analytical and bootstrap approximations to estimator distributions in l 1 regression abstract edgeworth and bootstrap approximations to estimator distributions in l1 regression are described analytic approximations based on edgeworth expansions that mix lattice and nonlattice components and allow for an intercept term in the regression are developed under mild conditions which do not even require a density for the error distribution under stronger assumptions on the error distribution the edgeworth expansion assumes a simpler form bootstrap approximations are described and the consistency of the bootstrap in the l 1 regression setting is established we show how the slow rate n 1 4 of convergence in this context of the standard unsmoothed bootstrap that resamples for the raw residuals may be improved to rate n 2 5 by two methods a smoothed bootstrap approach based on resampling from an appropriate kernel estimator of the error density and a normal approximation that uses a kernel estimator of the error density at a particular point its median 0 both of these methods require ch

a zero one law approach to the central limit theorem for the weighted bootstrap mean we develop a new technique to prove the conditional clt for the weighted bootstrap mean through 0 1 laws we show that this conditional clt can be derived from an unconditional one which easily arises conditioning with respect to the weights from the standard lindeberg clt

higher cohomology of divisors on a projective variety if l is ample or merely nef then of course hi x ox ml o md for i 0 in general the converse is false for instance it can happen that h i x ox ml 0 for m 0 and all i even if l is not nef or for that matter pseudoeffective however our main result shows that if one considers also small perturbations of the divisor in question then in fact the maximal growth of higher cohomology characterizes non ample divisors

inference for local distributions at high sampling frequencies a bootstrap approach abstract we study inference for the local innovations of ito semimartingales specifically we construct a resampling procedure for the empirical cdf of high frequency innovations that have been standardized using a nonparametric estimate of its stochastic scale volatility and truncated to rid the effect of large jumps our locally dependent wild bootstrap ldwb accommodate issues related to the stochastic scale and jumps as well as account for a special block wise dependence structure induced by sampling errors we show that the ldwb replicates first and second order limit theory from the usual empirical process and the stochastic scale estimate respectively in addition to an asymptotic bias moreover we design the ldwb sufficiently general to establish asymptotic equivalence between it and a nonparametric local block bootstrap also introduced here up to second order distribution theory finally we introduce ldwb aided kolmogorov smirnov tests for local gaussianity as well as local von mises statistics with and without bootstrap inference and establish their asymptotic validity using the second order distribution theory the finite sample performance of clt and ldwb aided local gaussianity tests is assessed in a simulation study and an empirical application whereas the clt test is oversized even in large samples the size of the ldwb tests is accurate even in small samples the empirical analysis verifies this pattern in addition to providing new insights about the fine scale distributional properties of innovations to equity indices commodities and exchange rates

bootstrapping a regression equation some empirical results abstract the bootstrap like the jackknife is a technique for estimating standard errors the idea is to use monte carlo simulation based on a nonparametric estimate of the underlying error distribution the main object of this article is to present the bootstrap in the context of an econometric equation describing the demand for energy by industry as it turns out the conventional asymptotic formulas for estimating standard errors are too optimistic by factors of nearly three when applied to a particular finite sample problem in a simpler context this finding can be given a mathematical proof

efficiency and robustness in resampling it is known that the standard delete 1 jackknife and the classical bootstrap are in general equally efficient for estimating the mean square error of a statistic in the i i d setting however this equivalence no longer holds in the linear regression model it turns out that the bootstrap is more efficient when error variables are homogeneous and the jackknife is more robust when they are heterogeneous in fact we can divide all the commonly used resampling procedures for linear regression models into two types the e type the efficient ones like the bootstrap and the r type the robust ones like the jackknife thus the theory presented here provides a unified view of all the known resampling procedures in linear regression

bootstrap inference for instrumental variable models with many weak instruments this study s main contribution is to theoretically analyze the application of bootstrap methods to instrumental variable models when the available instruments may be weak and the number of instruments goes to infinity with the sample size we demonstrate that a standard residual based bootstrap procedure cannot consistently estimate the distribution of the limited information maximum likelihood estimator or fuller 1977 estimator under many many weak instrument sequence the primary reason is that the standard procedure fails to capture the instrument strength in the sample adequately in addition we consider the restricted efficient re bootstrap of davidson and mackinnon 2008 2010 2014 that generates bootstrap data under the null restricted and uses an efficient estimator of the coefficient of the reduced form equation efficient we find that the re bootstrap is also invalid however it effectively mimics more key features in the limiting distributions of interest and thus is less distorted in finite samples than the standard bootstrap procedure finally we propose modified bootstrap procedures that provide a valid distributional approximation for the two estimators with many many weak instruments a monte carlo experiment shows that hypothesis testing based on the asymptotic normal approximation can have severe size distortions in finite samples instead our modified bootstrap procedures greatly reduce these distortions

wild bootstrap for quantile regression the existing theory of the wild bootstrap has focused on linear estimators in this note we broaden its validity by providing a class of weight distributions that is asymptotically valid for quantile regression estimators as most weight distributions in the literature lead to biased variance estimates for nonlinear estimators of linear regression we propose a modification of the wild bootstrap that admits a broader class of weight distributions for quantile regression a simulation study on median regression is carried out to compare various bootstrap methods with a simple finite sample correction the wild bootstrap is shown to account for general forms of heteroscedasticity in a regression model with fixed design points

bootstrapping m estimators in generalized autoregressive conditional heteroscedastic models summarywe consider the weighted bootstrap approximation to the distribution of a class of m estimators for the parameters of the generalized autoregressive conditional heteroscedastic model we prove that the bootstrap distribution given the data is a consistent estimate in probability of the distribution of the m estimator which is asymptotically normal we propose an algorithm for the computation of m estimates which at the same time is useful for computing bootstrap replicates from the given data our simulation study indicates superior coverage rates for various weighted bootstrap schemes compared with the rates based on the normal approximation and existing bootstrap methods for the generalized autoregressive conditional heteroscedastic model such as percentile t subsampling schemes since some familiar bootstrap schemes are special cases of the weighted bootstrap this paper thus provides a unified theory and algorithm for bootstrapping in generalized autoregressive conditional heteroscedastic models

