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

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

berry esseen theorems under weak dependence let xk k z be a stationary sequence given p 2 3 moments and a mild weak dependence condition we show a berry esseen theorem with optimal rate np 2 1 for p 4 we also show a convergence rate of n1 2 in lq norm where q 1 up to logn factors we also obtain nonuniform rates for any p 2 this leads to new optimal results for many linear and nonlinear processes from the time series literature but also includes examples from dynamical system theory the proofs are based on a hybrid method of characteristic functions coupling and conditioning arguments and ideal metrics

nmr flow velocity mapping in random percolation model objects evidence for a power law dependence of the volume averaged velocity on the probe volume radius two or three dimensional lacunar objects were fabricated using computer simulated random sitepercolation networks as templates the flow of water pumped through the pore space was studied with the aid of nuclear magnetic resonance nmr microscopy modified for mapping of the velocity vector field the percolation backbones of the objects were determined by exclusion of all pixels or voxels of the spin density images with velocities below the noise level an evaluation procedure was established which reliably renders the fractal dimensions of the whole cluster and of its backbone the volume averaged velocity magnitude as a function of the probe volume radius r was found to obey a power lawv v r 2l in the range a r jv where a is the voxel edge length and jv the velocity correlation length the exponents turned out to be l50 3260 04 and l50 8260 03 for the two and three dimensional objects respectively in order to test the time dependence of the mean squared displacement on fractals r 2 5at 2 dw expected for random walks of a fractal dimension dw self diffusion of gaseous methane was examined in the pore spaces of the same objects with the aid of field gradient nmr diffusometry the results are in accordance with the theoretical predictions for anomalous diffusion on percolation clusters this finding is supported by studies of incoherent water flow in the percolation network s1063 651x 96 06510 5

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

finite difference spectral approximations for the fractional cable equation fujian nsf s0750017 national nsf of china 10531080 ministry of education of china 973 high performance scientific computation research program 2005cb321703

on the bootstrap for moran s i test for spatial dependence this paper is concerned with the use of the bootstrap for statistics in spatial econometric models with a focus on the test statistic for moran s i test for spatial dependence we show that for many statistics in spatial econometric models the bootstrap can be studied based on linear quadratic lq forms of disturbances by proving the uniform convergence of the cumulative distribution function for lq forms to that of a normal distribution we show that the bootstrap is generally consistent for test statistics that can be approximated by lq forms including moran s i possible asymptotic refinements of the bootstrap are most commonly studied using edgeworth expansions for spatial econometric models we may establish asymptotic refinements of the bootstrap based on asymptotic expansions of lq forms when the disturbances are normal we prove the existence of the usual edgeworth expansions for lq forms when the disturbances are not normal we establish an asymptotic expansion of lq forms based on martingales these results are applied to show the second order correctness of the bootstrap for moran s i test

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

bootstrap confidence regions for the intensity of a poisson point process abstract bootstrap methods are developed for constructing confidence regions for the intensity function of a nonstationary poisson process several different resampling algorithms are suggested ranging from resampling a poisson process with intensity equal to that estimated nonparametrically from the data to resampling the data points themselves in the same manner that the bootstrap is used in problems involving independent and identically distributed observations for each different bootstrap method various percentile t ways of constructing confidence bands are described the effectiveness of these different approaches is demonstrated both theoretically and numerically for real and simulated data issues such as bias correction are addressed

bootstrap prediction intervals for autoregression abstract the nonparametric bootstrap is applied to the problem of prediction in autoregression let y t t 0 1 2 be a stationary autoregressive process of known order p ar p given a realization of the series up to time t y 1 y 2 y t a 100 prediction interval for y t k is desired standard forecasting techniques which assume that the error sequence of the process y t is gaussian rely on the fact that the conditional distribution of y t k given the data is gaussian as well as a nonparametric alternative the bootstrap provides an estimate of the conditional distribution of y t k the method is similar to other applications of the bootstrap for linear models because the residuals are resampled the proposed methodology represents a different approach since an alternative representation for ar p series is used allowing for bootstrap replicates generated backward in time it follows that the resulting replicates all have the same conditionally fixed values at the end

berry esseen theorems for quadratic forms of gaussian stationary processes letx1 x2 be a gaussian stationary process with spectral densityf and let z j frac 1 sqrt t left sum limits l 1 t j 1 left x l x l j 1 ex l x l j 1 right right j 1 p the present paper gives a berry esseen type theorem for the joint distribution of z1 zp

