exact calculations for false discovery proportion with application to least favorable configurations in a context of multiple hypothesis testing we provide several new exact calculations related to the false discovery proportion fdp of step up and step down procedures for step up procedures we show that the number of erroneous rejections conditionally on the rejection number is simply a binomial variable which leads to explicit computations of the c d f the s th moment and the mean of the fdp the latter corresponding to the false discovery rate fdr for step down procedures we derive what is to our knowledge the first explicit formula for the fdr valid for any alternative c d f of the p values we also derive explicit computations of the power for both step up and step down procedures these formulas are explicit in the sense that they only involve the parameters of the model and the c d f of the order statistics of i i d uniform variables the p values are assumed either independent or coming from an equicorrelated multivariate normal model and an additional mixture model for the true false hypotheses is used this new approach is used to investigate new results which are of interest in their own right related to least most favorable configurations for the fdr and the variance of the fdp

an adaptive step down procedure with proven fdr control under independence in this work we study an adaptive step down procedure for testing m hypotheses it stems from the repeated use of the false discovery rate controlling the linear step up procedure sometimes called bh and makes use of the critical constants iq m 1 i 1 q i 1 m motivated by its success as a model selection procedure as well as by its asymptotic optimality we are interested in its false discovery rate fdr controlling properties for a finite number of hypotheses we prove this step down procedure controls the fdr at level q for independent test statistics we then numerically compare it with two other procedures with proven fdr control under independence both in terms of power under independence and fdr control under positive dependence

on the benjamini hochberg method we investigate the properties of the benjamini hochberg method for multiple testing and of a variant of storey s generalization of it extending and complementing the asymptotic and exact results available in the literature results are obtained under two different sets of assumptions and include asymptotic and exact expressions and bounds for the proportion of rejections the proportion of incorrect rejections out of all rejections and two other proportions used to quantify the efficacy of the method

an improved bonferroni procedure for multiple tests of significance summary a modification of the bonferroni procedure for testing multiple hypotheses is presented the method based on the ordered p values of the individual tests is less conservative than the classical bonferroni procedure but is still simple to apply a simulation study shows that the probability of a type i error of the procedure does not exceed the nominal significance level a for a variety of multivariate normal and multivariate gamma test statistics for independent tests the procedure has type i error probability equal to a the method appears particularly advantageous over the classical bonferroni procedure when several highly correlated test statistics are involved

no gap second order optimality conditions for optimal control problems with a single state constraint and control the paper deals with optimal control problems with only one control variable and one state constraint of arbitrary order we consider the case of finitely many boundary arcs and touch times we obtain a no gap theory of second order conditions allowing to characterize second order quadratic growth

stability analysis of optimal control problems with a second order state constraint this paper gives stability results for nonlinear optimal control problems subject to a regular state constraint of second order the strengthened legendre clebsch condition is assumed to hold and no assumption on the structure of the contact set is made under a weak second order sufficient condition taking into account the active constraints we show that the solutions are lipschitz continuous w r t the perturbation parameter in the l 2 norm and holder continuous in the l infty norm we use a generalized implicit function theorem in metric spaces by dontchev and hager siam j control optim 36 1998 pp 698 718 the difficulty is that multipliers associated with second order state constraints have a low regularity they are only bounded measures we obtain lipschitz stability of a primitive of the state constraint multiplier

correlated z values and the accuracy of large scale statistical estimates we consider large scale studies in which there are hundreds or thousands of correlated cases to investigate each represented by its own normal variate typically a z value a familiar example is provided by a microarray experiment comparing healthy with sick subjects expression levels for thousands of genes this paper concerns the accuracy of summary statistics for the collection of normal variates such as their empirical cdf or a false discovery rate statistic it seems like we must estimate an n by n correlation matrix n the number of cases but our main result shows that this is not necessary good accuracy approximations can be based on the root mean square correlation over all n n 1 2 pairs a quantity often easily estimated a second result shows that z values closely follow normal distributions even under nonnull conditions supporting application of the main theorem practical application of the theory is illustrated for a large leukemia microarray study

a survey of the maximum principles for optimal control problems with state constraints this paper gives a survey of the various forms of pontryagin s maximum principle for optimal control problems with state variable inequality constraints the relations between the different sets of optimality conditions arising in these forms are shown furthermore the application of these maximum principle conditions is demonstrated by solving some illustrative examples

