simultaneous inference for high dimensional linear models abstractthis article proposes a bootstrap assisted procedure to conduct simultaneous inference for high dimensional sparse linear models based on the recent desparsifying lasso estimator our procedure allows the dimension of the parameter vector of interest to be exponentially larger than sample size and it automatically accounts for the dependence within the desparsifying lasso estimator moreover our simultaneous testing method can be naturally coupled with the margin screening to enhance its power in sparse testing with a reduced computational cost or with the step down method to provide a strong control for the family wise error rate in theory we prove that our simultaneous testing procedure asymptotically achieves the prespecified significance level and enjoys certain optimality in terms of its power even when the model errors are non gaussian our general theory is also useful in studying the support recovery problem to broaden the applicability we further extend our main results to general

tests for high dimensional covariance matrices we propose tests for sphericity and identity of high dimensional covariance matrices the tests are nonparametric without assuming a specific parametric distribution for the data they can accommodate situations where the data dimension is much larger than the sample size namely the large p small n situations we demonstrate by both theoretical and empirical studies that the tests have good properties for a wide range of dimensions and sample sizes we applied the proposed test on a microarray dataset on yorkshire gilts and tested for the covariance structure for the expression levels for sets of genes

to how many simultaneous hypothesis tests can normal student s t or bootstrap calibration be applied in the analysis of microarray data and in some other contemporary statistical problems it is not uncommon to apply hypothesis tests in a highly simultaneous way the number n say of tests used can be much larger than the sample sizes n to which the tests are applied yet we wish to calibrate the tests so that the overall level of the simultaneous test is accurate often the sampling distribution is quite different for each test so there may not be an opportunity to combine data across samples in this setting how large can n be as a function of n before level accuracy becomes poor here we answer this question in cases where the statistic under test is of student s t type we show that if either the normal or student t distribution is used for calibration then the level of the simultaneous test is accurate provided that log n increases at a strictly slower rate than n 1 3 as n diverges on the other hand if bootstrap methods are used for calibration then we may choose log n almost as large as

testing a single regression coefficient in high dimensional linear models abstract in linear regression models with high dimensional data the classical z test or t test for testing the significance of each single regression coefficient is no longer applicable this is mainly because the number of covariates exceeds the sample size in this paper we propose a simple and novel alternative by introducing the correlated predictors screening cps method to control for predictors that are highly correlated with the target covariate accordingly the classical ordinary least squares approach can be employed to estimate the regression coefficient associated with the target covariate in addition we demonstrate that the resulting estimator is consistent and asymptotically normal even if the random errors are heteroscedastic this enables us to apply the z test to assess the significance of each covariate based on the p value obtained from testing the significance of each covariate we further conduct multiple hypothesis testing by controlling the false discovery rate at the nominal level then we show that the multiple hypothesis testing achieves consistent model selection simulation studies and empirical examples are presented to illustrate the finite sample performance and the usefulness of the proposed method respectively

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

type i error robustness of anova and anova on ranks when the number of treatments is large summary agricultural screening trials often involve a large number t of treatments in a complete block design with limited replication b 3 or 4 blocks the null hypothesis of interest is that of no differences between treatments for the commonly used analysis of variance anova procedure most texts do not discuss agreement between actual and nominal type i error rates in the presence of nonnormality in this small b large t situation similarly for the friedman and the increasingly popular anova on ranks procedures it is not easy to find results concerning null performance given b small and t large in this article we therefore present results from two different bodies of theory that provide useful insight concerning null performance of these anova and rank procedures when t is large the two types of theory are i the classical approach based on moment approximations to the permutation distribution and ii central limit theory based asymptotics in the nonstandard t x situation both approaches demonstrate the validity of standard anova and of anova on within block ranks under nonnormality when t is large choice of the procedure to be used on a given data set should therefore be based on consideration of power properties in general anova on ranks will be superior to standard anova for data with frequent extreme values

marginal asymptotics for the large p small n paradigm with applications to microarray data the large p small n paradigm arises in microarray studies image analysis high throughput molecular screening astronomy and in many other high dimensional applications false discovery rate fdr methods are useful for resolving the accompanying multiple testing problems in cdna microarray studies for example p values may be computed for each of p genes using data from n arrays where typically p is in the thousands and n is less than 30 for fdr methods to be valid in identifying differentially expressed genes the p values for the nondifferentially expressed genes must simultaneously have uniform distributions marginally while feasible for permutation p values this uniformity is problematic for asymptotic based p values since the number of p values involved goes to infinity and intuition suggests that at least some of the p values should behave erratically we examine this neglected issue when n is moderately large but p is almost exponentially large relative to n we show the somewhat surprising result that under very general dependence structures and for both mean and median tests the p values are simultaneously valid a small simulation study and data analysis are used for illustration

instrumental variable with competing risk model in this paper we discuss causal inference on the efficacy of a treatment or medication on a time to event outcome with competing risks although the treatment group can be randomized there can be confoundings between the compliance and the outcome unmeasured confoundings may exist even after adjustment for measured covariates instrumental variable methods are commonly used to yield consistent estimations of causal parameters in the presence of unmeasured confoundings on the basis of a semiparametric additive hazard model for the subdistribution hazard we propose an instrumental variable estimator to yield consistent estimation of efficacy in the presence of unmeasured confoundings for competing risk settings we derived the asymptotic properties for the proposed estimator the estimator is shown to be well performed under finite sample size according to simulation results we applied our method to a real transplant data example and showed that the unmeasured confoundings lead to significant bias in the estimation of the effect about 50 attenuated copyright 2017 john wiley sons ltd

