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

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 realized volatility we propose bootstrap methods for a general class of nonlinear transformations of realized volatility which includes the raw version of realized volatility and its logarithmic transformation as special cases we consider the independent and identically distributed i i d bootstrap and the wild bootstrap wb and prove their first order asymptotic validity under general assumptions on the log price process that allow for drift and leverage effects we derive edgeworth expansions in a simpler model that rules out these effects the i i d bootstrap provides a second order asymptotic refinement when volatility is constant but not otherwise the wb yields a second order asymptotic refinement under stochastic volatility provided we choose the external random variable used to construct the wb data appropriately none of these methods provides third order asymptotic refinements both methods improve upon the first order asymptotic theory in finite samples

rate of convergence for the wild bootstrap in nonparametric regression this paper concerns the distributions used to construct confidence intervals for the regression function in a nonparametric setup some rates of convergence for the normal limit its plug in approach and the wild bootstrap are obtained conditionally on the explanatory variable x and also unconditionally the bound found for the wild bootstrap approximation is slightly better by a factor n 1 45 than the bounds given by the plug in approach or the clt for the conditional probability on the contrary the unconditional bounds present a different feature the rate obtained when approximating by the clt improves the one given by the plug in approach by a factor of n8 45 while this last one performs better than the wild bootstrap approximation and the corresponding ratio is n1 45 it should be mentioned that these two sequences especially the last one tend to zero at an extremely slow rate

bootstrapping regression models the regression and correlation models are considered it is shown that the bootstrap approximation to the distribution of the least squares estimates is valid and some error bounds are given

testing the predictor effect on a functional response abstractthis article examines the problem of nonparametric testing for the no effect of a random covariate or predictor on a functional response this means testing whether the conditional expectation of the response given the covariate is almost surely zero or not without imposing any model relating response and covariate the covariate could be univariate multivariate or functional our test statistic is a quadratic form involving univariate nearest neighbor smoothing and the asymptotic critical values are given by the standard normal law when the covariate is multidimensional or functional a preliminary dimension reduction device is used which allows the effect of the covariate to be summarized into a univariate random quantity the test is able to detect not only linear but nonparametric alternatives the responses could have conditional variance of unknown form and the law of the covariate does not need to be known an empirical study with simulated and real data shows that the test performs

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

optimum nonlinear filtering this paper is composed of two parts the first part surveys the literature regarding optimum nonlinear filtering from the continuous time stochastic analysis point of view and the other part explores the impact of recent applications of neural networks in a discrete time context to nonlinear filtering in particular the results obtained by using a regularized form of radial basis function rbf networks are presented in fair detail

estimating subject specific dependent competing risk profile with censored event time observations in a longitudinal study suppose that the primary endpoint is the time to a specific event this response variable however may be censored by an independent censoring variable or by the occurrence of one of several dependent competing events for each study subject a set of baseline covariates is collected the question is how to construct a reliable prediction rule for the future subject s profile of all competing risks of interest at a specific time point for risk benefit decision makings in this paper we propose a two stage procedure to make inferences about such subject specific profiles for the first step we use a parametric model to obtain a univariate risk index score system we then estimate consistently the average competing risks for subjects which have the same parametric index score via a nonparametric function estimation procedure we illustrate this new proposal with the data from a randomized clinical trial for evaluating the efficacy of a treatment for prostate cancer the primary endpoint for this study was the time to prostate cancer death but had two types of dependent competing events one from cardiovascular death and the other from death of other causes

nested hierarchical functional data modeling and inference for the analysis of functional plant phenotypes abstractin a plant science root image study the process of seedling roots bending in response to gravity is recorded using digital cameras and the bending rates are modeled as functional plant phenotype data the functional phenotypes are collected from seeds representing a large variety of genotypes and have a three level nested hierarchical structure with seeds nested in groups nested in genotypes the seeds are imaged on different days of the lunar cycle and an important scientific question is whether there are lunar effects on root bending we allow the mean function of the bending rate to depend on the lunar day and model the phenotypic variation between genotypes groups of seeds imaged together and individual seeds by hierarchical functional random effects we estimate the covariance functions of the functional random effects by a fast penalized tensor product spline approach perform multi level functional principal component analysis fpca using the best linear unbiased predictor of the pri

