double block bootstrap confidence intervals for dependent data the block bootstrap confidence interval for dependent data can outperform the conventional normal approximation only with nontrivial studentization which in the case of complicated statistics calls for specialist treatment and often results in unstable endpoints we propose two double block bootstrap approaches for improving the accuracy of the block bootstrap confidence interval under very general conditions the first approach calibrates the nominal coverage level and the second calculates studentizing factors directly from a block bootstrap series without the need for nontrivial analytical treatment we prove that the two approaches reduce the coverage error of the block bootstrap interval by an order of magnitude with simple tuning of block lengths at the two block bootstrapping levels empirical properties of the procedures are investigated by simulations and application to an econometric time series

block bootstraps for time series with fixed regressors this article examines block bootstrap methods in linear regression models with weakly dependent error variables and nonstochastic regressors contrary to intuition the tapered block bootstrap tbb with a smooth taper not only loses its superior bias properties but may also fail to be consistent in the regression problem a similar problem albeit at a smaller scale is shown to exist for the moving and the circular block bootstrap mbb and cbb respectively as a remedy an additional block randomization step is introduced that balances out the effects of nonuniform regression weights and restores the superiority of the modified tbb the randomization step also improves the mbb or cbb interestingly the stationary bootstrap sb automatically balances out regression weights through its probabilistic blocking mechanism without requiring any modification and enjoys a kind of robustness optimal block sizes are explicitly determined for block bootstrap variance estimators under regression finite sa

edgeworth correction by bootstrap in autoregressions we prove that the distribution of least squares estimates in autoregressions can be bootstrapped with accuracy o n 1 2 a s thereby improving the normal approximation error of o n

the error in rejection probability of simple autocorrelation robust tests a new class of autocorrelation robust test statistics is introduced the class of tests generalizes the kiefer vogelsang and bunzel 2000 test in a manner analogous to anderson and darling s 1952 generalization of the cramer von mises goodness of fit test in a gaussian location model the error in rejection probability of the new tests is found to be o t 1 log t where t denotes the sample size

internal resonances effects simulation versus analytical methods results abstract in this study of internal resonance effects in a two degree of freedom system with cubic non linearity attention is focussed on behaviour of the system as examined with the aid of an analog computer and on an adequacy of approximate analytical methods the averaging method and the ritz method it is shown that while the averaging method fails to predict the behaviour of the system the ritz method gives results surprisingly close to those obtained by the simulation method the results indicate that the internal resonance can be responsible for occurrence of two frequency oscillations with the amplitude of the subharmonic tone nearly 10 times higher than the amplitude of the fundamental harmonic component

texture synthesis and nonparametric resampling of random fields this paper introduces a nonparametric algorithm for bootstrapping a stationary random field and proves certain consistency properties of the algorithm for the case of mixing random fields the motivation for this paper comes from relating a heuristic texture synthesis algorithm popular in computer vision to general nonparametric bootstrapping of stationary random fields we give a formal resampling scheme for the heuristic texture algorithm and prove that it produces a consistent estimate of the joint distribution of pixels in a window of certain size under mixing and regularity conditions on the random field the joint distribution of pixels is the quantity of interest here because theories of human perception of texture suggest that two textures with the same joint distribution of pixel values in a suitably chosen window will appear similar to a human thus we provide theoretical justification for an algorithm that has already been very successful in practice and suggest an explanation for its perceptually good results

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

the moving blocks bootstrap and robust inference for linear least squares and quantile regressions abstract this paper presents the moving blocks bootstrap mbb as a new inference procedure for linear regression estimation which is robust to heteroskedasticity and autocorrelation of unknown forms the mbb covariance estimator is shown to provide heteroskedasticity and autocorrelation consistent hac standard errors for least squares ls and quantile regression qr coefficient estimators the mbb covariance estimator is shown to be asymptotically equivalent to the bartlett kernel estimator suggested by newey and west 1987 and the asymptotically optimal choice of the blocksize is discussed a monte carlo study is included showing that the mbb fares well in comparison to standard hac inference procedures considering strong mixing data generating processes the paper extends existing asymptotic results for the qr estimator the analogy to the ls case is stressed the paper analyzes both the cases of stochastic and of nonstochastic regressors and suggests two new grenander like conditions for the latter case the use of the mbb approach is illustrated for a practical example using a standard econometric package

the bethe ansatz for the six vertex model with rotated boundary conditions a method is suggested for derivation of the bethe ansatz equations for the six vertex model on a square lattice rotated at an arbitrary angle with respect to the coordinate axes the method is based on the random walk representation for configurations of the model the equations for the ice model on the rotated lattice are derived and some numerical results are obtained

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

fixed smoothing asymptotics for time series 1 in this paper we derive higher order edgeworth expansions for the finite sample distributions of the subsampling based t statistic and the wald statistic in the gaussian location model under the so called fixed smoothing paradigm in particular we show that the error of asymptotic approximation is at the order of the reciprocal of the sample size and obtain explicit forms for the leading error terms in the expansions the results are used to justify the second order correctness of a new bootstrap method the gaussian dependent bootstrap in the context of gaussian location model 1 introduction many economic and financial applications involve time series data with autocorrelation and heteroskedasticity properties often the unknown dependence structure is not the chief object of interest but the inference on the parameter of interest involves the estimation of unknown dependence in stationary time series models estimated by generalized method of moments gmm robust inference is typically accomplished by consistently estimating the asymptotic covariance matrix which is proportional to the long run variance lrv matrix of the estimating equations or moment conditions defining the estimator using a kernel smoothing method in the econometrics and statistics literature the bandwidth parameter truncation lag involved in the kernel smoothing method is assumed to grow slowly with sample size in order to achieve consistency the inference is conducted by plugging in a covariance matrix estimator that is consistent under heteroskedasticity and autocorrelation this approach dates back to newey and west 25 and andrews 1 recently kiefer and vogelsang 13 kv hereafter developed an alternative first order asymptotic theory for the hac heteroskedasticity and autocorrelation consistent based robust inference where the proportion of the bandwidth involved in the hac estimator to the sample size t denoted as b is held fixed in the asymptotics under the fixed b asymptotics the hac estimator converges to a nondegenerate yet nonstandard limiting distribution the tests based on the fi xed b asymptotic approximation were shown to enjoy better finite sample properties than the tests based on the small b asymptotic theory

