on the rate of convergence in wasserstein distance of the empirical measure let mu n be the empirical measure associated to a n sample of a given probability distribution mu on mathbb r d we are interested in the rate of convergence of mu n to mu when measured in the wasserstein distance of order p 0 we provide some satisfying non asymptotic l p bounds and concentration inequalities for any values of p 0 and d ge 1 we extend also the non asymptotic l p bounds to stationary rho mixing sequences markov chains and to some interacting particle systems

mean rates of convergence of empirical measures in the wasserstein metric abstract an upper bound is given for the mean square wasserstein distance between the empirical measure of a sequence of i i d random vectors and the common probability law of the sequence the same result holds for an infinite exchangeable sequence and its directing measure similarly for an i i d sequence of stochastic processes an upper bound is obtained for the mean square of the maximum over 0 t t of the wasserstein distance between the empirical measure of the sequence at time t and the common marginal law at t these upper bounds are derived under weak assumptions and are not very far from the known rate of convergence pertaining to an i i d sequence of uniform random vectors on the unit cube our approach however allows us to get results for arbitrary distributions under moment conditions and also gives results for processes an application is given to so called diffusions with jumps moment estimates for these processes are derived which may be of independent interest

remarks on the solution of the position dependent mass schrodinger equation an approximate method is proposed to solve the position dependent mass pdm schrodinger equation the procedure suggested here leads to the solution of the pdm schrodinger equation without transforming the potential function to the mass space or vice versa the method based on the asymptotic taylor expansion of the function produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the pdm schrodinger equation the results show that the pdm and constant mass schrodinger equations are not isospectral the calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm

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

rates of convergence for partial mass problems we consider a class of partial mass problems in which a fraction of the mass of a probability measure is allowed to be changed trimmed to maximize fit to a given pattern this includes the problem of optimal partial transportation of mass where a part of the mass need not be transported and also trimming procedures which are often used in statistical data analysis to discard outliers in a sample the data with lowest agreement to a certain pattern this results in a modified trimmed version of the original probability which is closer to the pattern we focus on the case of the empirical measure and analyze to what extent its optimally trimmed version is closer to the true random generator in terms of rates of convergence we deal with probabilities on mathbb r k and measure agreement through probability metrics our choices include transportation cost metrics associated to optimal partial transportation and the kolmogorov distance we show that partial transportation as opposed to classical complete transportation results in a sharp decrease of costs only in low dimension in contrast for the kolmogorov metric this decrease is seen in any dimension

permutation based simultaneous confidence bounds for the false discovery proportion when multiple hypotheses are tested interest is often in ensuring that the proportion of false discoveries fdp is small with high confidence in this paper confidence upper bounds for the fdp are constructed which are simultaneous over all rejection cut offs in particular this allows the user to select a set of hypotheses post hoc such that the fdp lies below some constant with high confidence our method uses permutations to account for the dependence structure in the data so far only meinshausen provided an exact permutation based and computationally feasible method for simultaneous fdp bounds we provide an exact method which uniformly improves this procedure further we provide a generalization of this method it lets the user select the shape of the simultaneous confidence bounds this gives the user more freedom in determining the power properties of the method interestingly several existing permutation methods such as significance analysis of microarrays sam and westfall and young s maxt method are obtained as special cases

inference for empirical wasserstein distances on finite spaces summary the wasserstein distance is an attractive tool for data analysis but statistical inference is hindered by the lack of distributional limits to overcome this obstacle for probability measures supported on finitely many points we derive the asymptotic distribution of empirical wasserstein distances as the optimal value of a linear programme with random objective function this facilitates statistical inference e g confidence intervals for sample based wasserstein distances in large generality our proof is based on directional hadamard differentiability failure of the classical bootstrap and alternatives are discussed the utility of the distributional results is illustrated on two data sets