a central limit theorem for generalized quadratic forms random variables of the form w n mathop sum limits 1 leqq i leqq n mathop sum limits text 1 leqq j leqq n w ijn x i x j are considered with x i independent not necessarily identically distributed and w ijn borel functions such that w ijn x i x j is square integrable and has vanishing conditional expectations e w ijn x i x j x i e w ijn x i x j x j 0 text a text s text a central limit theorem is proved under the condition that the normed fourth moment tends to 3 under some restrictions the condition is also necessary finally conditions on the individual tails of w ijn x i x j and an eigenvalue condition are given that ensure asymptotic normality of w n

generalized quasi likelihood ratio tests for semiparametric analysis of covariance models in longitudinal data abstractwe model generalized longitudinal data from multiple treatment groups by a class of semiparametric analysis of covariance models which take into account the parametric effects of time dependent covariates and the nonparametric time effects in these models the treatment effects are represented by nonparametric functions of time and we propose a generalized quasi likelihood ratio test procedure to test if these functions are identical our estimation procedure is based on profile estimating equations combined with local linear smoothers we find that the much celebrated wilks phenomenon which is well established for independent data still holds for longitudinal data if a working independence correlation structure is assumed in the test statistic however this property does not hold in general especially when the working variance function is misspecified our empirical study also shows that incorporating correlation into the test statistic does not necessarily improve the power of the test the

optimal estimation for the satellite attitude using star tracker measurements an optimal estimation scheme is presented which determines the satellite attitude using the gyro readings and the star tracker measurements of a commonly used satellite attitude measuring unit the scheme is mainly based on the exponential fourier densities that have the desirable closure property under conditioning by updating a finite and fixed number of parameters the conditional probability density which is an exponential fourier density is recursively determined simulation results indicate that the scheme is more accurate and robust than extended kalman filtering it is believed that this approach is applicable to many other attitude measuring units as no linearization and approximation are necessary in the approach it is ideal for systems involving high levels of randomness and or low levels of observability and systems for which accuracy is of overriding importance

asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity consider a sample of size n from a regular exponential family in p n dimensions let hat theta n denote the maximum likelihood estimator and consider the case where p n tends to infinity with n and where theta n is a sequence of parameter values in r p n moment conditions are provided under which hat theta n theta n o p sqrt p n n and hat theta n theta n overline x n o p p n n where overline x n is the sample mean the latter result provides normal approximation results when p 2 n n rightarrow 0 it is shown by example that even for a single coordinate of hat theta n theta n p 2 n n rightarrow 0 may be needed for normal approximation however if p 3 2 n n rightarrow 0 the likelihood ratio test statistic lambda for a simple hypothesis has a chi square approximation in the sense that 2 log lambda p n sqrt 2p n rightarrow d mathscr n 0 1

on continuous dependence of solutions of dynamic equations the main goal of the paper is to present a new approach to the problem of continuous dependence of solutions of differential or dynamic problems on their domains this is of particular interests when we use dynamic difference in particular equations as discretization of a given one we cover a standard construction based of difference approximations for the continuous one but we are not restricted only to this case for a given differential equation we take a sequence of time scales and we study the convergence of time scales to the domain of the considered problem we choose a kind of convergence of such approximated solutions to the exact solution this is a step for creating numerical analysis on time scales and we propose to replace in such a situation the difference equations by dynamic ones in the proposed approach we are not restricted to the case of classical numerical algorithms moreover this allows us to find an exact solution for considered problems as a limit of a sequence of solutions for appropriate time scales instead of solving it analytically or calculating approximated solutions for the original problems

a note on the validity of statistical bootstrapping for estimating the uncertainty of tensor parameters in diffusion tensor images diffusion tensors are estimated from magnetic resonance images mris that are diffusion weighted and those images inherently contain noise therefore noise in the diffusion weighted images produces uncertainty in estimation of the tensors and their derived parameters which include eigenvalues eigenvectors and the trajectories of fiber pathways that are reconstructed from those eigenvalues and eigenvectors although repetition and wild bootstrap methods have been widely used to quantify the uncertainty of diffusion tensors and their derived parameters we currently lack theoretical derivations that would validate the use of these two bootstrap methods for the estimation of statistical parameters of tensors in the presence of noise the aim of this paper is to examine theoretically and numerically the repetition and wild bootstrap methods for approximating uncertainty in estimation of diffusion tensor parameters under two different schemes for acquiring diffusion weighted images whether these bootstrap methods can be used to quantify uncertainty in some diffusion tensor parameters such as fractional anisotropy fa depends critically on the morphology of the diffusion tensor that is being estimated the wild and repetition bootstrap methods in particular cannot quantify uncertainty in the principal direction pd of isotropic or oblate tensor we also examine the use of bootstrap methods in estimating tensors in a voxel containing multiple tensors demonstrating their limitations when quantifying the uncertainty of tensor parameters in those locations simulation studies are also used to understand more thoroughly our theoretical results our findings raise serious concerns about the use of bootstrap methods to quantify the uncertainty of fiber pathways when those pathways pass through voxels that contain either isotropic tensors oblate tensors or multiple tensors

