering algorithm provides new insights on the way gene clustering patterns differ between the two groups the proposed methods have been implemented in an r package hdtest and are available on cran

continuous time random walks and traveling fronts we present a geometric approach to the problem of propagating fronts into an unstable state valid for an arbitrary continuous time random walk with a fisher kolmogorov petrovski piskunov growth reaction rate we derive an integral hamilton jacobi type equation for the action functional determining the position of reaction front and its speed our method does not rely on the explicit derivation of a differential equation for the density of particles in particular we obtain an explicit formula for the propagation speed for the case of anomalous transport involving non markovian random processes

detecting case control expression quantitative trait loci using locally most powerful or maximin robust rank tests in testing genome wide gene expression quantitative trait loci efficiency robust statistical methods and their computational convenience are most relevant for this purpose we propose to use a modified locally most powerful rank test for the analysis of case control expression data this modified rank test statistic is computationally simple robust for non normally distributed expression data and asymptotically locally most powerful it depends on the specification of a location distribution form for data but is not sensitive to misspecifications when such a location distribution form cannot be specified we apply gastwirth s maximin efficiency robust rank test to gene expression data to maximize the worst pitman asymptotic relative efficiency among a family of location distributions we conduct simulation studies to assess their performance and use an application to real data for illustration copyright 2011 john wiley sons ltd

affine invariant multivariate multisample sign tests summary invariant sign type tests for the multivariate multisample location models are developed the definition of sign is based on the oja criterion function emphasis is on the twosample location model and the associated equivariant estimate of shift asymptotic theory is developed to provide large sample approximations for the null distribution of the test and to provide a framework for a discussion of efficiency tests are then developed for the one way and two way lay outs

inference on multivariate m estimators based on bivariate censored data consider two random variables subject to random right censoring for example the survival times of twins or two recurrence times of a certain disease recorded on the same individual of interest is the estimation of a multivariate m functional of these two random variables asymptotic results for the proposed estimator are established under general conditions on the criterion function of the m estimator the obtained results include the asymptotic normality of the estimator its local power and construction of a confidence region our primary example of interest is the bivariate l1 median we compare the small sample behavior of this estimator with that of the vector of marginal medians by means of a simulation study finally we consider a dataset on kidney dialysis patients and estimate the median time to two different infections for these individuals

affine invariant multivariate one sample signed rank tests abstract brown and hettmansperger introduced affine invariant bivariate analogs of the sign rank and signed rank tests based on the oja median in this article affine invariant k variate extensions of the one sample signed rank test and the hodges lehmann estimate are considered the necessary distribution theory is developed and asymptotic pitman efficiencies with respect to hotelling s t 2 test under multivariate t distributions are tabulated an application of the signed rank tests to a repeated measurement setting is presented

test for high dimensional regression coefficients using refitted cross validation variance estimation testing a hypothesis for high dimensional regression coefficients is of fundamental importance in the statistical theory and applications in this paper we develop a new test for the overall significance of coefficients in high dimensional linear regression models based on an estimated u statistics of order two with the aid of the martingale central limit theorem we prove that the asymptotic distributions of the proposed test are normal under two different distribution assumptions refitted cross validation rcv variance estimation is utilized to avoid the overestimation of the variance and enhance the empirical power we examine the finite sample performances of the proposed test via monte carlo simulations which show that the new test based on the rcv estimator achieves higher powers especially for the sparse cases we also demonstrate an application by an empirical analysis of a microarray data set on yorkshire gilts

nonparametric analysis of clustered multivariate data there is wide interest in extending univariate and multivariate nonparametric procedures to clustered and hierarchical data traditionally parametric mixed models have been used to account for the correlation structures among the dependent observational units in this work we extend multivariate nonparametric procedures for one sample and several sample location problems to clustered data settings the results are given for a general score function but with an emphasis on spatial sign and rank methods mixed models notation involving design matrices for fixed and random effects is used throughout the asymptotic variance formulas and limiting distributions of the test statistics under the null hypothesis and under a sequence of alternatives are derived as are the limiting distributions for the corresponding estimates the approach based on a general score function also shows for example how m estimates behave with clustered data efficiency studies demonstrate the practical advantages and disadvantag