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

malliavin calculus geometric mixing and expansion of diffusion functionals under geometric mixing condition we presented asymptotic expansion of the distribution of an additive functional of a markov or an e markov process with finite autoregression including markov type semimartingales and time series models with discrete time parameter the emphasis is put on the use of the malliavin calculus in place of the conditional type cramer condition whose verification is in most case not easy for continuous time processes without such an infinite dimensional approach in the second part by means of the perturbation method and the operational calculus we proved the geometric mixing property for non symmetric diffusion processes and presented a sufficient condition which is easily checked in practice accordingly we obtained asymptotic expansion of diffusion functionals and proved the validity of it under mild conditions e g without the strong contractivity condition

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

on resampling and uncertainty estimation in linear system identification linear system identification yields a nominal model parameter which minimizes a specific criterion based on the single input output data set here we investigate the utility of various methods for estimating the probability distribution of this nominal parameter using only the data from this single experiment the results are compared to the actual parameter distribution generated by many monte carlo runs of the data collection experiment the methods considered are collectively known as resampling schemes which include subsampling the jackknife and the bootstrap the broad aim is to generate an empirical parameter distribution function via the construction of a large number of new data records from the original single set of data based on an assumption that this data is representative of all possible data and then to run the parameter estimator on each of these new records to develop the distribution function the performance of these schemes is evaluated on a difficult almost unidentifiable system and compared to the standard results based on asymptotic normality in addition to the exploration of this example as a means to evaluate the strengths and weaknesses of these resampling schemes some new theoretical results are proven and demonstrated for subsampling schemes

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

analysis of aberrations in public health surveillance data estimating variances on correlated samples the detection of unusual patterns in health data presents an important challenge to health workers interested in early identification of epidemics or important risk factors a useful procedure for detection of aberrations is the ratio of a current report to some historic baseline this work addresses the problem of finding the variance of such a ratio when the surveillance reports are correlated results show that when estimating this variance or the variance of the sample mean from a series of observations with an estimated correlation structure bootstrap and jackknife estimates may be overly optimistic the delta method or a classical method may be more useful when such model dependence is inappropriate

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

asymptotic expansions in the central limit theorem for compound and markov processes for independent identically distributed bivariate random vectors x 1 y 1 x 2 y 2 and for large t the distribution of x 1 x n t is approximated by asymptotic expansions here n t is the counting process with lifetimes y 1 y 2 similar expansions are derived for multivariate x 1 furthermore local asymptotic expansions are valid for the distribution of f x 1 f x n when n is large and nonrandom and x i i 1 2 is a discrete strongly mixing markov chain

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

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

approximations for densities of sufficient estimators abstract a simple method of obtaining asymptotic expansions for the densities of sufficient estimators is described it is an extension of the one developed by o barndorff nielsen and d r cox 1979 for exponential families a series expansion in powers of n 1 is derived of which the first term has an error of order n 1 which can effectively be reduced to n 3 2 by renormalization the results obtained are similar to those given by h e daniels s 1954 saddlepoint method but the derivations are simpler a brief treatment of approximations to conditional densities is given theorems are proved which extend the validity of the multivariate edgeworth expansion to parametric families of densities of statistics which need not be standardized sums of independent and identically distributed vectors these extensions permit the treatment of problems arising in time series analysis the technique is used by j durbin 1980 to obtain approximations to the densities of partial serial correlation coefficients

stein s method and exact berry esseen asymptotics for functionals of gaussian fields we show how to detect optimal berry esseen bounds in the normal approximation of functionals of gaussian fields our techniques are based on a combination of malliavin calculus stein s method and the method of moments and cumulants and provide de facto local one term edgeworth expansions the findings of the present paper represent a further refinement of the main results proven in nourdin and peccati probab theory related fields 145 2009 75 118 among several examples we discuss three crucial applications i to toeplitz quadratic functionals of continuous time stationary processes extending results by ginovyan probab theory related fields 100 1994 395 406 and ginovyan and sahakyan probab theory related fields 138 2007 551 579 ii to exploding quadratic functionals of a brownian sheet and iii to a continuous time version of the breuer major clt for functionals of a fractional brownian motion