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

modal interaction of resonantly forced oscillations of two degree of freedom structure an analysis is presented for simultaneous primary resonances of two degree of freedom system with quadratic and cubic nonlinearities to external excitations having the frequencies w 1 and w 2 the relations between the external frequencies of the system are such that they are excited into external resonances in two cases the first case without internal resonances and the second case in the presence of different possibilities of internal resonances one to one two to one three to one one to two and one to three the method of multiple scales mms is used to determine four first order ordinary differential equations which describe the modulation of the amplitudes and phases of both modes in order to use the mms properly we introduce new detuning parameters of both modes it is found analytically that two mode responses can occur at each primary resonance in all cases the steady state solutions and their stability are determined numerical results depicting the resonances are presented

application of the incremental harmonic balance method to cubic non linearity systems abstract in this paper the formulation of the incremental harmonic balance ihb method is derived for a general system of differential equations with cubic non linearity which governs a wide range of engineering problems such as large amplitude vibration of beams or plates an incremental arc length method combined with a cubic extrapolation technique is adopted to trace the response curve automatically the stability of its periodic solutions can also be evaluated from the ihb formulation by multi variable floquet theory hsu s method is adopted for computing the transition matrix at the end of one period two numerical examples are presented which demonstrate the effectiveness of the ihb method and hsu s method in treating the non linear vibration of a strongly non linear multiple degree of freedom system

graphical procedures for multiple comparisons under general dependence abstractit has been more than half a century since tukey first introduced graphical displays that relate nonoverlap of confidence intervals to statistically significant differences between parameter estimates in this article we show how tukey s graphical overlap procedure can be modified to accommodate general forms of dependence within and across samples we also develop a procedure that can be used to more effectively resolve rankings within the tails of the distributions of parameter values thereby generalizing existing methods for multiple comparisons with the best we show that these new procedures retain the simplicity of tukey s original procedure while maintaining asymptotic control of the familywise error rate under very general conditions simple examples are used throughout to illustrate the procedures supplementary materials for this article are available online

maximum likelihood and the bootstrap for nonlinear dynamic models we provide a unified framework for analyzing bootstrapped extremum estimators of nonlinear dynamic models for heterogeneous dependent stochastic processes we apply our results to the moving blocks bootstrap of kunsch ann stat 17 1989 1217 and liu and singh in r lepage l billiard eds exploring the limits of the bootstrap wiley new york 1992 and prove the first order asymptotic validity of the bootstrap approximation to the true distribution of quasi maximum likelihood estimators we also consider bootstrap testing in particular we prove the first order asymptotic validity of the bootstrap distribution of suitable bootstrap analogs of wald and lagrange multiplier statistics for testing hypotheses

the use of subseries values for estimating the variance of a general statistic from a stationary sequence let z i infty i infty be a strictly stationary infty mixing sequence without specifying the dependence model giving rise to z i and without specifying the marginal distribution of z i we address the question of variance estimation for a general statistic t n t n z 1 z n for estimating var t n from just the available data z 1 z n we propose computing subseries values t m z i 1 z i 2 z i m 0 leq i i m leq n these subseries values are used as replicates to model the sampling variability of t n in particular we use adjacent nonoverlapping subseries of length m m n with m rightarrow infty and m n n rigtharrow 0 our variance estimator is just the usual sample variance computed amongst these subseries values after appropriate standardization this estimator is shown to be consistent under mild integrability conditions we present optimal i e minimum m s e choices of m n for the special case where t n overset z n and z i is a normal ar 1 sequence a simulation study is conducted showing that those same choices of m n are effective when t n is a robust estimator of location and z i is subject to contamination

parameter estimation for a stationary process on a d dimensional lattice summary we study asymptotic properties of various estimation procedures for a general stationary regular process on a d dimensional lattice differences between d 1 time series and d 2 spatial processes are pointed out we suppose that the process is observed on a set pn with cn points which tends to infinity with the same speed in all directions the relative edge effect is of order cn lid increasing with d this effect is without statistical importance if d 1 but is important if d 2 we give a lcn consistent asymptotically normal estimator of the underlying parameter the procedure being constructed by a modification of whittle s approximation to the log likelihood in the gaussian case this procedure is asymptotically efficient

a note on bootstrapping the sample median efron 1979 1982 in his treatment of the bootstrap discusses its use for estimation of the asymptotic variance of the sample median in the sampling situation of independent and identically distributed random variables with common distribution function f having a positive derivative continuous in a neighborhood of the true median mu the natural conjecture that the bootstrap variance estimator converges almost surely to the asymptotic variance is shown by an example to be false unless a tail condition is imposed on f we prove that such strong convergence does hold under the fairly nonrestrictive condition that e lbrack mid x alpha rbrack 0