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

asymptotics for analysis of variance when the number of levels is large abstract we study asymptotic results for f tests in analysis of variance models as the number of factor levels goes to but the number of observations for each factor combination is fixed asymptotic derivations of the type discussed in this article would be relevant whenever both the numerator and denominator degrees of freedom go to at the same rate we consider null and alternative distributions of f the usual f statistic for fixed effects and random effects balanced and unbalanced one way and two way and normal and nonnormal analysis of variance anova models the results may be most relevant for random effects and mixed models for example we may have an agricultural experiment in which the number of cows is quite large but the number of measurements on each cow is small the results would also be relevant for fixed effects models in which there are many factor levels but not many observations for each factor level

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

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

a two way semilinear model for normalization and analysis of cdna microarray data a basic question in analyzing cdna microarray data is normalization the purpose of which is to remove systematic bias in the observed expression values by establishing a normalization curve across the whole dynamic range a proper normalization procedure ensures that the normalized intensity ratios provide meaningful measures of relative expression levels we propose a two way semilinear model tw slm for normalization and analysis of microarray data this method does not make the usual assumptions underlying some of the existing methods for example it does not assume that the percentage of differentially expressed genes is small or that there is symmetry in the expression levels of up regulated and down regulated genes as required in the lowess normalization method the tw slm also naturally incorporates uncertainty due to normalization into significance analysis of microarrays we use a semiparametric approach based on polynomial splines in the tw slm to estimate the normalization curves and the no

an assumption free exact test for fixed design linear models with exchangeable errors we propose the cyclic permutation test cpt to test general linear hypotheses for linear models this test is non randomized and valid in finite samples with exact type i error alpha for an arbitrary fixed design matrix and arbitrary exchangeable errors whenever 1 alpha is an integer and n p ge 1 alpha 1 the test involves applying the marginal rank test to 1 alpha linear statistics of the outcome vector where the coefficient vectors are determined by solving a linear system such that the joint distribution of the linear statistics is invariant with respect to a non standard cyclic permutation group under the null hypothesis the power can be further enhanced by solving a secondary non linear travelling salesman problem for which the genetic algorithm can find a reasonably good solution extensive simulation studies show that the cpt has comparable power to existing tests when testing for a single contrast of coefficients an exact confidence interval can be obtained by inverting the test furthermore we provide a selective yet extensive literature review of the century long efforts on this problem highlighting the novelty of our test

asymptotic behavior of m estimators of p regression parameters when p 2 n is large ii normal approximation in a general linear model y x beta r with y and r n dimensional x a n times p matrix and beta p dimensional let hat beta be an m estimator of beta satisfying 0 sum x i psi y i x i beta let p rightarrow infty such that p log n 3 2 n rightarrow 0 then max i x i hat beta beta rightarrow p 0 and it is possible to find a uniform normal approximation for the distribution of hat beta under which arbitrary linear combinations a n hat beta beta are asymptotically normal when appropriately normalized and hat beta beta x x hat beta beta is approximately chi 2 p

integer linear programming models for the weighted total domination problem abstract a total dominating set of a graph g v e is a subset d of v such that every vertex in v including the vertices from d has at least one neighbour in d suppose that every vertex v v has an integer weight w v 0 and every edge e e has an integer weight w e 0 then the weighted total domination wtd problem is to find a total dominating set d which minimizes the cost f d u d w u e e d w e v v d min w u v u n v d in this paper we put forward three integer linear programming ilp models with a polynomial number of constraints and present some numerical results implemented on random graphs for wtd problem

forward regression for ultra high dimensional variable screening motivated by the seminal theory of sure independence screening fan and lv 2008 sis we investigate here another popular and classical variable screening method namely forward regression fr our theoretical analysis reveals that fr can identify all relevant predictors consistently even if the predictor dimension is substantially larger than the sample size in particular if the dimension of the true model is finite fr can discover all relevant predictors within a finite number of steps to practically select the best candidate from the models generated by fr the recently proposed bic criterion of chen and chen 2008 can be used the resulting model can then serve as an excellent starting point from where many existing variable selection methods e g scad and adaptive lasso can be applied directly fr s outstanding finite sample performances are confirmed by extensive numerical studies

box type approximations in nonparametric factorial designs abstract linear rank statistics in nonparametric factorial designs are asymptotically normal and in general heteroscedastic in a comprehensive simulation study the asymptotic chi squared law of the corresponding quadratic forms is shown to be a rather poor approximation of the finite sample distribution motivated by this problem we propose simple finite sample size approximations for the distribution of quadratic forms in factorial designs under a normal heteroscedastic error structure these approximations are based on an f distribution with estimated degrees of freedom that generalizes ideas of patnaik and box simulation studies show that the nominal level is maintained with high accuracy and in most cases the power is comparable to the asymptotic maximin wald test data driven guidelines are given to select the most appropriate test procedure these ideas are finally transferred to nonparametric factorial designs where the same quadratic forms as in the parametric case are applied to the vector

testing super diagonal structure in high dimensional covariance matrices abstract the covariance matrices are essential quantities in econometric and statistical applications including portfolio allocation asset pricing and factor analysis testing the entire covariance under high dimensionality endures large variability and causes a dilution of the signal to noise ratio and hence a reduction in the power we consider a more powerful test procedure that focuses on testing along the super diagonals of the high dimensional covariance matrix which can infer more accurately on the structure of the covariance we show that the test is powerful in detecting sparse signals and parametric structures in the covariance the properties of the test are demonstrated by theoretical analyses simulation and empirical studies

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