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

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

bootstrapping explosive autoregressive processes on etablit la validite asymptotique du bootstrap pour l estimation d apres les moindres carres du parametre d un processus autoregressif de premier ordre explosif on obtient des lois limites non normales pour les estimations bootstrap et traditionnelles les resultats theoriques de validite du bootstrap sont soutenus par une simulation appropriee

saddlepoint expansions for sums of markov dependent variables on a continuous state space based on the conjugate kernel studied in iscoe et al 1985 we derive saddlepoint expansions for either the density or distribution function of a sumf x1 f x n where thex i s constitute a markov chain the chain is assumed to satisfy a strong recurrence condition which makes the results here very similar to the classical results for i i d variables in particular we establish also conditions under which the expansions hold uniformly over the range of the saddlepoint expansions are also derived for sums of the formf x1 x0 f x2 x1 f x n xn 1 although the uniformity result just mentioned does not generalize

bootstrap statistical tests of rank determination for system identification identification of multivariate state space models involves the specification of the dimension of the state vector this note evaluates the performance of some asymptotic statistical tests of rank determination together with their bootstrapped versions against information criteria ic methods results show that certain bootstrapped procedures improve upon the performance of the corresponding asymptotic tests and statistical tests have in general a better performance than ic methods

bootstrap critical values for tests based on generalized method of moments estimators monte carlo experiments have shown that tests based on generalized method ofmoments estimators often have true levels that differ greatly from their nominal levels when asymptotic critical values are used this paper gives conditions under which the bootstrap provides asymptotic refinements to the critical values of t tests and the test of overidentifying restrictions particular attention is given to the case of dependent data it is shown that with such data the bootstrap must sample blocks of data and that the formulae for the bootstrap versions of test statistics differ from the formulae that apply with the original data the results of monte carlo experiments on the numerical performance of the bootstrap show that it usually reduces the errors in level that occur when critical values based on first order asymptotic theory are used the bootstrap also provides an indication of the accuracy of critical values obtained from first order asymptotic theory

refinements in asymptotic expansions for sums of weakly dependent random vectors let s0 denote the nth normalized partial sum of a sequence of mean zero weakly dependent random vectors this paper gives asymptotic expansions for ef sn under weaker moment conditions than those of g6tze and hipp 1983 it is also shown that an expansion for ef sn with an error term o n s 2 2 is valid without any cramer type condition if f has partial derivatives of order s 1 only this settles a conjecture of g6tze and hipp in their 1983 paper

error bands for impulse responses we show how correctly to extend known methods for generating error bands in reduced form var s to overidentified models we argue that the conventional pointwise bands common in the literature should be supplemented with measures of shape uncertainty and we show how to generate such measures we focus on bands that characterize the shape of the likelihood such bands are not classical confidence regions we explain that classical confidence regions mix information about parameter location with information about model fit and hence can be misleading as summaries of the implications of the data for the location of parameters because classical confidence regions also present conceptual and computational problems in multivariate time series models we suggest that likelihood based bands rather than approximate confidence bands based on asymptotic theory be standard in reporting results for this type of model

partial mixing and edgeworth expansion introducing a conditional mixing property gotze and hipp s theory is generalized to a continuous time conditional markov process satisfying this property the malliavin calculus for jump processes applies to random coefficient stochastic differential equations with jumps with the aid of the support theorem to verify the non degeneracy condition i e a conditional type cramer condition

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

a space time spectral method for the time fractional diffusion equation in this paper we consider the numerical solution of the time fractional diffusion equation essentially the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term in the former case the first order time derivative is replaced by a fractional derivative making the problem global in time we propose a spectral method in both temporal and spatial discretizations for this equation the convergence of the method is proven by providing a priori error estimate numerical tests are carried out to confirm the theoretical results thanks to the spectral accuracy in both space and time of the proposed method the storage requirement due to the global time dependence can be considerably relaxed and therefore calculation of the long time solution becomes possible

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

limit theorems for toeplitz quadratic functionals of continuous time stationary processes let x t t in mathbb r be a centered real valued stationary gaussian process with spectral density f the paper considers a question concerning asymptotic distribution of toeplitz type quadratic functional qt of the process x t generated by an integrable even function g sufficient conditions in terms of f and g ensuring central limit theorems for standard normalized quadratic functionals qt are obtained extending the results of fox and taqqu prob theory relat fields 74 213 240 1987 avram prob theory relat fields 79 37 45 1988 giraitis and surgailis prob theory relat fields 86 87 104 1990 ginovian and sahakian theory prob appl 49 612 628 2004 for discrete time processes

calibrating prediction regions abstract suppose that the variable x to be predicted and the learning sample yn that was observed have a joint distribution which depends on an unknown parameter the parameter can be finite or infinite dimensional a prediction region dn for x is a random set depending on yn that contains x with prescribed probability this article studies methods for controlling simultaneously the conditional coverage proability of dn given yn and the overall unconditional coverage probability of dn the basic construction yields a prediction region dn which has the following properties in regular models both the conditional and overall coverage probabilities of dn converge to as the size n of the learning sample increases the convergence of the former is in probability moreover the asymptotic distribution of the conditional coverage probability about is typically normal and the overall coverage probability tends to at rate n 1 can one reduce the dispersion of the conditional coverage pr