a stagewise rejective multiple test procedure based on a modified bonferroni test summary simes 1986 has proposed a modified bonferroni procedure for the test of an overall hypothesis which is the combination of n individual hypotheses in contrast to the classical bonferroni procedure it is not obvious how statements about individual hypotheses are to be made for this procedure in the present paper a multiple test procedure allowing statements on individual hypotheses is proposed it is based on the principle of closed test procedures marcus peritz gabriel 1976 and controls the multiple level a

some probability inequalities for ordered rm mtp sb 2 random variables a proof of the simes conjecture some new probability inequalities involving the ordered components of an mtp 2 random vector are derived which provide an analytical proof of an important conjecture in the field of multiple hypothesis testing this conjecture has been mostly validated so far using simulation

robustness of multiple testing procedures against dependence an important aspect of multiple hypothesis testing is controlling the significance level or the level of type i error when the test statistics are not independent it can be particularly challenging to deal with this problem without resorting to very conservative procedures in this paper we show that in the context of contemporary multiple testing problems where the number of tests is often very large the difficulties caused by dependence are less serious than in classical cases this is particularly true when the null distributions of test statistics are relatively light tailed for example when they can be based on normal or student s t approximations there if the test statistics can fairly be viewed as being generated by a linear process an analysis founded on the incorrect assumption of independence is asymptotically correct as the number of hypotheses diverges in particular the point process representing the null distribution of the indices at which statistically significant test results occur is approximately poisson just as in the case of independence the poisson process also has the same mean as in the independence case and of course exhibits no clustering of false discoveries however this result can fail if the null distributions are particularly heavy tailed there clusters of statistically significant results can occur even when the null hypothesis is correct we give an intuitive explanation for these disparate properties in light and heavy tailed cases and provide rigorous theory underpinning the intuition

the exact solution of the schrodinger equation with a polynomially spatially varying mass the schrodinger equation with a position dependent mass sepdm is employed in many areas of quantum physics exact solutions for the sepdm lie at the center of interest of the professional public because it helps us to understand the behavior of quantum particles in the cases in which their mass varies spatially for this purpose we used the mass function represented by a quartic polynomial and a quadratic potential function which extends the current class of exact solutions of the sepdm the exact analytical solution of the problem is expressed as a linear combination of local heun functions heun s equation contains many parameters resulting in its general nature we studied how limit changes in some of these parameters will affect the solution of the sepdm the obtained solutions are particularly suitable for the transfer matrix method and solutions of scattering problems this is demonstrated by the calculation of bound states the schrodinger equation with a position dependent mass sepdm is employed in many areas of quantum physics exact solutions for the sepdm lie at the center of interest of the professional public because it helps us to understand the behavior of quantum particles in the cases in which their mass varies spatially for this purpose we used the mass function represented by a quartic polynomial and a quadratic potential function which extends the current class of exact solutions of the sepdm the exact analytical solution of the problem is expressed as a linear combination of local heun functions heun s equation contains many parameters resulting in its general nature we studied how limit changes in some of these parameters will affect the solution of the sepdm the obtained solutions are particularly suitable for the transfer matrix method and solutions of scattering problems this is demonstrated by the calculation of bound states

conditional association and unidimensionality in monotone latent variable models latent variable models represent the joint distribution of observable variables in terms of a simple structure involving unobserved or latent variables usually assuming the conditional independence of the observable variables given the latent variables these models play an important role in educational measurement and psychometrics in sociology and in population genetics and are implicit in some work on systems reliability we study a broad class of latent variable models namely the monotone unidimensional models in which the latent variable is a scalar the observable variables are conditionally independent given the latent variable and the conditional distribution of the observables given the latent variable is stochastically increasing in the latent variable all models in this class imply a new strong form of positive dependence among the observable variables namely conditional positive association this positive dependence condition may be used to test whether any model in this class can provide an adequate fit to observed data various applications generalizations and a numerical example are discussed

