multiple comparison of several linear regression models research on multiple comparison during the past 50 years or so has focused mainly on the comparison of several population means several years ago spurrier considered the multiple comparison of several simple linear regression lines he constructed simultaneous confidence bands for all of the contrasts of the simple linear regression lines over the entire range when the models have the same design matrices this article extends spurrier s work in several directions first multiple linear regression models are considered and the design matrices are allowed to be different second the predictor variables are either unconstrained or constrained to finite intervals third the types of comparison allowed can be very flexible including pairwise many one and successive two simulation methods are proposed for the calculation of critical constants the methodologies are illustrated with examples

influential features pca for high dimensional clustering we consider a clustering problem where we observe feature vectors x i in r p i 1 2 ldots n from k possible classes the class labels are unknown and the main interest is to estimate them we are primarily interested in the modern regime of p gg n where classical clustering methods face challenges we propose influential features pca if pca as a new clustering procedure in if pca we select a small fraction of features with the largest kolmogorov smirnov ks scores obtain the first k 1 left singular vectors of the post selection normalized data matrix and then estimate the labels by applying the classical k means procedure to these singular vectors in this procedure the only tuning parameter is the threshold in the feature selection step we set the threshold in a data driven fashion by adapting the recent notion of higher criticism as a result if pca is a tuning free clustering method we apply if pca to 10 gene microarray data sets the method has competitive performance in clustering especially in three of the data sets the error rates of if pca are only 29 or less of the error rates by other methods we have also rediscovered a phenomenon on empirical null by efron j amer statist assoc 99 2004 96 104 on microarray data with delicate analysis especially post selection eigen analysis we derive tight probability bounds on the kolmogorov smirnov statistics and show that if pca yields clustering consistency in a broad context the clustering problem is connected to the problems of sparse pca and low rank matrix recovery but it is different in important ways we reveal an interesting phase transition phenomenon associated with these problems and identify the range of interest for each

adaptive robot under fuzzy control an arc welding industrial robot isotron 251 is treated its adaptation is considered as an attribute of the robot to trace an arbitrary pretaught welding seam by means of a noncontact laser sensor the control strategy is proposed to use a fuzzy model describing uncertainty of working conditions the experience shows how the notion of fuzzy sets powers is used for on line estimation prediction and control of robot motions the control algorithm has been tested by a three level microprocessor controller

false discovery control in large scale spatial multiple testing summary the paper develops a unified theoretical and computational framework for false discovery control in multiple testing of spatial signals we consider both pointwise and clusterwise spatial analyses and derive oracle procedures which optimally control the false discovery rate false discovery exceedance and false cluster rate a data driven finite approximation strategy is developed to mimic the oracle procedures on a continuous spatial domain our multiple testing procedures are asymptotically valid and can be effectively implemented using bayesian computational algorithms for analysis of large spatial data sets numerical results show that the procedures proposed lead to more accurate error control and better power performance than conventional methods we demonstrate our methods for analysing the time trends in tropospheric ozone in eastern usa

tweedie s formula and selection bias we suppose that the statistician observes some large number of estimates zi each with its own unobserved expectation parameter i the largest few of the zi s are likely to substantially overestimate their corresponding i s this being an example of selection bias or regression to the mean tweedie s formula first reported by robbins in 1956 offers a simple empirical bayes approach for correcting selection bias this article investigates its merits and limitations in addition to the methodology tweedie s formula raises more general questions concerning empirical bayes theory discussed here as relevance and empirical bayes information there is a close connection between applications of the formula and james stein estimation

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

size power and false discovery rates modern scientific technology has provided a new class of large scale simultaneous inference problems with thousands of hypothesis tests to consider at the same time microarrays epitomize this type of technology but similar situations arise in proteomics spectroscopy imaging and social science surveys this paper uses false discovery rate methods to carry out both size and power calculations on large scale problems a simple empirical bayes approach allows the false discovery rate fdr analysis to proceed with a minimum of frequentist or bayesian modeling assumptions closed form accuracy formulas are derived for estimated false discovery rates and used to compare different methodologies local or tail area fdr s theoretical permutation or empirical null hypothesis estimates two microarray data sets as well as simulations are used to evaluate the methodology the power diagnostics showing why nonnull cases might easily fail to appear on a list of significant discoveries