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

consistency of the jackknife after bootstrap variance estimator for the bootstrap quantiles of a studentized statistic efron j roy statist soc ser b 54 1992 83 1111 proposed a computationally efficient method called the jackknife after bootstrap for estimating the variance of a bootstrap estimator for independent data for dependent data a version of the jackknife after bootstrap method has been recently proposed by lahiri econometric theory 18 2002 79 98 in this paper it is shown that the jackknife after bootstrap estimators of the variance of a bootstrap quantile are consistent for both dependent and independent data results from a simulation study are also presented

infinitesimal robustness for autoregressive processes we define the influence function and construct optimal robust estimators for autoregressive processes we show that the asymptotic bias caused by small contaminations of the marginals can be written as the integral of a certain function with respect to the contamination this function is called the influence function it is unique only up to an equivalence relation but there is a natural unique version which describes the limiting influence of an additional observation given the previous observations moreover with this version the asymptotic variance at the true model can be expressed in a simple form optimal robust estimators minimize this asymptotic variance under a constraint on the influence function as in the i i d case they are found by truncating a multiple of the influence function of the maximum likelihood estimator

asymptotic expansions for sums of block variables under weak dependence let x i be a sequence of random vectors and y in f in x i l be zero mean block variables where x i l x i x i l 1 i 1 are overlapping blocks of length l and where f in are borel measurable functions this paper establishes valid joint asymptotic expansions of general orders for the joint distribution of the sums n i 1 x i and n i 1 y in under weak dependence conditions on the sequence x i i when the block length t grows to infinity in contrast to the classical edgeworth expansion results where the terms in the expansions are given by powers of n 1 2 the expansions derived here are mixtures of two series one in powers of n 1 2 and the other in powers of n l 1 2 applications of the main results to i expansions for studentized statistics of time series data and ii second order correctness of the blocks of blocks bootstrap method are given

cluster robust bootstrap inference in quantile regression models abstractin this article i develop a wild bootstrap procedure for cluster robust inference in linear quantile regression models i show that the bootstrap leads to asymptotically valid inference on the entire quantile regression process in a setting with a large number of small heterogeneous clusters and provides consistent estimates of the asymptotic covariance function of that process the proposed bootstrap procedure is easy to implement and performs well even when the number of clusters is much smaller than the sample size an application to project star data is provided supplementary materials for this article are available online

asymptotic expansions for sums of weakly dependent random vectors it is shown that formal edgeworth expansions are valid for sums of weakly dependent random vectors the error of approximation has ordero n s 2 2 if the strong mixing coefficients in iii are decreasing at an exponential rate the above conditions can easily be checked and are often satisfied when the sequence of random vectors is a gaussian or a markov or an autoregressive process explicit formulas are given for the distribution of finite fourier transforms of a strictly stationary time series

resampling based empirical prediction an application to small area estimation best linear unbiased prediction is well known for its wide range of applications including small area estimation while the theory is well established for mixed linear models and under normality of the error and mixing distributions the literature is sparse for nonlinear mixed models under nonnormality of the error distribution or of the mixing distributions we develop a resampling based unified approach for predicting mixed effects under a generalized mixed model set up second order accurate nonnegative estimators of mean squared prediction errors are also developed given the parametric model the proposed methodology automatically produces estimators of the small area parameters and their mean squared prediction errors without requiring explicit analytical expressions for the mean squared prediction errors

asymptotic distribution of statistics in time series veritable conditions are given for the validity of formal edgeworth expansions for the distribution of sums x 1 x n where x i f z i z i p 1 and z 1 z 2 is a strict sense stationary sequence that can be written as z j g e j k k 0 with an iid sequence e i of innovations these models include nonlinear functions of arma processes z i as well as certain nonlinear ar processes the results apply to many statistics in nonlinear time series models

extended lindstedt poincare method for non stationary resonances of dynamical systems with cubic nonlinearities this paper presents the extended lindstedt poincare el p method which applies multiple time variables to treat non stationary oscillations arising in dynamical systems with cubic nonlinearities due to the slowly varied excitation parameters the method is applied extensively in research of non stationary vibrations of clamped hinged beams recognizing the aperiodic nature of non stationary oscillations the new formulation is presented by adding an additional slow time scale beside time scales of the nonlinear system which generally correspond to the incommensurate nonlinear frequencies of the response using this concept a generalized approach of the study to the passage through fundamental superharmonic and subharmonic resonances is presented in the paper effects of slowly varying excitation frequency and slowly varying excitation amplitude on the non stationary oscillations are studied with the computation of deviations from the stationary response although the method is formulated for n dof dynamical systems having weak cubic nonlinearities it is applied for non stationary vibrations where two mode shape approximation of damped and undamped clamped hinged beam respectively is used and the simultaneous appearance of internal resonance is taken into account stability analysis of stationary solutions is performed and comparisons of stationary resonance curves by results obtained with the incremental harmonic balance ihb method show good agreement the passage through the fundamental resonance of damped and undamped clamped hinged beam respectively is investigated in detail

subsampling for heteroskedastic time series abstract in this article a general theory for the construction of confidence intervals or regions in the context of heteroskedastic dependent data is presented the basic idea is to approximate the sampling distribution of a statistic based on the values of the statistic computed over smaller subsets of the data this method was first proposed by politis and romano 1994b for stationary observations we extend their results to heteroskedastic observations and prove a general asymptotic validity result under minimal conditions in contrast the usual bootstrap and moving blocks bootstrap are typically valid only for asymptotically linear statistics and their justification requires a case by case analysis our general asymptotic results are applied to a regression setting with dependent heteroskedastic errors

automatic optimal batch size selection for recursive estimators of time average covariance matrix abstractthe time average covariance matrix tacm k z k where k is the auto covariance function is an important quantity for the inference of the mean of an rd valued stationary process d 1 this article proposes two recursive estimators for with optimal asymptotic mean square error amse under different strengths of serial dependence the optimal estimator involves a batch size selection which requires knowledge of a smoothness parameter k z k k for some this article also develops recursive estimators for combining these two estimators we obtain a fully automatic procedure for optimal online estimation for consistency and convergence rates of the proposed estimators are derived applications to confidence region construction and markov chain monte carlo convergence diagnosis are discussed supplementary materials for this article are available online