generalized bootstrap for estimating equations we introduce a generalized bootstrap technique for estimators obtained by solving estimating equations some special cases of this generalized bootstrap are the classical bootstrap of efron the delete d jackknife and variations of the bayesian bootstrap the use of the proposed technique is discussed in some examples distributional consistency of the method is established and an asymptotic representation of the resampling variance estimator is obtained

the sorted effects method discovering heterogeneous effects beyond their averages the partial ceteris paribus effects of interest in nonlinear and interactive linear models are heterogeneous as they can vary dramatically with the underlying observed or unobserved covariates despite the apparent importance of heterogeneity a common practice in modern empirical work is to largely ignore it by reporting average partial effects or at best average effects for some groups while average effects provide very convenient scalar summaries of typical effects by definition they fail to reflect the entire variety of the heterogeneous effects in order to discover these effects much more fully we propose to estimate and report sorted effects a collection of estimated partial effects sorted in increasing order and indexed by percentiles by construction the sorted effect curves completely represent and help visualize the range of the heterogeneous effects in one plot they are as convenient and easy to report in practice as the conventional average partial effects they also serve as a basis for classification analysis where we divide the observational units into most or least affected groups and summarize their characteristics we provide a quantification of uncertainty standard errors and confidence bands for the estimated sorted effects and related classification analysis and provide confidence sets for the most and least affected groups the derived statistical results rely on establishing key new mathematical results on hadamard differentiability of a multivariate sorting operator and a related classification operator which are of independent interest we apply the sorted effects method and classification analysis to demonstrate several striking patterns in the gender wage gap we find that this gap is particularly strong for married women ranging from 60 to 0 between the 2 and 98 percentiles as a function of observed and unobserved characteristics while the gap for never married women ranges from 40 to 20 the most adversely affected women tend to be married do not have college degrees work in sales and have high levels of potential experience

a nadel vanishing theorem for metrics with minimal singularities on big line bundles abstract the purpose of this paper is to establish a nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic zariski decomposition such as metrics with minimal singularities and siu s metrics for this purpose we apply the theory of harmonic integrals and generalize enoki s proof of kollar s injectivity theorem moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics

solving matrix inequalities whose unknowns are matrices this paper provides algorithms for numerical solution of convex matrix inequalities in which the variables naturally appear as matrices this includes for instance many systems and control problems to use these algorithms no knowledge of linear matrix inequalities is required however as tools they preserve many advantages of the linear matrix inequality framework our method has two components 1 a numerical algorithm that solves a large class of matrix optimization problems and 2 a symbolic convexity checker that automatically provides a region which if convex guarantees that the solution from 1 is a global optimum on that region the algorithms are partly numerical and partly symbolic and since they aim at exploiting the matrix structure of the unknowns the symbolic part requires the development of new computer techniques for treating noncommutative algebra

quantile regression with censoring and endogeneity in this paper we develop a new censored quantile instrumental variable cqiv estimator and describe its properties and computation the cqiv estimator combines powell 1986 censored quantile regression cqr to deal with censoring with a control variable approach to incorporate endogenous regressors the cqiv estimator is obtained in two stages that are nonadditive in the unobservables the first stage estimates a nonadditive model with infinite dimensional parameters for the control variable such as a quantile or distribution regression model the second stage estimates a nonadditive censored quantile regression model for the response variable of interest including the estimated control variable to deal with endogeneity for computation we extend the algorithm for cqr developed by chernozhukov and hong 2002 to incorporate the estimation of the control variable we give generic regularity conditions for asymptotic normality of the cqiv estimator and for the validity of resampling methods to approximate its asymptotic distribution we verify these conditions for quantile and distribution regression estimation of the control variable our analysis covers two stage uncensored quantile regression with nonadditive first stage as an important special case we illustrate the computation and applicability of the cqiv estimator with a monte carlo numerical example and an empirical application on estimation of engel curves for alcohol

are discoveries spurious distributions of maximum spurious correlations and their applications over the last two decades many exciting variable selection methods have been developed for finding a small group of covariates that are associated with the response from a large pool can the discoveries by such data mining approaches be spurious due to high dimensionality and limited sample size can our fundamental assumptions on exogeneity of covariates needed for such variable selection be validated with the data to answer these questions we need to derive the distributions of the maximum spurious correlations given certain number of predictors namely the distribution of the correlation of a response variable y with the best s linear combinations of p covariates x even when x and y are independent when the covariance matrix of x possesses the restricted eigenvalue property we derive such distributions for both finite s and diverging s using gaussian approximation and empirical process techniques however such a distribution depends on the unknown covariance matrix of x hence we use the multiplier bootstrap procedure to approximate the unknown distributions and establish the consistency of such a simple bootstrap approach the results are further extended to the situation where residuals are from regularized fits our approach is then applied to construct the upper confidence limit for the maximum spurious correlation and testing exogeneity of covariates the former provides a baseline for guarding against false discoveries due to data mining and the latter tests whether our fundamental assumptions for high dimensional model selection are statistically valid our techniques and results are illustrated by both numerical examples and real data analysis