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

estimating the null and the proportion of nonnull effects in large scale multiple comparisons an important issue raised by efron in the context of large scale multiple comparisons is that in many applications the usual assumption that the null distribution is known is incorrect and seemingly negligible differences in the null may result in large differences in subsequent studies this suggests that a careful study of estimation of the null is indispensable in this article we consider the problem of estimating a null normal distribution and a closely related problem estimation of the proportion of nonnull effects we develop an approach based on the empirical characteristic function and fourier analysis the estimators are shown to be uniformly consistent over a wide class of parameters we investigate the numerical performance of the estimators using both simulated and real data in particular we apply our procedure to the analysis of breast cancer and human immunodeficiency virus microarray datasets the estimators perform favorably compared with existing methods

the empirical distribution of a large number of correlated normal variables motivated by the advent of high dimensional highly correlated data this work studies the limit behavior of the empirical cumulative distribution function ecdf of standard normal random variables under arbitrary correlation first we provide a necessary and sufficient condition for convergence of the ecdf to the standard normal distribution next under general correlation we show that the ecdf limit is a random possible infinite mixture of normal distribution functions that depends on a number of latent variables and can serve as an asymptotic approximation to the ecdf in high dimensions we provide conditions under which the dimension of the ecdf limit defined as the smallest number of effective latent variables is finite estimates of the latent variables are provided and their consistency proved we demonstrate these methods in a real high dimensional data example from brain imaging where it is shown that while the study exhibits apparently strongly significant results they can be entirely explained by correlation as captured by the asymptotic approximation developed here

local shrinkage rules levy processes and regularized regression we use l evy processes to generate joint prior distributions and therefore penalty functions for a location parameter b b1 bp as p grows large this generalizes the class of local global shrinkage rules based on scale mixtures of normals illuminates new connections among disparate methods and leads to new results for computing posterior means and modes under a wide class of priors we extend this framework to large scale regularized regression problems where p n and provide comparisons with other methodologies

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

optimal rates of convergence for estimating the null density and proportion of nonnull effects in large scale multiple testing an important estimation problem that is closely related to large scale multiple testing is that of estimating the null density and the proportion of nonnull effects a few estimators have been introduced in the literature however several important problems including the evaluation of the minimax rate of convergence and the construction of rate optimal estimators remain open in this paper we consider optimal estimation of the null density and the proportion of nonnull effects both minimax lower and upper bounds are derived the lower bound is established by a two point testing argument where at the core is the novel construction of two least favorable marginal densities f 1 and f 2 the density f 1 is heavy tailed both in the spatial and frequency domains and f 2 is a perturbation of f 1 such that the characteristic functions associated with f 1 and f 2 match each other in low frequencies the minimax upper bound is obtained by constructing estimators which rely on the empirical characteristic function and fourier analysis the estimator is shown to be minimax rate optimal compared to existing methods in the literature the proposed procedure not only provides more precise estimates of the null density and the proportion of the nonnull effects but also yields more accurate results when used inside some multiple testing procedures which aim at controlling the false discovery rate fdr the procedure is easy to implement and numerical results are given

nonuniqueness of the positive dirichlet problem for parabolic equations in cylinders abstract we investigate uniqueness of nonnegative solutions to parabolic equations in cylinders of r n 1 with zero dirichlet boundary condition where bases of cylinders are unbounded domains we give necessary and or sufficient conditions for the uniqueness in terms of geometric properties of the bases