edgeworth expansions of suitably normalized sample mean statistics for atomic markov chains this paper is devoted to the problem of estimating functionals of type f fd from observations drawn from a positive recurrent atomic markov chain open image in new window with stationary distribution the properties of different estimators are studied beyond an accurate estimation of their bias the estimation of their asymptotic variance is considered we also show that the results of malinovskii 1987 on the validity of the formal edgeworth expansion for sample mean statistics of type t n n 1 n i 1 f x i extend to their studentized versions normalized by the asymptotic variance estimates we consider

second order correctness of the blockwise bootstrap for stationary observations we show that the blockwise bootstrap approximation for the distribution of a studentized statistic computed from dependent data is second order correct provided we choose an appropriate variance estimator we also show how to adapt the bc a confidence interval of efron to the dependent case for the proofs we extend the results of gotze and hipp on the validity of the formal edgeworth expansion for a sum to the studentized mean

the model confidence set this paper introduces the model confidence set mcs and applies it to the selection of models a mcs is a set of models that is constructed such that it will contain the best model with a given level of confidence the mcs is in this sense analogous to a confidence interval for a parameter the mcs acknowledges the limitations of the data such that uninformative data yield a mcs with many models whereas informative data yield a mcs with only a few models the mcs procedure does not assume that a particular model is the true model in fact the mcs procedure can be used to compare more general objects beyond the comparison of models we apply the mcs procedure to two empirical problems first we revisit the inflation forecasting problem posed by stock and watson 1999 and compute the mcs for their set of inflation forecasts second we compare a number of taylor rule regressions and determine the mcs of the best regression in terms of in sample likelihood criteria

on a geometric notion of quantiles for multivariate data abstract an extension of the concept of quantiles in multidimensions that uses the geometry of multivariate data clouds has been considered the approach is based on blending as well as generalization of the key ideas used in the construction of spatial median and regression quantiles both of which have been extensively studied in the literature these geometric quantiles are potentially useful in constructing trimmed multivariate means as well as many other l estimates of multivariate location and they lead to a directional notion of central and extreme points in a multidimensional setup such quantiles can be defined as meaningful and natural objects even in infinite dimensional hilbert and banach spaces and they yield an effective generalization of quantile regression in multiresponse linear model problems desirable equivariance properties are shown to hold for these multivariate quantiles and issues related to their computation for data in finite dimensional spaces are discussed n 1 2 consistenc

estimated wold representation and spectral density driven bootstrap for time series the second order dependence structure of purely non deterministic stationary processes is described by the coefficients of the famous wold representation these coefficients can be obtained by factorizing the spectral density of the process this relationship together with some spectral density estimator is used to obtain consistent estimators of these coefficients a spectral density driven bootstrap for time series is then developed which uses the entire sequence of estimated moving average coefficients together with appropriately generated pseudoinnovations to obtain a bootstrap pseudo time series it is shown that if the underlying process is linear and if the pseudoinnovations are generated by means of an independent and identically distributed wild bootstrap which mimics to the extent necessary the moment structure of the true innovations this bootstrap proposal asymptotically works for a wide range of statistics the relationships of the proposed bootstrap procedure to some other bootstrap procedures including the auto regressive sieve bootstrap are discussed it is shown that the latter is a special case of the spectral density driven bootstrap if a parametric auto regressive spectral density estimator is used simulations investigate the performance of the new bootstrap procedure in finite sample situations furthermore a real life data example is presented

a general theory for jackknife variance estimation the delete 1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles this well known deficiency can be rectified by using a more general jackknife with d the number of observations deleted depending on a smoothness measure of the point estimator our general theory explains why jackknife works or fails it also shows that i for sufficiently smooth estimators the jackknife variance estimators with bounded d are consistent and asymptotically unbiased and ii for nonsmooth estimators d has to go to infinity at a rate explicitly determined by a smoothness measure to ensure consistency and asymptotic unbiasedness improved results are obtained for several classes of estimators in particular for the sample p quantiles the jackknife variance estimators with d satisfying n 1 2 d rightarrow 0 and n d rightarrow infty are consistent and asymptotically unbiased

fixed b subsampling and the block bootstrap improved confidence sets based on p value calibration summary subsampling and block based bootstrap methods have been used in a wide range of inference problems for time series to accommodate the dependence these resampling methods involve a bandwidth parameter such as the subsampling window width and block size in the block based bootstrap in empirical work using different bandwidth parameters could lead to different inference results but traditional first order asymptotic theory does not capture the choice of the bandwidth we propose to adopt the fixed b approach as advocated by kiefer and vogelsang in the heteroscedasticity auto correlation robust testing context to account for the influence of the bandwidth on inference under the fixed b asymptotic framework we derive the asymptotic null distribution of the p values for subsampling and the moving block bootstrap and further propose a calibration of the traditional small b based confidence intervals regions or bands and tests our treatment is fairly general as it includes both finite dimensional parameters and infinite dimensional parameters such as the marginal distribution function simulation results show that the fixed b approach is more accurate than the traditional small b approach in terms of approximating the finite sample distribution and that the calibrated confidence sets tend to have smaller coverage errors than the uncalibrated counterparts

a review of bootstrap confidence intervals on passe en revue plusieurs methodes distinctes basees sur le bootstrap pour construire des intervalles de confiance