gaussian and bootstrap approximations for high dimensional u statistics and their applications this paper studies the gaussian and bootstrap approximations for the probabilities of a nondegenerate u statistic belonging to the hyperrectangles in mathbb r d when the dimension d is large a two step gaussian approximation procedure that does not impose structural assumptions on the data distribution is proposed subject to mild moment conditions on the kernel we establish the explicit rate of convergence uniformly in the class of all hyperrectangles in mathbb r d that decays polynomially in sample size for a high dimensional scaling limit where the dimension can be much larger than the sample size we also provide computable approximation methods for the quantiles of the maxima of centered u statistics specifically we provide a unified perspective for the empirical bootstrap the randomly reweighted bootstrap and the gaussian multiplier bootstrap with the jackknife estimator of covariance matrix as randomly reweighted quadratic forms and we establish their validity we show that all three methods are inferentially first order equivalent for high dimensional u statistics in the sense that they achieve the same uniform rate of convergence over all d dimensional hyperrectangles in particular they are asymptotically valid when the dimension d can be as large as o e n c for some constant c in 0 1 7 the bootstrap methods are applied to statistical applications for high dimensional non gaussian data including i principled and data dependent tuning parameter selection for regularized estimation of the covariance matrix and its related functionals ii simultaneous inference for the covariance and rank correlation matrices in particular for the thresholded covariance matrix estimator with the bootstrap selected tuning parameter we show that for a class of sub gaussian data error bounds of the bootstrapped thresholded covariance matrix estimator can be much tighter than those of the minimax estimator with a universal threshold in addition we also show that the gaussian like convergence rates can be achieved for heavy tailed data which are less conservative than those obtained by the bonferroni technique that ignores the dependency in the underlying data distribution

bayesian bootstrap for proportional hazards models we propose two bayesian bootstrap extensions the binomial and poisson forms for proportional hazards models the binomial form bayesian bootstrap is the limit of the posterior distribution with a beta process prior as the amount of the prior information vanishes and thus can be considered as a default nonparametric bayesian analysis it is also the same as lo s bayesian bootstrap for censored data when covariates are absent the poisson form bayesian bootstrap is equivalent to the bayesian analysis with cox s profile likelihood when the baseline distribution is discrete thus when the data set has many ties simulation study suggests that the binomial form bayesian bootstrap performs better than standard frequentist procedures in the frequentist sense an advantage of the proposed bayesian bootstrap procedures over the standard bayesian analysis is conceptual and computational simplicity finally it is shown that both bayesian bootstrap posteriors are asymptotically equivalent to the sampling distribution of the maximum likelihood estimator

nonparametric identification in panels using quantiles abstract this paper considers identification and estimation of ceteris paribus effects of continuous regressors in nonseparable panel models with time homogeneity the effects of interest are derivatives of the average and quantile structural functions of the model we find that these derivatives are identified with two time periods for stayers i e for individuals with the same regressor values in two time periods we show that the identification results carry over to models that allow location and scale time effects we propose nonparametric series methods and a weighted bootstrap scheme to estimate and make inference on the identified effects the bootstrap proposed allows inference for function valued parameters such as quantile effects uniformly over a region of quantile indices and or regressor values an empirical application to engel curve estimation with panel data illustrates the results

smooth minimum distance estimation and testing with conditional estimating equations uniform in bandwidth theory abstract to study the influence of a bandwidth parameter in inference with conditional moments we propose a new class of estimators and establish an asymptotic representation of our estimator as a process indexed by a bandwidth which can vary within a wide range including bandwidths independent of the sample size we study its behavior under misspecification we also propose an efficient version of our estimator we develop a procedure based on a distance metric statistic for testing restrictions on parameters as well as a bootstrap technique to account for the bandwidth s influence our new methods are simple to implement apply to non smooth problems and perform well in our simulations

residual bootstrap tests in linear models with many regressors abstract this paper is concerned with bootstrap hypothesis testing in linear regression models with many regressors i show that bootstrap f l r and l m tests are asymptotically valid even when the numbers of estimated parameters and tested restrictions are not asymptotically negligible fractions of the sample size one of the conditions for these results is that the regressors come from an asymptotically balanced design depending on the number of restrictions tested and on the errors distribution violation of that condition might render the bootstrap tests asymptotically invalid in that case i propose bootstrapping calhoun s 2011 g statistic or modified versions of the l r and l m statistics and show that these procedures remain asymptotically valid monte carlo simulations indicate that the bootstrap tests often outperform asymptotic ones however analyzing the approximate third cumulant of the f statistic reveals that the bootstrap test does not generally provide the usual higher order asymptotic refinements nevertheless it is found that the bootstrap third cumulant partially matches the population third cumulant which might explain the bootstrap s good finite sample performances especially when the errors come from a symmetric distribution

bootstrapping general empirical measures it is proved that the bootstrapped central limit theorem for empirical processes indexed by a class of functions mathscr f and based on a probability measure p holds a s if and only if mathscr f in mathrm clt p and int f 2 dp infty where f sup f in mathscr f f and it holds in probability if and only if mathscr f in mathrm clt p thus for a large class of statistics no local uniformity of the clt about p is needed for the bootstrap to work consistency of the bootstrap the bootstrapped law of large numbers is also characterized these results are proved under certain weak measurability assumptions on mathscr f

invariance of plurigenera in this paper we give a proof of the following long conjectured result on the invariance of the plurigenera main theorem let p x d be a smooth projective family of compact complex manifolds parametrized by the open unit 1 disk d assume that the bers xt py1 t t 2 d are of general type then for every positive integer m the plurigenus dimc c xt mkxt is independent of t 2 d where kxt is the canonical line bundle of xt

asymptotics for high dimensional regression m estimates fixed design results we investigate the asymptotic distributions of coordinates of regression m estimates in the moderate p n regime where the number of covariates p grows proportionally with the sample size n under appropriate regularity conditions we establish the coordinate wise asymptotic normality of regression m estimates assuming a fixed design matrix our proof is based on the second order poincare inequality and leave one out analysis some relevant examples are indicated to show that our regularity conditions are satisfied by a broad class of design matrices we also show a counterexample namely an anova type design to emphasize that the technical assumptions are not just artifacts of the proof finally numerical experiments confirm and complement our theoretical results