detecting simultaneous changepoints in multiple sequences we discuss the detection of local signals that occur at the same location in multiple one dimensional noisy sequences with particular attention to relatively weak signals that may occur in only a fraction of the sequences we propose simple scan and segmentation algorithms based on the sum of the chi squared statistics for each individual sample which is equivalent to the generalized likelihood ratio for a model where the errors in each sample are independent the simple geometry of the statistic allows us to derive accurate analytic approximations to the significance level of such scans the formulation of the model is motivated by the biological problem of detecting recurrent dna copy number variants in multiple samples we show using replicates and parent child comparisons that pooling data across samples results in more accurate detection of copy number variants we also apply the multisample segmentation algorithm to the analysis of a cohort of tumour samples containing complex nested and overlapping copy number aberrations for which our method gives a sparse and intuitive cross sample summary

on the conditional distributions of low dimensional projections from high dimensional data we study the conditional distribution of low dimensional projections from high dimensional data where the conditioning is on other low dimensional projections to fix ideas consider a random d vector z that has a lebesgue density and that is standardized so that ez 0 and ezz id moreover consider two projections defined by unit vectors and namely a response y z and an explanatory variable x z it has long been known that the conditional mean of y given x is approximately linear in x under some regularity conditions cf hall and li 1993 however a corresponding result for the conditional variance has not been available so far we here show that the conditional variance of y given x is approximately constant in x again under some regularity conditions these results hold uniformly in and for most s provided only that the dimension of z is large in that sense we see that most linear submodels of a high dimensional overall model are approximately correct our findings provide new insights in a variety of modeling scenarios we discuss several examples including sparse linear modeling generalized linear models under potential link violation sliced inverse regression sliced average variance estimation and kernel learning machines

projected principal component analysis in factor models this paper introduces a projected principal component analysis projected pca which employees principal component analysis to the projected smoothed data matrix onto a given linear space spanned by covariates when it applies to high dimensional factor analysis the projection removes noise components we show that the unobserved latent factors can be more accurately estimated than the conventional pca if the projection is genuine or more precisely when the factor loading matrices are related to the projected linear space when the dimensionality is large the factors can be estimated accurately even when the sample size is finite we propose a flexible semi parametric factor model which decomposes the factor loading matrix into the component that can be explained by subject specific covariates and the orthogonal residual component the covariates effects on the factor loadings are further modeled by the additive model via sieve approximations by using the newly proposed projected pca the rates of convergence of the smooth factor loading matrices are obtained which are much faster than those of the conventional factor analysis the convergence is achieved even when the sample size is finite and is particularly appealing in the high dimension low sample size situation this leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings the proposed method is illustrated by both simulated data and the returns of the components of the s p 500 index

fuzzy controller for a sensor guided robot manipulator sensor guided robots require according to nonlinear characteristics of certain interaction processes between robot effector and object appropriate robust control algorithms for the external sensory control loop this paper deals with a fuzzy controller which bases on linguistic production rules following the pattern of the suboptimal control method its structure is available in a large field of applications simulations and real time experiments concerning the contour tracking problem show compared with a conventional pd controller the high tracking quality of the controller

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

the effect of correlation in false discovery rate estimation the objective of this paper is to quantify the effect of correlation in false discovery rate analysis specifically we derive approximations for the mean variance distribution and quantiles of the standard false discovery rate estimator for arbitrarily correlated data this is achieved using a negative binomial model for the number of false discoveries where the parameters are found empirically from the data we show that correlation may increase the bias and variance of the estimator substantially with respect to the independent case and that in some cases such as an exchangeable correlation structure the estimator fails to be consistent as the number of tests becomes large

a heuristic self tuning fuzzy controller abstract in this paper a self tuning fuzzy controller based on the tuning of scaling factors is presented the tuning schemes are tuning the scaling factors is individual tuning rules the tuning rules are transformed into numeric tuning tables by applying suitable membership functions so that the adjustings of the scaling factors are merely matrix mappings since the tuning rules are constructed with the control signal and the output error and change in output error of the plant as their antecedent parts the tuning processes have the advantage of real time property simulation results with a perturbed servo and the inverted pendulum system as the controlled plants show the performance and adaptivity of the proposed controller