edgeworth expansions for studentized statistics under weak dependence in this paper we derive valid edgeworth expansions for studentized versions of a large class of statistics when the data are generated by a strongly mixing process under dependence the asymptotic variance of such a statistic is given by an infinite series oflag covariances and therefore studentizing factors i e estimators of the asymptotic standard error typically involve an increasing number say l of lag covariance estimators which are themselves quadratic functions of the observations the unboundedness of the dimension l of these quadratic functions makes the derivation and the form of the expansions nonstandard it is shown that in contrast to the case of the studentized means under independence the derived edgeworth expansion is a superposition of three distinct series respectively given by one in powers of n 1 2 one in powers of n l 1 2 resulting from the standard error of the studentizing factor and one in powers of the bias of the studentizing factor where n denotes the sample size

six vertex model with rotated boundary conditions we investigate the six vertex model on a square lattice rotated through an arbitrary angle with respect to the coordinate axes a model recently introduced by litvin and priezzhev auxiliary vertices are used to define an inhomogeneous system which leads to a one parameter family of commuting transfer matrices a product of commuting transfer matrices can be interpreted as a transfer matrix acting on zigzag walls in the rotated system using an equation for commuting transfer matrices we calculate their eigenvalues finite size properties of the model are discussed from the viewpoint of the conformal field theory

sieve bootstrap with variable length markov chains for stationary categorical time series a bootstrap for stationary categorical time series based on the method of sieves is studied here the data generating process is approximated by the so called variable length markov chain vlmc a flexible class of markov models that allows for parsimonious structure then the resampling is given by simulating from the fitted model it is shown that for a whole class of stationary categorical time series that is more general than vlmc the vlmc sieve has faster rate of convergence for variance estimation than the more general block bootstrap results are illustrated from a theoretical and empirical perspective for the latter a real data application about in homogeneity classification of a dna strand is also presented finally the vlmc sieve scheme enjoys an implementational advantage of using the plug in rule for bootstrapping a statistical procedure which generally is not the case for the block method

a vertex model with an unbounded number of components a model which is an infinite component generalization of the six vertex model is proposed the method of solution is based on a random walk representation of states the bethe ansatz equations for periodic boundary conditions are obtained the partition function of the finite model with special arrow configurations at the boundary of a square is given in contrast to analogous solutions of korepin 1982 and baxter 1982 it has the form of an evaluable determinant

heteroskedasticity autocorrelation and spatial correlation robust inference in linear panel models with fixed effects abstract this paper develops an asymptotic theory for test statistics in linear panel models that are robust to heteroskedasticity autocorrelation and or spatial correlation two classes of standard errors are analyzed both are based on nonparametric heteroskedasticity autocorrelation hac covariance matrix estimators the first class is based on averages of hac estimators across individuals in the cross section i e averages of hacs this class includes the well known cluster standard errors analyzed by arellano 1987 as a special case the second class is based on the hac of cross section averages and was proposed by driscoll and kraay 1998 the hac of averages standard errors are robust to heteroskedasticity serial correlation and spatial correlation but weak dependence in the time dimension is required the averages of hacs standard errors are robust to heteroskedasticity and serial correlation including the nonstationary case but they are not valid in the presence of spatial correlation the main contribution of the paper is to develop a fixed b asymptotic theory for statistics based on both classes of standard errors in models with individual and possibly time fixed effects dummy variables the asymptotics is carried out for large time sample sizes for both fixed and large cross section sample sizes extensive simulations show that the fixed b approximation is usually much better than the traditional normal or chi square approximation especially for the driscoll kraay standard errors the use of fixed b critical values will lead to more reliable inference in practice especially for tests of joint hypotheses

effects of block lengths on the validity of block resampling methods resampling methods for dependent variables typically involve selecting blocks of observations in this paper we investigate the effects of different block lengths on the consistency of approximations generated by block bootstrap and subsampling methods a complete description is provided for the case of sample mean it is shown that both methods are consistent if the block length grows at a rate slower than the sample size when the growth rate of block lengths is comparable to the sample size the resulting approximations are no longer consistent in the latter case we present a detailed account of the limit behavior of these resampling distribution approximations considered as random elements in the space of all probability measures on the real line

exact convergence rate of bootstrap quantile variance estimator it is shown that the relative error of the bootstrap quantile variance estimator is of precise order n 1 4 when n denotes sample size likewise the error of the bootstrap sparsity function estimator is of precise order n 1 4 therefore as point estimators these estimators converge more slowly than the bloch gastwirth estimator and kernel estimators which typically have smaller error of order at most n 2 5

bootstrap confidence regions computed from autoregressions of arbitrary order summary given a linear time series e g an autoregression of infinite order we may construct a finite order approximation and use that as the basis for bootstrap confidence regions the sieve or autoregressive bootstrap as this method is often called is generally seen as a competitor with the better understood block bootstrap approach however in the present paper we argue that for linear time series the sieve bootstrap has significantly better performance than blocking methods and offers a wider range of opportunities in particular since it does not corrupt second order properties then it may be used in a double bootstrap form with the second bootstrap application being employed to calibrate a basic percentile method confidence interval this approach confers secondorder accuracy without the need to estimate variance that offers substantial benefits since variances of statistics based on time series can be difficult to estimate reliably and partly because of the relatively small amount of information contained in a dependent process are notorious for causing problems when used to studentize other advantages of the sieve bootstrap include considerably greater robustness against variations in the choice of the tuning parameter here equal to the autoregressive order and the fact that in contradistinction to the case of the block bootstrap the percentile t version of the sieve bootstrap may be based on the raw estimator of standard error in the process of establishing these properties we show that the sieve bootstrap is second order correct