regression adjustment in completely randomized experiments with a diverging number of covariates extending r a fisher and d a freedman s results on the analysis of covariance lin 2013 proposed an ordinary least squares adjusted estimator of the average treatment effect in completely randomized experiments we further study its statistical properties under the potential outcomes model in the asymptotic regimes allowing for a diverging number of covariates we show that lin 2013 s estimator is consistent when kappa log p rightarrow 0 and asymptotically normal when kappa p rightarrow 0 under mild moment conditions where kappa is the maximum leverage score of the covariate matrix in the favorable case where leverage scores are all close together his estimator is consistent when p o n log n and is asymptotically normal when p o n 1 2 in addition we propose a bias corrected estimator that is consistent when kappa log p rightarrow 0 and is asymptotically normal with the same variance in the fixed p regime when kappa 2 p log p rightarrow 0 in the favorable case the latter condition reduces to p o n 2 3 log n 1 3 similar to lin 2013 our results hold for non random potential outcomes and covariates without any model specification our analysis requires novel analytic tools for sampling without replacement which complement and potentially enrich the theory in other areas such as survey sampling matrix sketching and transductive learning

evaluating prediction rules for t year survivors with censored regression models suppose that we are interested in establishing simple but reliable rules for predicting future t year survivors through censored regression models in this article we present inference procedures for evaluating such binary classification rules based on various prediction precision measures quantified by the overall misclassification rate sensitivity and specificity and positive and negative predictive values specifically under various working models we derive consistent estimators for the above measures through substitution and cross validation estimation procedures furthermore we provide large sample approximations to the distributions of these nonsmooth estimators without assuming that the working model is correctly specified confidence intervals for example for the difference of the precision measures between two competing rules can then be constructed all of the proposals are illustrated with real examples and their finite sample properties are evaluated through a simulation study

bootstrapping robust estimates for clustered data in mixed models the use of robust estimates raises several interesting inferential challenges one of these challenges arises from the realization that the effect of contamination is to increase the variability in the data but robust estimates of variance components are usually smaller than their nonrobust counterparts the robust estimates reflect the variability of the bulk of the data which is not the same as the variability in the data generating process this means that the naive implementation of bootstrap procedures might not work in this article we consider several bootstrap procedures including random effect transformation and weighted bootstraps we give conditions for the asymptotic validity of the bootstraps and assess their performance via a small simulation study both the transformation and generalized cluster bootstrap perform well and are asymptotically valid under reasonable conditions

the bayesian bootstrap the bayesian bootstrap is the bayesian analogue of the bootstrap instead of simulating the sampling distribution of a statistic estimating a parameter the bayesian bootstrap simulates the posterior distribution of the parameter operationally and inferentially the methods are quite similar because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions neither method should be applied without some consideration of the reasonableness of these model assumptions in this sense neither method is a true bootstrap procedure yielding inferences unaided by external assumptions

some nonasymptotic results on resampling in high dimension i confidence regions we study generalized bootstrap confidence regions for the mean of a random vector whose coordinates have an unknown dependency structure the random vector is supposed to be either gaussian or to have a symmetric and bounded distribution the dimensionality of the vector can possibly be much larger than the number of observations and we focus on a non asymptotic control of the confidence level following ideas inspired by recent results in learning theory we consider two approaches the first based on a concentration principle valid for a large class of resampling weights and the second on a direct resampled quantile specifically using rademacher weights several intermediate results established in the approach based on concentration principles are of self interest we also discuss the question of accuracy when using monte carlo approximations of the resampled quantities we present an application of these results to the one sided and two sided multiple testing problem in which we derive several resampling based step down procedures providing a non asymptotic fwer control we compare our different procedures in a simulation study and we show that they can outperform bonferroni s or holm s procedures as soon as the observed vector has sufficiently correlated coordinates

an eigenvalue estimate for the bar partial laplacian we study the bar partial laplacian on forms taking values in lk a high power of a semipositive line bundle over a compact complex manifold and give an estimate of the number of eigenvalues smaller than examples show that the estimate gives the right order of magnitude in terms of the two spectral parameters k and

weighted likelihood estimation under two phase sampling we develop asymptotic theory for weighted likelihood esti mators wle under two phase stratied sampling without replacement we also consider several variants of wle s involving estimated weights and calibration a set of empirical process tools are developed including a glivenko cantelli theorem a theorem for rates of convergence of z estimators and a donsker theorem for the inverse probability weighted em pirical processes under two phase sampling and sampling without replace ment at the second phase using these general results we derive asymptotic distributions of the wle of a nite dimensional parameter in a general semiparametric model where an estimator of a nuisance parameter is es timable either at regular or non regular rates we illustrate these results and methods in the cox model with right censoring and interval censor ing we compare the methods via their asymptotic variances under both sampling without replacement and the more usual and easier to analyze assumption of bernoulli sampling at the second phase ams 2000 subject classications primary 62e20 secondary 62g20 62d99 62n01

intentionally biased bootstrap methods summary a class of weighted bootstrap techniques called biased bootstrap or b bootstrap methods is introduced it is motivated by the need to adjust empirical methods such as the uniform bootstrap in a surgical way to alter some of their features while leaving others unchanged depending on the nature of the adjustment the b bootstrap can be used to reduce bias or to reduce variance or to render some characteristic equal to a predetermined quantity examples of the last application include a b bootstrap approach to hypothesis testing in nonparametric contexts where the b bootstrap enables simulation under the null hypothesis even when the hypothesis is false and a b bootstrap competitor to tibshirani s variance stabilization method an example of the bias reduction application is adjustment of nadaraya watson kernel estimators to make them competitive with local linear smoothing other applications include density estimation under constraints outlier trimming sensitivity analysis skewness or kurtosis reduction and shrinkage