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

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

a direct approach to sparse discriminant analysis in ultra high dimensions summary sparse discriminant methods based on independence rules such as the nearest shrunken centroids classifier tibshirani et al 2002 and features annealed independence rules fan fan 2008 have been proposed as computationally attractive tools for feature selection and classification with high dimensional data a fundamental drawback of these rules is that they ignore correlations among features and thus could produce misleading feature selection and inferior classification we propose a new procedure for sparse discriminant analysis motivated by the least squares formulation of linear discriminant analysis to demonstrate our proposal we study the numerical and theoretical properties of discriminant analysis constructed via lasso penalized least squares our theory shows that the method proposed can consistently identify the subset of discriminative features contributing to the bayes rule and at the same time consistently estimate the bayes classification direction even when the dimension can grow faster than any polynomial order of the sample size the theory allows for general dependence among features simulated and real data examples show that lassoed discriminant analysis compares favourably with other popular sparse discriminant proposals

variance of the number of false discoveries summary in high throughput genomic work a very large number d of hypotheses are tested based on n d data samples the large number of tests necessitates an adjustment for false discoveries in which a true null hypothesis was rejected the expected number of false discoveries is easy to obtain dependences between the hypothesis tests greatly affect the variance of the number of false discoveries assuming that the tests are independent gives an inadequate variance formula the paper presents a variance formula that takes account of the correlations between test statistics that formula involves o d2 correlations and so a naive implementation has cost o nd2 a method based on sampling pairs of tests allows the variance to be approximated at a cost that is independent of d

the approximation of piecewise linear membership functions and łukasiewicz operators in this paper we propose an approximation of piecewise linear membership functions with the help of sigmoid functions and certain arithmetic operations the gradient based tuning of piecewise linear membership functions can be achieved with the proposed efficient approximation because it has simple continuous derivatives with this construction we can even approximate the lukasiewicz operator family which plays an important role in fuzzy logic first of all from the theoretical point of view although in practice in optimization and learning it is rarely used because the lack of good analytical properties e g a continuous gradient the proposed approximation enlarges the applicability of fuzzy methods to the operators and membership functions where the differentiability is desirable

an intrinsic metric approach to uniqueness of the positive cauchy neumann problem for parabolic equations abstract we study uniqueness of nonnegative solutions of the cauchy neumann problem for parabolic equations in unbounded domains and give a sufficient condition for the uniqueness of nonnegative solutions to hold we also give a parabolic harnack inequality with neumann boundary conditions

bootstrap confidence intervals for a class of parametric problems summary we consider the following class of problems having observed a multivariate normal data vector y with unknown mean vector q covariance matrix the identity find an approximate confidence interval for 0 t q a real valued function of q a simple geometric construction is given which leads to highly accurate solutions this construction shows that the standard approximation based on maximum likelihood theory 04 cz00 can be quite misleading when 0 is nonlinear in q we discuss bootstrap based confidence intervals which remove most of the error in the standard approximation at the expense of considerably more calculation the bootstrap intervals are invariant under transformation of both y and i and so they automatically produce accurate solutions in problems which can be transformed to multivariate normality without requiring knowledge of the normalizing transformation

optimal classification in sparse gaussian graphic model consider a two class classification problem where the number of features is much larger than the sample size the features are masked by gaussian noise with mean zero and covariance matrix sigma where the precision matrix omega sigma 1 is unknown but is presumably sparse the useful features also unknown are sparse and each contributes weakly i e rare and weak to the classification decision by obtaining a reasonably good estimate of omega we formulate the setting as a linear regression model we propose a two stage classification method where we first select features by the method of innovated thresholding it and then use the retained features and fisher s lda for classification in this approach a crucial problem is how to set the threshold of it we approach this problem by adapting the recent innovation of higher criticism thresholding hct we find that when useful features are rare and weak the limiting behavior of hct is essentially just as good as the limiting behavior of ideal threshold the threshold one would choose if the underlying distribution of the signals is known if only somewhat surprisingly when omega is sufficiently sparse its off diagonal coordinates usually do not have a major influence over the classification decision compared to recent work in the case where omega is the identity matrix proc natl acad sci usa 105 2008 14790 14795 philos trans r soc lond ser a math phys eng sci 367 2009 4449 4470 the current setting is much more general which needs a new approach and much more sophisticated analysis one key component of the analysis is the intimate relationship between hct and fisher s separation another key component is the tight large deviation bounds for empirical processes for data with unconventional correlation structures where graph theory on vertex coloring plays an important role