bootstrap inference for linear dynamic panel data models with individual fixed effects abstract this paper s main contribution is to propose and theoretically justify the application of bootstrap methods for inference in autoregressive panel data models with fixed effects whereas the focus of the existing literature has been on bias correcting the standard fixed effects ols estimator due to the well known incidental parameter bias our focus here is on improving the quality of inference by relying on the bootstrap instead of the standard normal distribution when computing critical values for test statistics in particular we show by simulation that confidence intervals based on the normal distribution can be very distorted in finite samples instead a bootstrap that resamples the residuals and generates the bootstrap observations recursively using the estimated autoregressive panel data model greatly reduces these distortions we show that this recursive design residual based bootstrap fixed effects ols estimator contains a built in bias correction term that mimics the incidental parameter bias thus this method can be used to approximate the bias as well as the entire distribution of the biased fixed effects ols estimator this is in contrast with two other methods we consider a fixed design residual based bootstrap and a pairs bootstrap whose distributions are incorrectly centered at zero as it turns out both the recursive design and the pairs bootstrap are asymptotically valid when applied to the bias corrected estimator but the fixed design bootstrap is not in the simulations the recursive design bootstrap is the method that does best overall

bootstrapping gmm estimators for time series abstract this paper considers the bootstrap for the gmm estimator of overidentified linear models when autocorrelation structures of moment functions are unknown when moment functions are uncorrelated after finite lags hall and horowitz 1996 bootstrap critical values for tests based on generalized method of moments estimators econometrica 64 891 916 showed that errors in the rejection probabilities of the bootstrap tests are o t 1 however this rate cannot be obtained with the hac covariance matrix estimator since it converges at a nonparametric rate by incorporating the hac covariance matrix estimator in the edgeworth expansion of the distribution we show that the bootstrap provides asymptotic refinements when the characteristic exponent of the kernel function is greater than two

gmm gel serial correlation and asymptotic bias for stationary time series models with serial correlation we consider generalized method of moments gmm estimators that use heteroskedasticity and autocorrelation consistent hac positive definite weight matrices and generalized empirical likelihood gel estimators based on smoothed moment conditions following the analysis of newey and smith 2004 for independent observations we derive second order asymptotic biases of these estimators the inspection of bias expressions reveals that the use of smoothed gel in contrast to gmm removes the bias component associated with the correlation between the moment function and its derivative while the bias component associated with third moments depends on the employed kernel function we also analyze the case of no serial correlation and find that the seemingly unnecessary smoothing and hac estimation can reduce the bias for some of the estimators

bootstrap standard error estimates for linear regression standard errors of parameter estimates are widely used in empirical work the bootstrap often can provide a convenient means of estimating standard errors the conditions under which bootstrap standard error estimates are theoretically justified have not received much attention however this article establishes conditions for the consistency of the moving blocks bootstrap estimators of the variance of the least squares estimator in linear dynamic models with dependent data we discuss several applications of this result in particular the use of bootstrap standard error estimates for bootstrapping studentized statistics a simulation study shows that inference based on bootstrap standard error estimates may be considerably more accurate in small samples than inference based on closed form asymptotic estimates

the dependent wild bootstrap we propose a new resampling procedure the dependent wild bootstrap for stationary time series as a natural extension of the traditional wild bootstrap to time series setting the dependent wild bootstrap offers a viable alternative to the existing block based bootstrap methods whose properties have been extensively studied over the last two decades unlike all of the block based bootstrap methods the dependent wild bootstrap can be easily extended to irregularly spaced time series with no implementational difficulty furthermore it preserves the favorable bias and mean squared error property of the tapered block bootstrap which is the state of the art block based method in terms of asymptotic accuracy of variance estimation and distribution approximation the consistency of the dependent wild bootstrap in distribution approximation is established under the framework of the smooth function model in addition we obtain the bias and variance expansions of the dependent wild bootstrap variance estimat

prediction of spatial cumulative distribution functions using subsampling abstract the spatial cumulative distribution function scdf is a random function that provides a statistical summary of a random field over a spatial domain of interest in this article we develop a spatial subsampling method for predicting an scdf based on observations made on a hexagonal grid similar to the one used in the environmental monitoring and assessment program of the u s environmental protection agency we show that under quite general conditions the proposed subsampling method provides accurate data based approximations to the sampling distributions of various functionals of the scdf predictor in particular it produces estimators of different population characteristics such as the quantiles and weighted mean integrated squared errors of the empirical predictor as an illustration we apply the subsampling method to construct large sample prediction bands for the scdf of an ecological index for foliage condition of red maple trees in the state of maine

the block block bootstrap improved asymptotic refinements the asymptotic refinements attributable to the block bootstrap for time series are not as large as those of the nonparametric iid bootstrap or the parametric bootstrap one reason is that the independence between the blocks in the block bootstrap sample does not mimic the dependence structure of the original sample this is the join point problem in this paper we propose a method of solving this problem the idea is not to alter the block bootstrap instead we alter the original sample statistics to which the block bootstrap is applied we introduce block statistics that possess join point features that are similar to those of the block bootstrap versions of these statistics we refer to the application of the block bootstrap to block statistics as the block block bootstrap the asymptotic refinements of the block block bootstrap are shown to be greater than those obtained with the block bootstrap and close to those obtained with the nonparametric iid bootstrap and parametric bootstrap

estimating the asymptotic covariance matrix for quantile regression models a monte carlo study abstract this monte carlo study examines several estimation procedures of the asymptotic covariance matrix in the quantile and censored quantile regression models design matrix bootstrap error bootstrapping order statistic sigma bootstrap homoskedastic kernel and heteroskedastic kernel the monte carlo samples are drawn from two alternative data sets 1 a the unaltered current population survey cps for 1987 and 2 b this cps data with independence between error term and regressors imposed this special setup allows one to evaluate the estimators under various realistic scenarios the results favor the design bootstrap for the general case but also support the order statistic when the error term is independent of the regressors

general m estimates for contaminated p th order autoregressive processes consistency and asymptotic normality this work is concerned with simultaneous estimation of coefficients and a scale parameter of a p th order autoregressive process x t the observations are y t v t z t 1 v t x t where z t is a contaminating process and v t represents the proportion of contamination if x t or z t have heavy tails both least squares estimates and ordinary m estimates are seriously affected under general conditions we prove consistency and asymptotic normality of a general class of m estimates which contains some m estimates studied by denby and martin 6