a large sample study of the bayesian bootstrap on donne une justification asymptotique du bootstrap bayesien on obtient des resultats lies a l article de rubin 1981 dans le meme sens que bickel et freedman 1981 et singh 1981 se rapportant a l article d efron 1979

functional central limit theorems for single stage sampling designs for a joint model based and design based inference we establish functional central limit theorems for the horvitz thompson empirical process and the h ajek empirical process centered by their finite population mean as well as by their super population mean in a survey sampling framework the results apply to single stage unequal probability sampling designs and essentially only require conditions on higher order correlations we apply our main results to a hadamard differentiable statistical functional and illustrate its limit behavior by means of a computer simulation

central limit theorems and bootstrap in high dimensions this paper derives central limit and bootstrap theorems for probabilities that sums of centered high dimensional random vectors hit hyperrectangles and sparsely convex sets specifically we derive gaussian and bootstrap approximations for probabilities p n 1 2 ni 1xi a p n 1 2 i 1nxi a where x1 xnx1 xn are independent random vectors in rprp and aa is a hyperrectangle or more generally a sparsely convex set and show that the approximation error converges to zero even if p pn p pn as n n and p np n in particular pp can be as large as o ecnc o ecnc for some constants c c 0c c 0 the result holds uniformly over all hyperrectangles or more generally sparsely convex sets and does not require any restriction on the correlation structure among coordinates of xixi sparsely convex sets are sets that can be represented as intersections of many convex sets whose indicator functions depend only on a small subset of their arguments with hyperrectangles being a special case

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

relative critical exponents non vanishing and metrics with minimal singularities in this article we establish a very general non vanishing statement as well as several properties of metrics with minimal singularities of adjoint bundles our arguments combine the proof of the classical non vanishing theorem due to v shokurov as well as many ideas from y t siu s analytic proof of the finite generation of the canonical ring an important tool is the notion of relative critical exponent of two closed positive currents with respect to a measure

distributional consistency of the lasso by perturbation bootstrap summarythe lasso is a popular estimation procedure in multiple linear regression we develop and establish the validity of a perturbation bootstrap method for approximating the distribution of the lasso estimator in a heteroscedastic linear regression model we allow the underlying covariates to be either random or nonrandom and show that the proposed bootstrap method works irrespective of the nature of the covariates we also investigate finite sample properties of the proposed bootstrap method in a moderately large simulation study

analysis of least absolute deviation summary we develop a unified l 1 based analysis of variance type method for testing linear hypotheses like the classical l2 based analysis of variance the method is coordinate free in the sense that it is invariant under any linear transformation of the covariates or regression parameters moreover it allows singular design matrices and heterogeneous error terms a simple approximation using stochastic perturbation is proposed to obtain cut offvalues for the resulting test statistics both test statistics and distributional approximations can be computed using standard linear programming an asymptotic theory is derived for the method special cases of one and multi way analysis of variance and analysis of covariance models are worked out in detail the main results of this paper can be extended to general quantile regression extensive simulations show that the method works well in practical settings the method is also applied to a dataset from general social surveys

generic inference on quantile and quantile effect functions for discrete outcomes quantile and quantile effect qe functions are important tools for descriptive and causal analysis due to their natural and intuitive interpretation existing inference methods for these functions

statistical consistency and asymptotic normality for high dimensional robust m estimators we study theoretical properties of regularized robust m estimators applicable when data are drawn from a sparse high dimensional linear model and contaminated by heavy tailed distributions and or outliers in the additive errors and covariates we first establish a form of local statistical consistency for the penalized regression estimators under fairly mild conditions on the error distribution when the derivative of the loss function is bounded and satisfies a local restricted curvature condition all stationary points within a constant radius of the true regression vector converge at the minimax rate enjoyed by the lasso with sub gaussian errors when an appropriate nonconvex regularizer is used in place of an ell 1 penalty we show that such stationary points are in fact unique and equal to the local oracle solution with the correct support hence results on asymptotic normality in the low dimensional case carry over immediately to the high dimensional setting this has important implications for the efficiency of regularized nonconvex m estimators when the errors are heavy tailed our analysis of the local curvature of the loss function also has useful consequences for optimization when the robust regression function and or regularizer is nonconvex and the objective function possesses stationary points outside the local region we show that as long as a composite gradient descent algorithm is initialized within a constant radius of the true regression vector successive iterates will converge at a linear rate to a stationary point within the local region furthermore the global optimum of a convex regularized robust regression function may be used to obtain a suitable initialization the result is a novel two step procedure that uses a convex m estimator to achieve consistency and a nonconvex m estimator to increase efficiency we conclude with simulation results that corroborate our theoretical findings

bootstrap and wild bootstrap for high dimensional linear models in this paper two bootstrap procedures are considered for the estimation of the distribution of linear contrasts and of f test statistics in high dimensional linear models an asymptotic approach will be chosen where the dimension p of the model may increase for sample size n rightarrow infty the range of validity will be compared for the normal approximation and for the bootstrap procedures furthermore it will be argued that the rates of convergence are different for the bootstrap procedures in this asymptotic framework this is in contrast to the usual asymptotic approach where p is fixed

computer algebra tailored to matrix inequalities in control a major advance in linear systems theory over the last decade has been a formalism for converting systems problems to matrix inequalities in this tutorial paper we describe computer algebra algorithms methodology and implementation which allows users to convert many systems problems to linear matrix inequalities lmis we shall focus on computer algebra methodology which can assist the user in producing lmis for control design we provide a step by step computer derivation of lmi formulas for the design of linear time invariant dynamic controllers that achieve a prespecified performance measured by the h norm of a certain closed loop transfer function