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 empirical process of a short range dependent stationary sequence under gaussian subordination consider the stationary sequencex 1 g z 1 x 2 g z 2 whereg is an arbitrary borel function andz 1 z 2 is a mean zero stationary gaussian sequence with covariance functionr k e z 1 z k 1 satisfyingr 0 1 and 1 r k m where withi denoting the indicator function andf the continuous marginal distribution function of the sequence x n the integerm is the hermite rank of the family i g x f x x r letf n be the empirical distribution function ofx 1 x n we prove that asn the empirical processn 1 2 f n f converges in distribution to a gaussian process in the spaced

noisy matrix completion understanding statistical guarantees for convex relaxation via nonconvex optimization this paper studies noisy low rank matrix completion given partial and noisy entries of a large low rank matrix the goal is to estimate the underlying matrix faithfully and efficiently arguably o

on false discovery control under dependence a popular framework for false discovery control is the random effects model in which the null hypotheses are assumed to be independent this paper generalizes the random effects model to a conditional dependence model which allows dependence between null hypotheses the dependence can be useful to characterize the spatial structure of the null hypotheses asymptotic properties of false discovery proportions and numbers of rejected hypotheses are explored and a large sample distributional theory is obtained

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

blow up and global existence for the porous medium equation with reaction on a class of cartan hadamard manifolds abstract we consider the porous medium equation with power type reaction terms u p on negatively curved riemannian manifolds and solutions corresponding to bounded nonnegative and compactly supported data if p m small data give rise to global in time solutions while solutions associated to large data blow up in finite time if p m large data blow up at worst in infinite time and under the stronger restriction p 1 1 m 2 all data give rise to solutions existing globally in time whereas solutions corresponding to large data blow up in infinite time the results are in several aspects significantly different from the euclidean ones as has to be expected since negative curvature is known to give rise to faster diffusion properties of the porous medium equation

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

penalized normal likelihood and ridge regularization of correlation and covariance matrices high dimensionality causes problems in various areas of statistics a particular situation that rarely has been considered is the testing of hypotheses about multivariate regression models in which

false discovery rates and copy number variation copy number changes the gains and losses of chromosome segments are a common type of genetic variation among healthy individuals as well as an important feature in tumour genomes microarray technology enables us to simultaneously measure with moderate accuracy copy number variation at more than a million chromosome locations and for hundreds of subjects this leads to massive data sets and complicated inference problems concerning which locations are more likely to vary in this paper we consider a relatively simple false discovery rate approach to copy number analysis more careful parametric change point methods can then be focused on promising regions of the genome

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

false discovery rate adjusted multiple confidence intervals for selected parameters often in applied research confidence intervals cis are constructed or reported only for parameters selected after viewing the data we show that such selected intervals fail to provide the assumed coverage probability by generalizing the false discovery rate fdr approach from multiple testing to selected multiple cis we suggest the false coverage statement rate fcr as a measure of interval coverage following selection a general procedure is then introduced offering fcr control at level q under any selection rule the procedure constructs a marginal ci for each selected parameter but instead of the confidence level 1 q being used marginally q is divided by the number of parameters considered and multiplied by the number selected if we further use the fdr controlling testing procedure of benjamini and hochberg for selecting the parameters the newly suggested procedure offers cis that are dual to the testing procedure and are shown to be optimal in the independent case under the positive re