a comparison of the variance estimation methods for heteroscedastic nonlinear models heteroscedasticity is commonly encountered when fitting nonlinear regression models in practice we discuss eight different variance estimation methods for nonlinear regression models with heterogeneous response variances and present a simulation study to compare the performance of the eight methods in terms of estimating the standard errors of the fitted model parameters the simulation study suggests that when the true variance is a function of the mean model the power of the mean variance function estimation method and the transform both sides method are the best choices for estimating the standard errors of the estimated model parameters in general the wild bootstrap estimator and two modified versions of the standard sandwich variance estimator are reasonably accurate with relatively small bias especially when the heterogeneity is nonsystematic across values of the covariate furthermore we note that the two modified sandwich estimators are appealing choices in practice considering the computational advantage of these two estimation methods relative to the variance function estimation method and the transform both sides approach copyright 2016 john wiley sons ltd

non strange weird resampling for complex survival data this paper introduces a new data dependent multiplier bootstrap for nonparametric analysis of survival data possibly subject to competing risks the nw procedure includes the general wild bootstrap and the weird bootstrap as special cases the data may be subject to independent right censoring and left truncation the asymptotic correctness of the proposed resampling procedure is proven under standard assumptions simulation results on time simultaneous inference suggest that the weird bootstrap performs better than the standard normal multiplier approach

regularization in the selection of radial basis function centers subset selection and regularization are two well known techniques that can improve the generalization performance of nonparametric linear regression estimators such as radial basis function networks this paper examines regularized forward selection rfs a combination of forward subset selection and zero order regularization an efficient implementation of rfs into which either delete 1 or generalized cross validation can be incorporated and a reestimation formula for the regularization parameter are also discussed simulation studies are presented that demonstrate improved generalization performance due to regularization in the forward selection of radial basis function centers

a heteroskedasticity robust breusch pagan test for contemporaneous correlation in dynamic panel data models abstract this paper proposes a heteroskedasticity robust breusch pagan test of the null hypothesis of zero cross section or contemporaneous correlation in linear panel data models without necessarily assuming independence of the cross sections the procedure allows for either fixed strictly exogenous and or lagged dependent regressor variables as well as quite general forms of both non normality and heteroskedasticity in the error distribution the asymptotic validity of the test procedure is predicated on the number of time series observations t being large relative to the number of cross section units n in that either i n is fixed as t or ii n 2 t 0 as both t and n diverge jointly to infinity given this it is not expected that asymptotic theory would necessarily provide an adequate guide to finite sample performance when t n is small because of this we also propose and establish asymptotic validity of a number of wild bootstrap schemes designed to provide improved inference when t n is small across a variety of experimental designs a monte carlo study suggests that the predictions from asymptotic theory do in fact provide a good guide to the finite sample behaviour of the test when t is large relative to n however when t and n are of similar orders of magnitude discrepancies between the nominal and empirical significance levels occur as predicted by the first order asymptotic analysis on the other hand for all the experimental designs the proposed wild bootstrap approximations do improve agreement between nominal and empirical significance levels when t n is small with a recursive design wild bootstrap scheme performing best in general and providing quite close agreement between the nominal and empirical significance levels of the test even when t and n are of similar size moreover in comparison with the wild bootstrap version of the original breusch pagan test godfrey and yamagata 2011 our experiments indicate that the corresponding version of the heteroskedasticity robust breusch pagan test appears reliable as an illustration the proposed tests are applied to a dynamic growth model for a panel of 20 oecd countries

on forward h k splitting for a class of noninvertible differential equations with impulse effect abstract we consider a class of linear differential equations with impulse effect at a given set of times we show that under general assumptions on the existence of forward solutions a splitting can be given in terms of an h k forward dichotomy which generalizes the hyperbolicity notion for dynamical systems

specification analysis of linear quantile models this paper introduces a nonparametric test for the correct specification of a linear conditional quantile function over a continuum of quantile levels these tests may be applied to assess the validity of post estimation inferences regarding the effect of conditioning variables on the distribution of outcomes we show that the use of an orthogonal projection on the tangent space of nuisance parameters at each quantile index both improves power and facilitates the simulation of critical values via the application of a simple multiplier bootstrap procedure monte carlo evidence and an application to the empirical analysis of age earnings curves are included

comparing nonparametric versus parametric regression fits in general there will be visible differences between a parametric and a nonparametric curve estimate it is therefore quite natural to compare these in order to decide whether the parametric model could be justified an asymptotic quantification is the distribution of the integrated squared difference between these curves we show that the standard way of bootstrapping this statistic fails we use and analyse a different form of bootstrapping for this task we call this method the wild bootstrap and apply it to fitting engel curves in expenditure data analysis