effectiveness of demailly s strong openness conjecture and related problems in this article stimulated by the effectiveness in berndtsson s solution of the openness conjecture and continuing our solution of demailly s strong openness conjecture we discuss conditions to guarantee the effectiveness of the conjecture and establish such an effectiveness result we explicitly point out a lower semicontinuity property of plurisubharmonic functions with a multiplier which is implicitly contained in guan and zhou 2014 arxiv 1401 7158 we also obtain optimal effectiveness of the conjectures of demailly kollar and jonsson mustat respectively

estimation of a multiplicative covariance structure in the large dimensional case abstract we propose a kronecker product model for correlation or covariance matrices in the large dimensional case the number of parameters of the model increases logarithmically with the dimension of the matrix we propose a minimum distance md estimator based on a log linear property of the model as well as a one step estimator which is a one step approximation to the quasi maximum likelihood estimator qmle we establish rates of convergence and central limit theorems clt for our estimators in the large dimensional case a specification test and tools for kronecker product model selection and inference are provided in a monte carlo study where a kronecker product model is correctly specified our estimators exhibit superior performance in an empirical application to portfolio choice for s p500 daily returns we demonstrate that certain kronecker product models are good approximations to the general covariance matrix

bootstrap confidence sets under model misspecification a multiplier bootstrap procedure for construction of likelihood based confidence sets is considered for finite samples and a possible model misspecification theoretical results justify the bootstrap consistency for a small or moderate sample size and allow to control the impact of the parameter dimension p the bootstrap approximation works if p n is small the main result about bootstrap consistency continues to apply even if the underlying parametric model is misspecified under the so called small modeling bias condition in the case when the true model deviates significantly from the considered parametric family the bootstrap procedure is still applicable but it becomes a bit conservative the size of the constructed confidence sets is increased by the modeling bias we illustrate the results with numerical examples for misspecified constant and logistic regressions jel classification codes c13 c15

validity of dependent bootstrapping in finite populations abstract the traditional bootstrap resamples with replacement from the original sample observations to form arrays of row wise independent and identically distributed bootstrap random variables there are situations for example when sampling from finite populations where resampling without replacement provides a more realistic bootstrap procedure and produces dependent bootstrap random variables the desired properties of consistency and asymptotic validity are shown to hold for certain nonparametric dependent bootstrap estimators in addition it is shown that the smaller variation in dependent bootstrap estimators can be used to increase precision in some of the estimators

on the impact of predictor geometry on the performance on high dimensional ridge regularized generalized robust regression estimators we study ridge regularized generalized robust regression estimators i e begin aligned widehat beta mathrm argmin beta in mathbb r p frac 1 n sum i 1 n rho i y i x i beta frac tau 2 beta 2 quad text where quad y i epsilon i x i beta 0 end aligned in the situation where p n tends to a finite non zero limit our study here focuses on the situation where the errors epsilon i s are heavy tailed and x i s have an elliptical like distribution our assumptions are quite general and we do not require homoskedasticity of epsilon i s for instance we obtain a characterization of the limit of widehat beta beta 0 as well as several other results including central limit theorems for the entries of widehat beta

jackknife bootstrap and other resampling methods in regression analysis motivated by a representation for the least squares estimator we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time they are unbiased for homoscedastic errors and a special case the delete one jackknife is almost unbiased for heteroscedastic errors the method is extended to cover nonlinear parameters regression m estimators nonlinear regression and generalized linear models interval estimators can be constructed from the jackknife histogram three bootstrap methods are considered two are shown to give biased variance estimators and one does not have the bias robustness property enjoyed by the weighted delete one jackknife a general method for resampling residuals is proposed it gives variance estimators that are bias robust several bias reducing estimators are proposed some simulation results are reported

augmented factor models with applications to validating market risk factors and forecasting bond risk premia abstract we study factor models augmented by observed covariates that have explanatory powers on the unknown factors in financial factor models the unknown factors can be reasonably well explained by a few observable proxies such as the fama french factors in diffusion index forecasts identified factors are strongly related to several directly measurable economic variables such as consumption wealth variable financial ratios and term spread with those covariates both the factors and loadings are identifiable up to a rotation matrix even only with a finite dimension to incorporate the explanatory power of these covariates we propose a smoothed or projected principal component analysis pca i regress the data onto the observed covariates and ii take the principal components of the fitted data to estimate the loadings and factors this allows us to more accurately estimate the percentage of both explained and unexplained components in factors and thus to assess the explanatory power of covariates we show that both the estimated factors and loadings can be estimated with improved rates of convergence compared to the benchmark method the degree of improvement depends on the strength of the signals representing the explanatory power of the covariates on the factors the proposed estimator is robust to possibly heavy tailed distributions we apply the model to forecast us bond risk premia and find that the observed macroeconomic characteristics contain strong explanatory powers of the factors the gain of forecast is more substantial when the characteristics are incorporated to estimate the common factors than directly used for forecasts

the two sample problem for poisson processes adaptive tests with a non asymptotic wild bootstrap approach considering two independent poisson processes we address the question of testing equality of their respective intensities we first propose single tests whose test statistics are u statistics based on general kernel functions the corresponding critical values are constructed from a non asymptotic wild bootstrap approach leading to level alpha tests various choices for the kernel functions are possible including projection approximation or reproducing kernels in this last case we obtain a parametric rate of testing for a weak metric defined in the rkhs associated with the considered reproducing kernel then we introduce in the other cases an aggregation procedure which allows us to import ideas coming from model selection thresholding and or approximation kernels adaptive estimation the resulting multiple tests are proved to be of level alpha and to satisfy non asymptotic oracle type conditions for the classical l2 norm from these conditions we deduce that they are adaptive in the minimax sense over a large variety of classes of alternatives based on classical and weak besov bodies in the univariate case but also sobolev and anisotropic nikol skii besov balls in the multivariate case