application of non parametric empirical bayes estimation to high dimensional classification we consider the problem of classification using high dimensional features space in a paper by bickel and levina 2004 it is recommended to use naive bayes classifiers that is to treat the features as if they are statistically independent consider now a sparse setup where only a few of the features are informative for classification fan and fan 2008 suggested a variable selection and classification method called fair the fair method improves the design of naive bayes classifiers in sparse setups the improvement is due to reducing the noise in estimating the features means this reduction is since that only the means of a few selected variables should be estimated we also consider the design of naive bayes classifiers we show that a good alternative to variable selection is estimation of the means through a certain non parametric empirical bayes procedure in sparse setups the empirical bayes implicitly performs an efficient variable selection it also adapts very well to non sparse setups and has the advantage of making use of the information from many weakly informative variables which variable selection type of classification procedures give up on using we compare our method with fair and other classification methods in simulation for sparse and non sparse setups and in real data examples involving classification of normal versus malignant tissues based on microarray data

robust variable and interaction selection for logistic regression and general index models under the logistic regression framework we propose a forward backward method soda for variable selection with both main and quadratic interaction terms in the forward stage soda adds in predic

approximate tail probabilities for the maxima of some random fields abstract hogan and siegmund 1986 adapt the method developed by pickands 1969 qualls and watanabe 1973 and bickel and rosenblatt 1973 to obtain explicit large deviation approximations for the maxima of several gaussian random fields arising in statistics using a special argument for one particular case they suggest a heuristic second order approximation for that case and they show by a monte carlo experiment that the second order approximation frequently gives considerably better numerical results the purpose of this paper is to show that the method developed by woodroofe 1976 1982 for problems in one dimensional time can be adapted to study maxima of random fields overall it involves simpler computations than the previous method and consequently seems potentially capable of delivering a genuine second order approximation should one seem desirable

the porous medium equation on riemannian manifolds with negative curvature the large time behaviour abstract we consider nonnegative solutions of the porous medium equation pme on cartan hadamard manifolds whose negative curvature can be unbounded we take compactly supported initial data because we are also interested in free boundaries we classify the geometrical cases we study into quasi hyperbolic quasi euclidean and critical cases depending on the growth rate of the curvature at infinity we prove sharp upper and lower bounds on the long time behaviour of the solutions in terms of corresponding bounds on the curvature in particular we estimate the location of the free boundary a global harnack principle follows we also present a change of variables that allows to transform radially symmetric solutions of the pme on model manifolds into radially symmetric solutions of a corresponding weighted pme on euclidean space and back this equivalence turns out to be an important tool of the theory

false discovery rates for spatial signals the problem of multiple testing for the presence of signal in spatial data can involve numerous locations traditionally each location is tested separately for signal presence but then the findings are reported in terms of clusters of nearby locations this is an indication that the units of interest for testing are clusters rather than individual locations the investigator may know a priori these more natural units or an approximation to them we suggest testing these cluster units rather than individual locations thus increasing the signal to noise ratio within the unit tested as well as reducing the number of hypothesis tests conducted because the signal may be absent from part of each cluster we define a cluster as containing a signal if the signal is present somewhere within the cluster we suggest controlling the false discovery rate fdr on clusters i e the expected proportion of clusters rejected erroneously out of all clusters rejected or its extension to general weights wfdr we int

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

empirical bayes estimates for large scale prediction problems classical prediction methods such as fisher s linear discriminant function were designed for small scale problems in which the number of predictors n is much smaller than the number of observations n modern scientific devices often reverse this situation a microarray analysis for example might include n 100 subjects measured on n 10 000 genes each of which is a potential predictor this article proposes an empirical bayes approach to large scale prediction where the optimum bayes prediction rule is estimated employing the data from all of the predictors microarray examples are used to illustrate the method the results demonstrate a close connection with the shrunken centroids algorithm of tibshirani et al 2002 a frequentist regularization approach to large scale prediction and also with false discovery rate theory