testing the conditional independence and monotonicity assumptions of item response theory when item characteristic curves are nondecreasing functions of a latent variable the conditional or local independence of item responses given the latent variable implies nonnegative conditional covariances between all monotone increasing functions of a set of item responses given any function of the remaining item responses this general result provides a basis for testing the conditional independence assumption without first specifying a parametric form for the nondecreasing item characteristic curves the proposed tests are simple have known asymptotic null distributions and possess certain optimal properties in an example the conditional independence hypothesis is rejected for all possible forms of monotone item characteristic curves

estimating the proportion of false null hypotheses among a large number of independently tested hypotheses we consider the problem of estimating the number of false null hypotheses among a very large number of independently tested hypotheses focusing on the situation in which the proportion of false null hypotheses is very small we propose a family of methods for establishing lower 100 1 a confidence bounds for this proportion based on the empirical distribution of the p values of the tests methods in this family are then compared in terms of ability to consistently estimate the proportion by letting 0 as the number of hypothesis tests increases and the proportion decreases this work is motivated by a signal detection problem that occurs in astronomy

novel relations and new properties of confluent heun s functions and their derivatives of arbitrary order the present paper reveals important properties of the confluent heun s functions we derive a set of novel relations for confluent heun s functions and their derivatives of arbitrary order specific new subclasses of confluent heun s functions are introduced and studied a new alternative derivation of confluent heun s polynomials is presented

asymptotic optimality of the westfall young permutation procedure for multiple testing under dependence test statistics are often strongly dependent in large scale multiple testing applications most corrections for multiplicity are unduly conservative for correlated test statistics resulting in a loss of power to detect true positives we show that the westfallyoung permutation method has asymptotically optimal power for a broad class of testing problems with a block dependence and sparsity structure among the tests when the number of tests tends to infinity

large scale multiple testing under dependence summary the paper considers the problem of multiple testing under dependence in a compound decision theoretic framework the observed data are assumed to be generated from an underlying two state hidden markov model we propose oracle and asymptotically optimal datadriven procedures that aim to minimize the false non discovery rate fnr subject to a constraint on the false discovery rate fdr it is shown that the performance of a multiple testing procedure can be substantially improved by adaptively exploiting the dependence structure among hypotheses and hence conventional fdr procedures that ignore this structural information are inefficient both theoretical properties and numerical performances of the procedures proposed are investigated it is shown that the procedures proposed control fdr at the desired level enjoy certain optimality properties and are especially powerful in identifying clustered non null cases the new procedure is applied to an influenza like illness surveillance study for detecting the timing of epidemic periods

the sequential rejection principle of familywise error control closed testing and partitioning are recognized as fundamental principles of familywise error control in this paper we argue that sequential rejection can be considered equally fundamental as a general principle of multiple testing we present a general sequentially rejective multiple testing procedure and show that many well known familywise error controlling methods can be constructed as special cases of this procedure among which are the procedures of holm shaffer and hochberg parallel and serial gatekeeping procedures modern procedures for multiple testing in graphs resampling based multiple testing procedures and even the closed testing and partitioning procedures themselves we also give a general proof that sequentially rejective multiple testing procedures strongly control the familywise error if they fulfill simple criteria of monotonicity of the critical values and a limited form of weak familywise error control in each single step the sequential rejection principle gives a novel theoretical perspective on many well known multiple testing procedures emphasizing the sequential aspect its main practical usefulness is for the development of multiple testing procedures for null hypotheses possibly logically related that are structured in a graph we illustrate this by presenting a uniform improvement of a recently published procedure

properties of higher criticism under strong dependence the problem of signal detection using sparse faint information is closely related to a variety of contemporary statistical problems including the control of false discovery rate and classification using very high dimensional data each problem can be solved by conducting a large number of simultaneous hypothesis tests the properties of which are readily accessed under the assumption of independence in this paper we address the case of dependent data in the context of higher criticism methods for signal detection short range dependence has no first order impact on performance but the situation changes dramatically under strong dependence there although higher criticism can continue to perform well it can be bettered using methods based on differences of signal values or on the maximum of the data the relatively inferior performance of higher criticism in such cases can be explained in terms of the fact that under strong dependence the higher criticism statistic behaves as though the data were partitioned into very large blocks with all but a single representative of each block being eliminated from the dataset