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

on robust regression with high dimensional predictors we study regression m estimates in the setting where p the number of covariates and n the number of observations are both large but p n we find an exact stochastic representation for the distribution of argmin p i 1 n y i x i at fixed p and n under various assumptions on the objective function and our statistical model a scalar random variable whose deterministic limit r can be studied when p n 0 plays a central role in this representation we discover a nonlinear system of two deterministic equations that characterizes r interestingly the system shows that r depends on through proximal mappings of as well as various aspects of the statistical model underlying our study several surprising results emerge in particular we show that when p n is large enough least squares becomes preferable to least absolute deviations for double exponential errors

anti concentration and honest adaptive confidence bands modern construction of uniform condence e and nickl 2010 this condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process equivalently for the supremum of a gaussian process with the same covariance function as that of the studentized empirical process the principal contribution of this paper is to remove the need for this classical condition we show that a considerably weaker sucient condi tion is derived from an anti concentration property of the supremum of the approximating gaussian process and we derive an inequality lead ing to such a property for separable gaussian processes we refer to the new condition as a generalized sbr condition our new result shows that the supremum does not concentrate too fast around any value we then apply this result to derive a gaussian multiplier bootstrap procedure for constructing honest condence bands for nonparametric density estimators this result can be applied in other nonparametric problems as well an essential advantage of our approach is that it ap plies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist or is un known this is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data driven fash ion which is needed for adaptive constructions of the condence bands furthermore our approach is asymptotically honest at a polynomial rate namely the error in coverage level converges to zero at a fast polynomial speed with respect to the sample size in sharp contrast the approach based on extreme value theory is asymptotically honest only at a logarithmic rate the error converges to zero at a slow loga rithmic speed finally of independent interest is our introduction of a new practical version of lepski s method which computes the optimal non conservative resolution levels via a gaussian multiplier bootstrap method

quantile treatment effects in difference in differences models under dependence restrictions and with only two time periods abstract this paper shows that the conditional quantile treatment effect on the treated is identified under i a conditional distributional difference in differences assumption and ii a new assumption that the dependence the copula between the change in untreated potential outcomes and the initial level of untreated potential outcomes is the same for the treated group and untreated group we consider estimation and inference with discrete covariates and propose a uniform inference procedure based on the exchangeable bootstrap finally we estimate the effect of increasing the minimum wage on the distribution of earnings for subgroups defined by race gender and education

on properties of sylvester and lyapunov operators abstract sylvester and lyapunov operators in real and complex matrix spaces are studied which include as particular cases the operators arising in the theory of linear time invariant systems let m f m n f p q be a linear operator where f r or f c the operator m is elementary if there exist matrices a f p m and b f q n such that m x axb each m can be represented as a sum of minimum number of elementary operators called the sylvester index of m an expression for the sylvester index of a general linear operator m is given an important tool here is a special permutation operator v p m f pq mn f pm nq such that the image v p m b t a of the matrix of a non zero elementary operator is equal to the rank 1 matrix vec a row b where vec x and row x are the column wise and row wise vector representation of the matrix x the application of v p m reduces a sum of kronecker products of matrices to the standard product of two matrices a linear operator l f n n f n n is a lyapunov operator if l x l x where the star denotes transposition in the real case and complex conjugate transposition in the complex case characterisations and parametrisations of the sets of real and complex lyapunov operators are given and their dimensions are found relevant lyapunov indexes for lyapunov operators are introduced and calculated similar results are given also for several classes of lyapunov like linear and pseudo linear operators the concept of lyapunov singular values of a lyapunov operator is introduced and the application of these values to the sensitivity and a posteriori error analysis of lyapunov equations is discussed

conditional quantile estimation for generalized autoregressive conditional heteroscedasticity models conditional quantile estimation is an essential ingredient in modern risk management although generalized autoregressive conditional heteroscedasticity garch processes have proven highly successful in modeling financial data it is generally recognized that it would be useful to consider a broader class of processes capable of representing more flexibly both asymmetry and tail behavior of conditional returns distributions in this article we study estimation of conditional quantiles for garch models using quantile regression quantile regression estimation of garch models is highly nonlinear we propose a simple and effective two step approach of quantile regression estimation for linear garch time series in the first step we use a quantile autoregression sieve approximation for the garch model by combining information over different quantiles then second stage estimation for the garch model is carried out based on the first stage minimum distance estimation of the scale process of the time series

on rank estimators in increasing dimensions abstract the family of rank estimators including han s maximum rank correlation han 1987 as a notable example has been widely exploited in studying regression problems for these estimators although the linear index is introduced for alleviating the impact of dimensionality the effect of large dimension on inference is rarely studied this paper fills this gap via studying the statistical properties of a larger family of m estimators whose objective functions are formulated as u processes and may be discontinuous in increasing dimension set up where the number of parameters p n in the model is allowed to increase with the sample size n first we find that often in estimation as p n n 0 p n n 1 2 rate of convergence is obtainable second we establish bahadur type bounds and study the validity of normal approximation which we find often requires a much stronger scaling requirement than p n 2 n 0 third we state conditions under which the numerical derivative estimator of asymptotic covariance matrix is consistent and show that the step size in implementing the covariance estimator has to be adjusted with respect to p n all theoretical results are further backed up by simulation studies

simplification of symbolic polynomials on non commutative variables abstract we address the issue of simplifying symbolic polynomials on non commutative variables the problem is motivated by applications in optimization and various problems in systems and control we develop theory for polynomials which are linear in a subset of the variables and develop algorithms to produce representations which have the minimal possible number of terms the results can handle polynomial matrices as well as block matrix variables