one dimensional schrodinger operators with symmetric zero range potentials non hermitian hamiltonians appearing as operator realizations of symmetric symmetric in physical literature zero range singular perturbations of one dimensional schrodinger operators are studied in particular hamiltonians with a real spectrum are described in terms of parameters of singular perturbations and moreover it is shown that only part of them are similar to hermitian ones in this case they can be used as exactly solvable models of symmetric quantum mechanics

operating characteristics and extensions of the false discovery rate procedure summary we investigate the operating characteristics of the benjamini hochberg false discovery rate procedure for multiple testing this is a distribution free method that controls the expected fraction of falsely rejected null hypotheses among those rejected the paper provides a framework for understanding more about this procedure we first study the asymptotic properties of the deciding point d that determines the critical p value from this we obtain explicit asymptotic expressions for a particular risk function we introduce the dual notion of false non rejections and we consider a risk function that combines the false discovery rate and false non rejections we also consider the optimal procedure with respect to a measure of conditional risk

application of optimal control method in buckling analysis of constrained elastica problems abstract main concern of the paper is to illustrate the generic application of the optimal control method in the analysis of a constrained buckling problem and its simple and elegant formulation when compared to other techniques thus an optimal control methodology is adopted to investigate the buckling behavior of thin elastic structures under the presence of unilateral constraints we particularly explore the post buckling response of a constant or variable length elastica constrained by rigid walls the equivalence between the calculus of variations and the optimal control is first demonstrated with a main focus on the post buckling behavior of a constant length elastica which is constrained by horizontal walls the gradual construction of the constrained buckling problem as an optimal control problem from its equivalent lagrangian form clearly shows its advantage when compared to the calculus of variations the necessary optimality conditions which constitute the pontryagin s minimum or maximum principle are also derived by considering a direct adjoining approach the solution of the optimal control problem is then performed by applying a direct method validation of the methodology is first achieved by reproducing examples available in the literature then the effects of factors such as the geometry of the walls and the variability in the bending stiffness of the elastica on its buckling response are analyzed these constrained buckling problems are investigated for the first time while through them it is also readily shown that the presence of geometric or material nonlinearity does not introduce any essential complexity in the buckling analysis when the optimal control methodology is adopted

empirical bayes analysis of a microarray experiment microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels a typical microarray experiment can produce millions of data points raising serious problems of data reduction and simultaneous inference we consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes a simple nonparametric empirical bayes model is introduced which is used to guide the efficient reduction of the data to a single summary statistic per gene and also to make simultaneous inferences concerning which genes were affected by the radiation although our focus is on one specific experiment the proposed methods can be applied quite generally the empirical bayes inferences are closely related to the frequentist false discovery rate fdr criterion

on the false discovery rate and an asymptotically optimal rejection curve in this paper we introduce and investigate a new rejection curve for asymptotic control of the false discovery rate fdr in multiple hypotheses testing problems we first give a heuristic motivation for this new curve and propose some procedures related to it then we introduce a set of possible assumptions and give a unifying short proof of fdr control for procedures based on simes critical values whereby certain types of dependency are allowed this methodology of proof is then applied to other fixed rejection curves including the proposed new curve among others we investigate the problem of finding least favorable parameter configurations such that the fdr becomes largest we then derive a series of results concerning asymptotic fdr control for procedures based on the new curve and discuss several example procedures in more detail a main result will be an asymptotic optimality statement for various procedures based on the new curve in the class of fixed rejection curves finally we briefly discuss strict fdr control for a finite number of hypotheses

dependency and false discovery rate asymptotics some effort has been undertaken over the last decade to provide conditions for the control of the false discovery rate by the linear step up procedure lsu for testing n hypotheses when test statistics are dependent in this paper we investigate the expected error rate eer and the false discovery rate fdr in some extreme parameter configurations when n tends to infinity for test statistics being exchangeable under null hypotheses all results are derived in terms of p values in a general setup we present a series of results concerning the interrelation of simes rejection curve and the limiting empirical distribution function of the p values main objects under investigation are largest limiting crossing points between these functions which play a key role in deriving explicit formulas for eer and fdr as specific examples we investigate equi correlated normal and t variables in more detail and compute the limiting eer and fdr theoretically and numerically a surprising limit behavior occurs if these models tend to independence

lipschitzian stability for state constrained nonlinear optimal control for a nonlinear optimal control problem with state constraints we give conditions under which the optimal control depends lipschitz continuously in the l2 norm on a parameter these conditions involve smoothness of the problem data uniform independence of active constraint gradients and a coercivity condition for the integral functional under these same conditions we obtain a new nonoptimal stability result for the optimal control in the l infty norm and under an additional assumption concerning the regularity of the state constraints a new tight l infty estimate is obtained our approach is based on an abstract implicit function theorem in nonlinear spaces