the schrodinger equation with position dependent mass it is shown that the traditional hartree second order shooting method for eigenvalues can be modified to give accurate energy levels for a schrodinger equation involving a position dependent mass some numerical applications are given to demonstrate the effectiveness of the method

adaptive linear step up procedures that control the false discovery rate we provide a new two stage procedure in which the linear step up procedure is used in stage one to estimate mo providing a new level q which is used in the linear step up procedure in the second stage we prove that a general form of the two stage procedure controls the false discovery rate at the desired level q this framework enables us to study analytically the properties of other procedures that exist in the literature a simulation study is presented that shows that two stage adaptive procedures improve in power over the original procedure mainly because they provide tighter control of the false discovery rate we further study the performance of the current suggestions some variations of the procedures and previous suggestions in the case where the test statistics are positively dependent a case for which the original procedure controls the false discovery rate in the setting studied here the newly proposed two stage procedure is the only one that controls the false discovery rate the procedures are illustrated with two examples of biological importance

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

false discovery control with p value weighting we present a method for multiple hypothesis testing that maintains control of the false discovery rate while incorporating prior information about the hypotheses the prior information takes the form of p value weights if the assignment of weights is positively associated with the null hypotheses being false the procedure improves power except in cases where power is already near one even if the assignment of weights is poor power is only reduced slightly as long as the weights are not too large we also provide a similar method to control false discovery exceedance

adaptive false discovery rate control under independence and dependence in the context of multiple hypothesis testing the proportion 0 of true null hypotheses in the pool of hypotheses to test often plays a crucial role although it is generally unknown a priori a testing procedure using an implicit or explicit estimate of this quantity in order to improve its efficency is called adaptive in this paper we focus on the issue of false discovery rate fdr control and we present new adaptive multiple testing procedures with control of the fdr in a first part assuming independence of the p values we present two new procedures and give a unified review of other existing adaptive procedures that have provably controlled fdr we report extensive simulation results comparing these procedures and testing their robustness when the independence assumption is violated the new proposed procedures appear competitive with existing ones the overall best though is reported to be storey s estimator albeit for a specific parameter setting that does not appear to have been considered before in a second part we propose adaptive versions of step up procedures that have provably controlled fdr under positive dependence and unspecified dependence of the p values respectively in the latter case while simulations only show an improvement over non adaptive procedures in limited situations these are to our knowledge among the first theoretically founded adaptive multiple testing procedures that control the fdr when the p values are not independent

p value adjustments for multiple tests in multivariate binomial models abstract data from rodent carcinogenicity preclinical and clinical studies involving new drugs may be modeled as having come from multivariate binomial distributions in two year rodent carcinogenicity studies there are typically 20 50 tissues examined for occurrence of any of several possible lesions for a particular treatment group the number of occurrences of a particular lesion at a particular tissue may be modeled as binomial and the vector of such frequencies may be considered multivariate binomial with unspecified dependence structure the same model may also apply to clinical side effects data in this case the marginal frequencies may represent occurrences of events ranging from headaches to ingrown toenails frequently the goal of such studies is to isolate site specific significant differences between treatment and control groups for example in rodent carcinogenicity analyses it is generally not sufficient to claim that a new compound causes an increase in tumors at some unspecified si

a lie algebraic approach to effective mass schrodinger equations we use lie algebraic techniques to obtain exact solutions of the effective mass schrodinger equation in particular we use the su 1 1 algebra both as a spectrum generating algebra and as a potential algebra to obtain exact solutions of effective mass schrodinger equations corresponding to a number of potentials we also discuss the construction of isospectral hamiltonians for which both the mass and the potential are different