multivariate sign based high dimensional tests for sphericity this article concerns tests for sphericity in cases where the data dimension is larger than the sample size the existing multivariate sign based procedure hallin paindaveine 2006 for sphericity is not robust with respect to high dimensionality producing tests with type i error rates that are much larger than the nominal levels this is mainly due to bias from estimating the location parameter we develop a correction that makes the existing test statistic robust with respect to high dimensionality and show that the proposed test statistic is asymptotically normal under elliptical distributions the proposed method allows the dimensionality to increase as the square of the sample size simulations demonstrate that it has good size and power in a wide range of settings

semiparametrically efficient inference based on signed ranks in symmetric independent component models we consider semiparametric location scatter models for which the p variate observation is obtained as x lambda z mu where mu is a p vector lambda is a full rank p p matrix and the unobserved random p vector z has marginals that are centered and mutually independent but are otherwise unspecified as in blind source separation and independent component analysis ica the parameter of interest throughout the paper is lambda on the basis of n i i d copies of x we develop under a symmetry assumption on z signed rank one sample testing and estimation procedures for lambda we exploit the uniform local and asymptotic normality ulan of the model to define signed rank procedures that are semiparametrically efficient under correctly specified densities yet as usual in rank based inference the proposed procedures remain valid correct asymptotic size under the null for hypothesis testing and root n consistency for point estimation under a very broad range of densities we derive the asymptotic properties of the proposed procedures and investigate their finite sample behavior through simulations

two sample test of high dimensional means under dependence summary the paper considers in the high dimensional setting a canonical testing problem in multivariate analysis namely testing the equality of two mean vectors we introduce a new test statistic that is based on a linear transformation of the data by the precision matrix which incorporates the correlations between the variables the limiting null distribution of the test statistic and the power of the test are analysed it is shown that the test is particularly powerful against sparse alternatives and enjoys certain optimality a simulation study is carried out to examine the numerical performance of the test and to compare it with other tests given in the literature the results show that the test proposed significantly outperforms those tests in a range of settings

the wave equation on singular space times we prove local unique solvability of the wave equation for a large class of weakly singular locally bounded space time metrics in a suitable space of generalised functions

on the efficiency of multivariate spatial sign and rank tests conditionally and asymptotically distribution free rotation but not scale invariant competitors of the well known hotelling s t 2 tests in this paper efficiency results are found in the one sample location case but they natu rally hold in the multivariate two sample multisample and general linear model case the lack of means of fully ordering or ranking observations in the multivariate setting seems at first sight to be an obstacle to generalizing the concepts of sign and rank and signed rank however these concepts can be defined also by using l criterion functions as follows consider k 1 variate observations y y utilizing the euclidean distance y s 1 n 2 2 1r2 y q qy the three objective functions 1 k t y s s y y i i i i i

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

a class of distribution free two sample tests based on placements abstract we consider a class of two sample distribution free tests that are appropriate for situations where one of the sample sizes is large relative to the other these procedures are based on the placements of the observations in the smaller sample among the ordered observations in the larger sample and this class of tests generalizes the mann whitney 1947 procedure in much the same way that the class of linear rank tests generalizes the equivalent wilcoxon 1945 rank sum form optimality criteria for choosing a test from this class are discussed and limiting distributions for the associated class of test statistics are determined for the case where only one of the sample sizes goes to infinity

semiparametrically efficient rank based inference for shape i optimal rank based tests for sphericity we propose a class of rank based procedures for testing that the shape matrix v of an elliptical distribution with unspecified center of symmetry scale and radial density has some fixed value v 0 this includes for v 0 i k the problem of testing for sphericity as an important particular case the proposed tests are invariant under translations monotone radial transformations rotations and reflections with respect to the estimated center of symmetry they are valid without any moment assumption for adequately chosen scores they are locally asymptotically maximin in the le cam sense at given radial densities they are strictly distribution free when the center of symmetry is specified and asymptotically so when it must be estimated the multivariate ranks used throughout are those of the distances in the metric associated with the null value v 0 of the shape matrix between the observations and the estimated center of the distribution local powers against elliptical alternatives and asymptotic relative efficiencies ares are derived with respect to the adjusted mauchly test a modified version of the gaussian likelihood ratio procedure proposed by muirhead and waternaux biometrika 67 1980 31 43 or equivalently with respect to an extension of the test for sphericity introduced by john biometrika 59 1972 169 173 for gaussian scores these ares are uniformly larger than one irrespective of the actual radial density necessary and or sufficient conditions for consistency under nonlocal possibly nonelliptical alternatives are given finite sample performance is investigated via a monte carlo study