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

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

prepivoting to reduce level error of confidence sets summary approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function s sampling distribution we call this function the root of the confidence set either asymptotic theory or bootstrap methods can be used to estimate the desired quantile when the root is not a pivot in the sense of classical statistics the actual level of the approximate confidence set may differ substantially from the intended level prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function prepivoting can be iterated bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root the first prepivoting is nearly equivalent to studentizing when that operation is appropriate further iterations of prepivoting make higher order corrections automatically

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

impulsive nonlocal differential equations through differential equations on time scales abstract we propose a non standard approach to impulsive differential equations in banach spaces by embedding this type of problems into differential dynamic problems on time scales we give an existence result for dynamic equations and as a consequence we obtain an existence result for impulsive differential equations

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

on the limit of the largest eigenvalue of the large dimensional sample covariance matrix in this paper the authors show that the largest eigenvalue of the sample covariance matrix tends to a limit under certain conditions when both the number of variables and the sample size tend to infinity the above result is proved under the mild restriction that the fourth moment of the elements of the sample sums of squares and cross products sp matrix exist

testing conditional independence via rosenblatt transforms this paper proposes new tests of conditional independence of two random variables given a single index involving an unknown finite dimensional parameter the tests employ rosenblatt transforms and are shown to be distribution free while retaining computational convenience some results from monte carlo simulations are presented and discussed

bootstrap simultaneous error bars for nonparametric regression simultaneous error bars are constructed for nonparametric kernel estimates of regression functions the method is based on the bootstrap where resampling is done from a suitably estimated residual distribution the error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints applications to an economic problem are given and comparison to both pointwise and bonferroni type bars is presented through a simulation study

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

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

tests and diagnostic plots for detecting lack of fit for circular linear regression models regression diagnostics and lack of fit tests mainly focus on linear linear regression models when the design points are distributed on the circumference of a circle difficulties arise as there is no natural starting point or origin most classical lack of fit tests require an arbitrarily chosen origin but different choices may result in different conclusions we propose a graphical diagnostic tool and a closely related lack of fit test which does not require a natural starting point the method is based on regional residuals which are defined on arcs of the circle the graphical method formally locates and visualizes subsets of poorly fitting observations on the circle a data example from the food technology is used to point out the before mentioned problems with conventional lack of fit tests and to illustrate the strength of the methodology based on regional residuals in detecting and localizing departures from the no effect hypothesis a small simulation study shows a good performance of the regional residual test in case of both global and local deviations from the null model finally the ideas are extended to the case of more than one predictor variable

a neural state estimator with bounded errors for nonlinear systems a neural state estimator is described acting on discrete time nonlinear systems with noisy measurement channels a sliding window quadratic estimation cost function is considered and the measurement noise is assumed to be additive no probabilistic assumptions are made on the measurement noise nor on the initial state novel theoretical convergence results are developed for the error bounds of both the optimal and the neural approximate estimators to ensure the convergence properties of the neural estimator a minimax tuning technique is used the approximate estimator can be designed offline in such a way as to enable it to process on line any possible measure pattern almost instantly

the wild bootstrap tamed at last the wild bootstrap is studied in the context of regression models with heteroskedastic disturbances we show that in one very specific case perfect bootstrap inference is possible and a substantial reduction in the error in the rejection probability of a bootstrap test is available much more generally however the version of the wild bootstrap with this desirable property is without the skewness correction afforded by the currently most popular version of the wild bootstrap simulation experiments show that this does not prevent the preferred version from having the smallest error in rejection probability in small and medium sized samples

inference in linear regression models with many covariates and heteroskedasticity the linear regression model is widely used in empirical work in economics statistics and many other disciplines researchers often include many covariates in their linear model specification in a

bootstrap procedures under some non i i d models it is shown in this article that the classical i i d bootstrap remains a valid procedure for estimating the sampling distributions of certain symmetric estimators of location as long as the random observations are independently drawn from distributions with essentially a common location this may be viewed as a robust property of the classical i i d bootstrap also included is a study of the second order properties of a different bootstrap procedure proposed by wu in the context of heteroscedasticity in regression