multiplier ideal sheaves and existence of kahler einstein metrics of positive scalar curvature abstract to study c0a priori estimates for solutions to certain complex monge ampere equations i introduce a coherent sheaf of ideals and show that it satisfies various global algebrogeometric conditions including a cohomology vanishing theorem this technique is used to establish the existence of kahler einstein metrics of positive scalar curvature on a very large class of compact complex manifolds

a nadel vanishing theorem via injectivity theorems the purpose of this paper is to establish nadel type vanishing theorems with multiplier ideal sheaves of singular metrics admitting an analytic zariski decomposition such as metrics with minimal singularities and siu s metrics for this purpose we generalize kollar s injectivity theorem to an injectivity theorem for line bundles equipped with singular metrics by making use of the theory of harmonic integrals moreover we give asymptotic cohomology vanishing theorems for high tensor powers of line bundles

bootstrap consistency for general semiparametric m estimation consider m estimation in a semiparametric model that is characterized by a euclidean parameter of interest and an infinite dimensional nuisance parameter as a general purpose approach to statistical inferences the bootstrap has found wide applications in semiparametric m estimation and because of its simplicity provides an attractive alternative to the inference approach based on the asymptotic distribution theory the purpose of this paper is to provide theoretical justifications for the use of bootstrap as a semiparametric inferential tool we show that under general conditions the bootstrap is asymptotically consistent in estimating the distribution of the m estimate of euclidean parameter that is the bootstrap distribution asymptotically imitates the distribution of the m estimate we also show that the bootstrap confidence set has the asymptotically correct coverage probability these general conclusions hold in particular when the nuisance parameter is not estimable at root n rate and apply to a broad class of bootstrap methods with exchangeable bootstrap weights this paper provides a first general theoretical study of the bootstrap in semiparametric models

a directional mixed effects model for compositional expenditure data abstractcompositional data are vectors of proportions defined on the unit simplex and this type of constrained data occur frequently in government surveys it is also possible for the compositional data to be correlated due to the clustering or grouping of the observations within small domains or areas we propose a new class of the mixed model for compositional data based on the kent distribution for directional data where the random effects also have kent distributions one useful property of the new directional mixed model is that the marginal mean direction has a closed form and is interpretable the random effects enter the model in a multiplicative way via the product of a set of rotation matrices and the conditional mean direction is a random rotation of the marginal mean direction in small area estimation settings the mean proportions are usually of primary interest and these are shown to be simple functions of the marginal mean direction for estimation we apply a quasi likelihood method whic

gee analysis of clustered binary data with diverging number of covariates clustered binary data with a large number of covariates have become increasingly common in many scientific disciplines this paper develops an asymptotic theory for generalized estimating equations gee analysis of clustered binary data when the number of covariates grows to infinity with the number of clusters in this large n diverging p framework we provide appropriate regularity conditions and establish the existence consistency and asymptotic normality of the gee estimator furthermore we prove that the sandwich variance formula remains valid even when the working correlation matrix is misspecified the use of the sandwich variance formula leads to an asymptotically valid confidence interval and wald test for an estimable linear combination of the unknown parameters the accuracy of the asymptotic approximation is examined via numerical simulations we also discuss the diverging p asymptotic theory for general gee the results in this paper extend the recent elegant work of xie and yang ann statist 31 2003 310 347 and balan and schiopu kratina ann statist 32 2005 522 541 in the fixed p setting

efficient estimation for rank based regression with clustered data rank based inference is widely used because of its robustness this article provides optimal rank based estimating functions in analysis of clustered data with random cluster effects the extensive simulation studies carried out to evaluate the performance of the proposed method demonstrate that it is robust to outliers and is highly efficient given the existence of strong cluster correlations the performance of the proposed method is satisfactory even when the correlation structure is misspecified or when heteroscedasticity in variance is present finally a real dataset is analyzed for illustration

gmm inference when the number of moment conditions is large abstract asymptotic theory typically presumes that the dimensionality of econometric models is independent of the sample size even though this presumption is often quite unrealistic in gmm estimation whenever optimal instruments are not available it can frequently be shown that adding over identifying restrictions moment conditions will increase asymptotic precision however the conventional asymptotics which underlies this view insists that the number of moment conditions remain finite even though the number of available moment conditions may grow without bound we consider the explicit dependence of the number of moment conditions or instruments q n on the sample size n and establish that under conventional regularity conditions for the estimation of a linear model with general heteroskedasticity q n 3 n 0 is a sufficient condition for the validity of conventional asymptotic inference about the gmm estimator

conditional quantile processes based on series or many regressors abstract quantile regression qr is a principal regression method for analyzing the impact of covariates on outcomes the impact is described by the conditional quantile function and its functionals in this paper we develop the nonparametric qr series framework covering many regressors as a special case for performing inference on the entire conditional quantile function and its linear functionals in this framework we approximate the entire conditional quantile function by a linear combination of series terms with quantile specific coefficients and estimate the function valued coefficients from the data we develop large sample theory for the qr series coefficient process namely we obtain uniform strong approximations to the qr series coefficient process by conditionally pivotal and gaussian processes based on these two strong approximations or couplings we develop four resampling methods pivotal gradient bootstrap gaussian and weighted bootstrap that can be used for inference on the entire qr series coefficient function we apply these results to obtain estimation and inference methods for linear functionals of the conditional quantile function such as the conditional quantile function itself its partial derivatives average partial derivatives and conditional average partial derivatives specifically we obtain uniform rates of convergence and show how to use the four resampling methods mentioned above for inference on the functionals all of the above results are for function valued parameters holding uniformly in both the quantile index and the covariate value and covering the pointwise case as a by product we demonstrate the practical utility of these results with an empirical example where we estimate the price elasticity function and test the slutsky condition of the individual demand for gasoline as indexed by the individual unobserved propensity for gasoline consumption