remarks on the distributional schwarzschild geometry this work is devoted to a mathematical analysis of the distributional schwarzschild geometry the schwarzschild solution is extended to include the singularity the energy momentum tensor becomes a distribution supported at r 0 using generalized distributional geometry in the sense of colombeau s special construction the nonlinearities are treated in a mathematically rigorous way moreover generalized function techniques are used as a tool to give a unified discussion of various approaches taken in the literature so far in particular we comment on geometrical issues

a two sample test for high dimensional data with applications to gene set testing we proposed a two sample test for means of high dimensional data when the data dimension is much larger than the sample size the classical hotelling s t 2 test does not work for this large p small n situation the proposed test does not require explicit conditions on the relationship between the data dimension and sample size this offers much flexibility in analyzing high dimensional data an application of the proposed test is in testing significance for sets of genes which we demonstrate in an empirical study on a leukemia data set

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

an exactly solvable model of subdiffusion reaction front propagation the propagation of a reaction front is considered in the framework of the seki lindenberg reaction subdiffusion model which is appropriate in the case of a diffusion limited reaction typically this kind of problem is solved using approximate approaches or numerical methods we apply a model piecewise linear reaction function which allows for exact analytical solutions for a front between two stable homogeneous states a unique value of the front velocity is found in the case of fronts between a stable and an unstable state a family of traveling wave solutions is revealed the conditions for the existence of a special pushed front are considered the influence of advection on the front velocity is also analyzed

fronts with continuous waiting time distributions theory and application to virus infections we generalize to arbitrary waiting time distributions some results which were previously derived for discrete distributions we show that for any two waiting time distributions with the same mean delay time that with higher dispersion will lead to a faster front experimental data on the speed of virus infections in a plaque are correctly explained by the theoretical predictions using a gaussian delay time distribution which is more realistic for this system than the dirac delta distribution considered previously j fort and v mendez phys rev lett 89 178101 2002

dynamical features of reaction diffusion fronts in fractals the speed of front propagation in fractals is studied by using i the reduction of the reaction transport equation into a hamilton jacobi equation and ii the local equilibrium approach different equations proposed for describing transport in fractal media together with logistic reaction kinetics are considered finally we analyze the main features of wave fronts resulting from this dynamic process i e why they are accelerated and what is the exact form of this acceleration

estimating sufficient reductions of the predictors in abundant high dimensional regressions we study the asymptotic behavior of a class of methods for sufficient dimension reduction in high dimension regressions as the sample size and number of predictors grow in various alignments it is demonstrated that these methods are consistent in a variety of settings particularly in abundant regressions where most predictors contribute some information on the response and oracle rates are possible simulation results are presented to support the theoretical conclusion

empirical likelihood test for a large dimensional mean vector summarythis paper is concerned with empirical likelihood inference on the population mean when the dimension p and the sample size n satisfy p n rightarrow c in 1 infty as shown in tsao 2004 the empirical likelihood method fails with high probability when p n 1 2 because the convex hull of the n observations in mathbb r p becomes too small to cover the true mean value moreover when p n the sample covariance matrix becomes singular and this results in the breakdown of the first sandwich approximation for the log empirical likelihood ratio to deal with these two challenges we propose a new strategy of adding two artificial data points to the observed data we establish the asymptotic normality of the proposed empirical likelihood ratio test the proposed test statistic does not involve the inverse of the sample covariance matrix furthermore its form is explicit so the test can easily be carried out with low computational cost our numerical comparison shows that the proposed test outperforms some existing tests for high dimensional mean vectors in terms of power we also illustrate the proposed procedure with an empirical analysis of stock data