bootstrap methods for markov processes the block bootstrap is the best known bootstrap method for time series data when the analyst does not have a parametric model that reduces the data generation process to simple random sampling however the errors made by the block bootstrap converge to zero only slightly faster than those made by first order asymptotic approximations this paper describes a bootstrap procedure for data that are generated by a markov process or a process that can be approximated by a markov process with sufficient accuracy the procedure is based on estimating the markov transition density nonparametrically bootstrap samples are obtained by sampling the process implied by the estimated transition density conditions are given under which the errors made by the markov bootstrap converge to zero more rapidly than those made by the block bootstrap

inference for linear models with dependent errors summary the paper is concerned with inference for linear models with fixed regressors and weakly dependent stationary time series errors theoretically we obtain asymptotic normality for the m estimator of the regression parameter under mild conditions and establish a uniform bahadur representation for recursive m estimators methodologically we extend the recently proposed self normalized approach of shao from stationary time series to the regression set up where the sequence of response variables is typically non stationary in mean since the limiting distribution of the self normalized statistic depends on the design matrix and its corresponding critical values are case dependent we develop a simulation based approach to approximate the critical values consistently through a simulation study we demonstrate favourable finite sample performance of our method in comparison with a block bootstrap based approach empirical illustrations using two real data sets are also provided

multidimensional lindstedt poincare method for nonlinear vibration of axially moving beams abstract the multidimensional lindstedt poincare mdlp method is extended to the nonlinear vibration analysis of axially moving systems galerkin method is used to discretize the governing equations the forced response of an axially moving beam with internal resonance between the first two transverse modes is studied the fundamental harmonic resonance is studied the response curves exhibit the same internal resonance characteristics as that of non transferring thin plates and beams because all these systems possess cubic nonlinearity and similar frequency distribution the examples show that the results of the mdlp method agree reasonably well with that obtained by the incremental harmonic balance ihb method however the former is more straightforward and efficient for obtaining the solution

periodic and non periodic combination resonance in kinematically excited system of rods a theoretical investigation has been made of periodic and non periodic combination resonances of the system of rods under vertical and horizontal kinematic excitations the elements of the system are connected with articulated points the coupling of the elements of the system through internal longitudinal forces which are transverse forces at the ends of neighboring rods are taken into account the equations of motion are obtained from the lagrange equations the mathematical analysis of the equations of motion is accomplished by using tondl s technique the steady state solutions and their stability are determined resonance curves for the stationary states have been determined numerically

empirical likelihood methods with weakly dependent processes this paper studies the method of empirical likelihood in models with weakly dependent processes in such cases if the likelihood function is formulated as if the data process were independent obviously empirical likelihood fails we propose to use empirical likelihood of blocks of observations to solve this problem in a nonparametric manner this method of blockwise empirical likelihood preserves the dependence of data and the resulting likelihood ratios can be used to construct asymptotically valid confidence intervals we consider general estimating equations for which an efficient estimator is derived by maximizing blockwise empirical likelihood we also introduce blocks of blocks empirical likelihood to conduct inference for parameters of the infinite dimensional joint distribution of data the smooth function model is used for such cases we show that blockwise empirical likelihood of the smooth function model with weakly dependent processes is bartlett correctable a wide variety of problems such as time series regressions and spectral densities can be treated using our methodology

nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities abstract this paper presents nonlinear vibration analysis of a curved beam subject to uniform base harmonic excitation with both quadratic and cubic nonlinearities the galerkin method is employed to discretize the governing equations a high dimensional model that can take nonlinear model coupling into account is derived and the incremental harmonic balance ihb method is employed to obtain the steady state response of the curved beam the cases investigated include softening stiffness hardening stiffness and modal energy transfer the stability of the periodic solutions for given parameters is determined by the multi variable floquet theory using hsu s method particular attention is paid to the anti symmetric response with and without excitation as the excitation frequency is close to the first and third natural frequencies of the system the results obtained with the ihb method compare very well with those obtained via numerical integration

a simple formula for mixing estimators with different convergence rates suppose that two estimators s n and n n are available for estimating an unknown parameter and are known to have convergence rates n1 2 and rn o n1 2 respectively based on a sample of size n typically the more efficient estimator s n is less robust than n n and a definitive choice cannot be easily made between them under practical circumstances we propose a simple mixture estimator in the form of a linear combination of s n and n n which successfully reaps the benefits of both estimators we prove that the mixture estimator possesses a kind of oracle property so that it captures the fast n1 2 convergence rate of s n when conditions are favorable and is at least rn consistent otherwise applications of the mixture estimator are illustrated with examples drawn from different problem settings including orthogonal function regression local polynomial regression density derivative estimation and bootstrap inferences for possibly dependent data

on the internal resonance of multi degree of freedom systems with cubic non linearity abstract a new method of multiple scales procedure is used to analyze the forced responses of multiple degrees of freedom systems with cubic non linearity in which the internal resonance takes place simultaneously together with the forced resonances attention is focused on the superharmonic and subharmonic resonance caused by internal resonance as is near 1 or 2 where is the forcing frequency and 1 and 2 are the first and the second natural frequencies it is found that the responses are usually excitation level dependent and there is a critical excitation amplitude c in the case of near 1 when c the out of phase superharmonic resonance disappears in the case of near 2 when 0 c there are two branches of response when c one branch of responses vanishes the critical excitation amplitudes in different forced vibration are determined analytically in this paper