the euler approximation in state constrained optimal control we analyze the euler approximation to a state constrained control problem we show that if the active constraints satisfy an independence condition and the lagrangian satisfies a coercivity condition then locally there exists a solution to the euler discretization and the error is bounded by a constant times the mesh size the proof couples recent stability results for state constrained control problems with results established here on discrete time regularity the analysis utilizes mappings of the discrete variables into continuous spaces where classical finite element estimates can be invoked

on the performance of fdr control constraints and a partial solution the false discovery rate fdr paradigm aims to attain certain control on type i errors with relatively high power for multiple hypothesis testing the benjamini hochberg bh procedure is a wellknown fdr controlling procedure under a random effects model we show that in general unlike the fdr the positive fdr pfdr of the bh procedure cannot be controlled at an arbitrarily low level due to the limited evidence provided by the observations to separate false and true nulls this results in a criticality phenomenon which is characterized by a transition of the procedure s power from being positive to asymptotically 0 without any reduction in the pfdr once the target fdr control level is below a positive critical value to address the constraints on the power and pfdr control imposed by the criticality phenomenon we propose a procedure which applies bh type procedures at multiple locations in the domain of p values both analysis and simulations show that the proposed procedure can attain substantially improved power and pfdr control

multiple hypotheses testing and expected number of type i errors the performance of multiple test procedures with respect to error control is an old issue assuming that all hypotheses are true we investigate the behavior of the expected number of type i errors ene as a characteristic of certain multiple tests controlling the familywise error rate fwer or the false discovery rate fdr at a prespecified level we derive explicit formulas for the distribution of the number of false rejections as well as for the ene for single step step down and step up procedures based on independent p values moreover we determine the corresponding asymptotic distributions of the number of false rejections as well as explicit formulae for the ene if the number of hypotheses tends to infinity in case of fwer control we mostly obtain poisson distributions and in one case a geometric distribution as limiting distributions in case of fdr control we obtain limiting distributions which are apparently not named in the literature surprisingly the ene is bounded by a small number regardless of the number of hypotheses under consideration finally it turns out that in case of dependent test statistics the ene behaves completely differently compared to the case of independent test statistics

some results on false discovery rate in stepwise multiple testing procedures the concept of false discovery rate fdr has been receiving increasing attention by researchers in multiple hypotheses testing this paper produces some theoretical results on the fdr in the context of stepwise multiple testing procedures with dependent test statistics it was recently shown by benjamini and yekutieli that the benjamini hochberg step up procedure controls the fdr when the test statistics are positively dependent in a certain sense this paper strengthens their work by showing that the critical values of that procedure can be used in a much more general stepwise procedure under similar positive dependency it is also shown that the fdr controlling benjamini liu step down procedure originally developed for independent test statistics works even when the test statistics are positively dependent in some sense an explicit expression for the fdr of a generalized stepwise procedure and an upper bound to the fdr of a step down procedure are obtained in terms of probability distributions of ordered components of dependent random variables before establishing the main results 1 introduction the main purpose of this paper is to present some formulas and inequalities providing theoretical insight into when and what stepwise procedures control the false discovery rate fdr at a prespecified level in multiple hypotheses testing the fdr first introduced in multiple testing by benjamini and hochberg 1995 is the expected proportion of erroneously rejected null hypotheses with v and r representing respectively the number of true null hypotheses rejected and the total number of null hypotheses rejected in a multiple testing procedure let q v r if r 0a nd 0i fr 0 then e q is the fdr of that procedure as they have argued it is an appropriate error rate to control in many practical situations particularly where a large number of null hypotheses are involved they also put forward an fdr controlling stepup procedure that is more powerful than comparable procedures controlling the traditional familywise error fwe rate when the underlying test statistics are independent benjamini and liu 1999 recently introduced a step down procedure with the same property the concept of fdr is now being used in a wide variety of applications abramovich and benjamini 1996 basford and tukey 1997 drigalenko

a stochastic process approach to false discovery control this paper extends the theory of false discovery rates fdr pioneered by benjamini and hochberg j roy statist soc ser b 57 1995 289 300 we develop a framework in which the false discovery proportion fdp the number of false rejections divided by the number of rejections is treated as a stochastic process after obtaining the limiting distribution of the process we demonstrate the validity of a class of procedures for controlling the false discovery rate the expected fdp we construct a confidence envelope for the whole fdp process from these envelopes we derive confidence thresholds for controlling the quantiles of the distribution of the fdp as well as controlling the number of false discoveries we also investigate methods for estimating the p value distribution