optimal and fast detection of spatial clusters with scan statistics we consider the detection of multivariate spatial clusters in the bernoulli model with n locations where the design distribution has weakly dependent marginals the locations are scanned with a rectangular window with sides parallel to the axes and with varying sizes and aspect ratios multivariate scan statistics pose a statistical problem due to the multiple testing over many scan windows as well as a computational problem because statistics have to be evaluated on many windows this paper introduces methodology that leads to both statistically optimal inference and computationally efficient algorithms the main difference to the traditional calibration of scan statistics is the concept of grouping scan windows according to their sizes and then applying different critical values to different groups it is shown that this calibration of the scan statistic results in optimal inference for spatial clusters on both small scales and on large scales as well as in the case where the cluster lives on one of the marginals methodology is introduced that allows for an efficient approximation of the set of all rectangles while still guaranteeing the statistical optimality results described above it is shown that the resulting scan statistic has a computational complexity that is almost linear in n

partial correlation screening for estimating large precision matrices with applications to classification given n samples x1 x2 xn from n 0 we are interested in estimating the p p precision matrix 1 we assume is sparse in that each row has relatively few nonzeros we propose partial correlation screening pcs as a new row by row approach to estimate the i th row of 1 i p pcs uses a screen step and a clean step in the

blow up and global existence for solutions to the porous medium equation with reaction and slowly decaying density abstract we study existence of global solutions and finite time blow up of solutions to the cauchy problem for the porous medium equation with a variable density x and a power like reaction term x u p with p 1 this is a mathematical model of a thermal evolution of a heated plasma see 29 the density decays slowly at infinity in the sense that x x q as x with q 0 2 we show that for large enough initial data solutions blow up in finite time for any p 1 on the other hand if the initial datum is small enough and p p for a suitable p depending on m n then global solutions exist in addition if p p for a suitable p p depending on m n then the solution blows up in finite time for any nontrivial initial datum we need the extra hypothesis that q 0 for 0 small enough when m p p observe that p p if x is a multiple of x q for x large enough such results are in agreement with those established in 48 where x 1 and are related to some results in 32 33 the case of fast decaying density at infinity i e q 2 is examined in 36

the porous medium equation on riemannian manifolds with negative curvature the superquadratic case we study the long time behaviour of nonnegative solutions of the porous medium equation posed on cartan hadamard manifolds having very large negative curvature more precisely when the sectional or ricci curvatures diverge at infinity more than quadratically in terms of the geodesic distance to the pole we find an unexpected separate variable behaviour that reminds one of dirichlet problems on bounded euclidean domains as a crucial step we prove existence of solutions to a related sublinear elliptic problem a result of independent interest uniqueness of solutions vanishing at infinity is also shown along with comparison principles both in the parabolic and in the elliptic case our results complete previous analyses of the porous medium equation flow on negatively curved riemannian manifolds which were carried out first for the hyperbolic space and then for general cartan hadamard manifolds with a negative curvature having at most quadratic growth we point out that no similar analysis seems to exist for the linear heat flow we also translate such results into some weighted porous medium equations in the euclidean space having special weights

the bayesian bridge summary we propose the bayesian bridge estimator for regularized regression and classification two key mixture representations for the bayesian bridge model are developed a scale mixture of normal distributions with respect to an stable random variable a mixture of bartlett fejer kernels or triangle densities with respect to a two component mixture of gamma random variables both lead to markov chain monte carlo methods for posterior simulation and these methods turn out to have complementary domains of maximum efficiency the first representation is a well known result due to west and is the better choice for collinear design matrices the second representation is new and is more efficient for orthogonal problems largely because it avoids the need to deal with exponentially tilted stable random variables it also provides insight into the multimodality of the joint posterior distribution which is a feature of the bridge model that is notably absent under ridge or lasso type priors we prove a theorem that extends this representation to a wider class of densities representable as scale mixtures of beta distributions and we provide an explicit inversion formula for the mixing distribution the connections with slice sampling and scale mixtures of normal distributions are explored on the practical side we find that the bayesian bridge model outperforms its classical cousin in estimation and prediction across a variety of data sets both simulated and real we also show that the markov chain monte carlo algorithm for fitting the bridge model exhibits excellent mixing properties particularly for the global scale parameter this makes for a favourable contrast with analogous markov chain monte carlo algorithms for other sparse bayesian models all methods described in this paper are implemented in the r package bayesbridge an extensive set of simulation results is provided in two on line supplemental files