higher order improvements of a computationally attractive k step bootstrap for extremum estimators this paper establishes the higher order equivalence of the k step bootstrap introduced recently by davidson and mackinnon 1999 and the standard bootstrap the k step bootstrap is a very attractive alternative computationally to the standard bootstrap for statistics based on nonlinear extremum estimators such as generalized method of moment and maximum likelihood estimators the paper also extends results of hall and horowitz 1996 to provide new results regarding the higher order improvements of the standard bootstrap and the k step bootstrap for extremum estimators compared to procedures based on first order asymptotics the results of the paper apply to newton raphson nr default nr line search nr and gauss newton k step bootstrap procedures the results apply to the nonparametric iid bootstrap and nonoverlapping and overlapping block bootstraps the results cover symmetric and equal tailed two sided t tests and confidence intervals one sided t tests and confidence intervals wald tests and confidence regions and j tests of over identifying restrictions

the incremental harmonic balance method for nonlinear vibration of axially moving beams in this paper the incremental harmonic balance ihb method is formulated for the nonlinear vibration analysis of axially moving beams the galerkin method is used to discretize the governing equations a high dimensional model that can take nonlinear model coupling into account is derived the forced response of an axially moving strip with internal resonance between the first two transverse modes is studied particular attention is paid to the fundamental superharmonic and subharmonic resonance as the excitation frequency is close to the first second or one third of the first natural frequency of the system numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear vibration of axially moving media

on sample reuse methods for dependent data we suggest a sample reuse method for dependent data based on a cross between the block bootstrap and richardson extrapolation instead of simulating a same size resample by resampling blocks and placing them end to end it analyses the blocks directly and employs a variant of richardson extrapolation to adjust for block size a simple empirical rule also based on richardson extrapolation is suggested for empirically selecting the block size performance in the contexts of distribution and bias estimation is discussed via theoretical analysis and numerical simulation

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

the stationary bootstrap abstract this article introduces a resampling procedure called the stationary bootstrap as a means of calculating standard errors of estimators and constructing confidence regions for parameters based on weakly dependent stationary observations previously a technique based on resampling blocks of consecutive observations was introduced to construct confidence intervals for a parameter of the m dimensional joint distribution of m consecutive observations where m is fixed this procedure has been generalized by constructing a blocks of blocks resampling scheme that yields asymptotically valid procedures even for a multivariate parameter of the whole i e infinite dimensional joint distribution of the stationary sequence of observations these methods share the construction of resampling blocks of observations to form a pseudo time series so that the statistic of interest may be recalculated based on the resampled data set but in the context of applying this method to stationary data it is natural

tapered empirical likelihood for time series data in time and frequency domains we investigate data tapering in two formulations of empirical likelihood for time series one empirical likelihood is formed from tapered data blocks in the time domain and a second is based on the tapered periodogram in the frequency domain limiting distributions are provided for both empirical likelihood versions under tapering theoretical and simulation evidence indicates that a data taper improves the coverage accuracy of empirical likelihood confidence intervals for time series parameters such as means and correlations

simple robust tests for scale differences in paired data summary the classical statistical test for assessing the difference between the two scale parameters in paired data due to pitman 1939 and morgan 1939 is not robust this paper explores robust alternatives to pitman s test using the framework of the one sample t test the asymptotic behaviour of the tests under the null hypothesis is examined in general it is easy to construct tests that are asymptotically distribution free provided the data come from a symmetric bivariate distribution versions of these tests that do not require symmetry are derived monte carlo simulation experiments are done to assess small sample behaviour and power characteristics the tests are applied to a small data set comparing cancerous and noncancerous lungs

subseries methods in regression abstract we extend subseries methods for stationary sequences to the regression setting to estimate sampling distributions the subseries approach computes the statistic of interest on all possible subseries of a shorter length than the original series and uses the distribution of these replicates to mimic the distribution of the original statistic for proper choice of subseries length the regression parameters can be estimated to order o n 2 3 thus improving on the normal approximation the replicates can also be used to reduce the bias of robust estimators simulation results demonstrate the finite sample effectiveness of the approach for both distribution function estimation and bias reduction applications to spatial data are also discussed

modeling heaped duration data an application to neonatal mortality in 2005 the indian government launched a conditional cash incentive program to encourage institutional delivery this paper studies the effects of the program on neonatal mortality using district level household survey data we model mortality using survival analysis paying special attention to substantial heaping a form of measurement error present in the data the main objective of this paper is to provide a set of sufficient conditions for identification and consistent estimation of the discretized baseline hazard accounting for heaping and unobserved heterogeneity our identification strategy requires neither administrative data nor multiple measurements but a correctly reported duration point and the presence of some flat segment s in the baseline hazard we establish the asymptotic properties of the maximum likelihood estimator and derive a set of specification tests that allow among other things to test for the presence of heaping and to compare different heaping mechanisms our empirical findings do not suggest a significant reduction of mortality in treated districts however they do indicate that accounting for heaping matters for the estimation of the hazard parameters

tapered block bootstrap summary we introduce and study tapered block bootstrap methodology that yields an improvement over the well known block bootstrap for time series of kiinsch 1989 the asymptotic validity and the favourable bias properties of the tapered block bootstrap are shown the important practical issues of optimally choosing the window shape and the block size are addressed in detail while some finite sample simulations are presented validating the good performance of the tapered block bootstrap

spatial anisotropy and rotational invariance of critical hard squares we develop a new method to analyse solvable models defined on a square lattice rotated through an arbitrary angle phi with respect to the coordinate axes in the method we introduce auxiliary faces into a rotated system to relate it with an inhomogeneous one in the natural orientation phi 0 the inhomogeneous system is investigated by commuting a transfer matrix argument as an application of the method we consider interacting hard squares at the critical point we discuss relations between spatial anisotropy and rotational or conformal invariance of the model in calculations of finite size properties