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

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

stability analysis for nonlinear optimal control problems subject to state constraints there is considered a family of nonlinear optimal control problems depending on a functional parameter the problems are subject to state constraints conditions are derived under which the solutions and lagrange multipliers are locally lipschitz continuous functions of the parameter those conditions consist of constraint qualifications and weakened coercivity conditions

asymptotic results on adaptive false discovery rate controlling procedures based on kernel estimators the false discovery rate fdr is a commonly used type i error rate in multiple testing problems it is defined as the expected false discovery proportion fdp that is the expected fraction of false positives among rejected hypotheses when the hypotheses are independent the benjamini hochberg procedure achieves fdr control at any pre specified level by construction fdr control offers no guarantee in terms of power or type ii error a number of alternative procedures have been developed including plug in procedures that aim at gaining power by incorporating an estimate of the proportion of true null hypotheses in this paper we study the asymptotic behavior of a class of plug in procedures based on kernel estimators of the density of the p values as the number m of tested hypotheses grows to infinity in a setting where the hypotheses tested are independent we prove that these procedures are asymptotically more powerful in two respects i a tighter asymptotic fdr control for any target fdr level and ii a broader range of target levels yielding positive asymptotic power we also show that this increased asymptotic power comes at the price of slower non parametric convergence rates for the fdp these rates are of the form m k 2k 1 where k is determined by the regularity of the density of the p value distribution or equivalently of the test statistics distribution these results are applied to one and two sided tests statistics for gaussian and laplace location models and for the student model

sensitivity analysis for relaxed optimal control problems with final state constraints in this article we compute a second order expansion of the value function of a family of relaxed optimal control problems with final state constraints parameterized by a perturbation variable the sensitivity analysis is performed for controls that we call r strong solutions they are optimal solutions with respect to the set of feasible controls with a uniform norm smaller than a given r and having an associated trajectory in a small neighborhood for the uniform norm in this framework relaxation enables us to consider a wide class of perturbations and therefore to derive sharp estimates of the value function

the positive false discovery rate a bayesian interpretation and the q value multiple hypothesis testing is concerned with controlling the rate of false positives when testing several hypotheses simultaneously one multiple hypothesis testing error measure is the false discovery rate fdr which is loosely defined to be the expected proportion of false positives among all significant hypotheses the fdr is especially appropriate for exploratory analyses in which one is interested in finding several significant results among many tests in this work we introduce a modified version of the fdr called the positive false discovery rate pfdr we discuss the advantages and disadvantages of the pfdr and investigate its statistical properties when assuming the test statistics follow a mixture distribution we show that the pfdr can be written as a bayesian posterior probability and can be connected to classification theory these properties remain asymptotically true under fairly general conditions even under certain forms of dependence also a new quantity called the q value is introduced and investigated which is a natural bayesian posterior p value or rather the pfdr analogue of the p value 1 introduction when testing a single hypothesis one is usually concerned with controlling the false positive rate while maximizing the probability of detecting an effect when one really exists in statistical terms we maximize the power conditional on the type i error rate being at or below some level the field of multiple hypothesis testing tries to extend this basic paradigm to the situation where several hypotheses are tested simultaneously one must define an appropriate compound error measure according to the rate of false positives one is willing to encounter then a procedure is developed that allows one to control the error rate at a desired level while maintaining the power of each test as much as possible the most commonly controlled quantity when testing multiple hypotheses is the family wise error rate fwer which is the probability of yielding one or more false positives out of all hypotheses tested the most familiar example of this is the bonferroni method if there are m hypothesis tests each test is controlled so that the probability of a false positive is less than or equal to m for some chosen value of it then follows that the overall fwer is less than or equal to m any

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

well posedness of the shooting algorithm for state constrained optimal control problems with a single constraint and control this paper deals with the shooting algorithm for optimal control problems with a scalar control and a regular scalar state constraint additional conditions are displayed under which the so called alternative formulation is equivalent to pontryagin s minimum principle the shooting algorithm appears to be well posed invertible jacobian iff i the no gap second order sufficient optimality condition holds and ii when the constraint is of order q geq 3 there is no boundary arc stability and sensitivity results without strict complementarity at touch points are derived using robinson s strong regularity theory under a minimal second order sufficient condition the directional derivatives of the control and state are obtained as solutions of a linear quadratic problem