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

iterates of expanding maps the iterates of expanding maps of the unit interval into itself have many of the properties of a more conventional stochastic process when the expanding map satisfies some regularity conditions and when the starting point is suitably chosen at random in this paper we show that the sequence of iterates can be closely tied to an m dependent process this enables us to prove good bounds on the accuracy of gaussian approximations our main tools are coupling and stein s method

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

spectral direction splitting methods for two dimensional space fractional diffusion equations a numerical method for a kind of time dependent two dimensional two sided space fractional diffusion equations is developed in this paper the proposed method combines a time scheme based on direction splitting approaches and a spectral method for the spatial discretization the direction splitting approach renders the underlying two dimensional equation into a set of one dimensional space fractional diffusion equations at each time step then these one dimensional equations are solved by using the spectral method based on weak formulations a time error estimate is derived for the semi discrete solution and the unconditional stability of the fully discretized scheme is proved some numerical examples are presented to validate the proposed method

interval forecasting an analysis based upon arch quantile estimators abstract in this paper we explore techniques for obtaining interval forecasts based on estimated time series models for processes which may exhibit autoregressive conditional heteroskedasticity arch to deal with the available variety of possible interval forecasts we propose a method for combining these forecasts based on quantile regression techniques our approach is practical rather than theoretical with attention focused directly on obtaining interval forecasts for two u s time series a measure of unemployment and a treasury bill rate we evaluate the performance of our procedures using a variety of diagnostics we find interval estimates which perform reasonably well judged by both in sample and out of sample criteria our experience suggests that a certain amount of care is required in order to obtain useful forecasts

edgeworth expansion for ergodic diffusions the edgeworth expansion for an additive functional of an ergodic diffusion is validated under fairly weak conditions the validation procedure does not depend on the stationarity or the geometric mixing property but exploits the strong markov property of the process in particular for an ito diffusion of dimension one verifiable conditions for the validity of the expansion are given in terms of the coefficients of the corresponding stochastic differential equation the maximum likelihood estimator for the cir process is treated as example

asymptotic expansions for potential functions of i i d random fields for sums of finite range potential functions of an iid random field we derive the validity of formal expansions of length two under standard conditions formal expansions are valid if and only if the characteristic functions of the sum converge to zero for all nonzero frequency parameters if this convergence fails the distribution of the sum can be approximated by a mixture of lattice distributions the result applies to m dependent random fields generated by independent random variables

inference in arch and garch models with heavy tailed errors arch and garch models directly address the dependency of conditional second moments and have proved particularly valuable in modelling processes where a relatively large degree of fluctuation is present these include financial time series which can be particularly heavy tailed however little is known about properties of arch or garch models in the heavy tailed setting and no methods are available for approximating the distributions of parameter estimators there in this paper we show that for heavy tailed errors the asymptotic distributions of quasi maximum likelihood parameter estimators in arch and garch models are nonnormal and are particularly difficult to estimate directly using standard parametric methods standard bootstrap methods also fail to produce consistent estimators to overcome these problems we develop percentile t subsample bootstrap approximations to estimator distributions studentizing is employed to approximate scale and the subsample bootstrap is used to estimate shape the good performance of this approach is demonstrated both theoretically and numerically

how to implement the bootstrap in static or stable dynamic regression models test statistic versus confidence region approach by combining two alternative formulations of a test statistic with two alternative resampling schemes we obtain four different bootstrap tests in the context of static linear regression models two of these are shown to have serious size and power problems whereas the remaining two are adequate and in fact equivalent the equivalence between the two valid implementations is shown to break down in dynamic regression models then the procedure based on the test statistic approach performs best at least in the ar 1 model similar finite sample phenomena are illustrated in the arma 1 1 model through a small scale monte carlo study and an empirical example

bootstrap model selection abstract in a regression problem typically there are p explanatory variables possibly related to a response variable and we wish to select a subset of the p explanatory variables to fit a model between these variables and the response a bootstrap variable model selection procedure is to select the subset of variables by minimizing bootstrap estimates of the prediction error where the bootstrap estimates are constructed based on a data set of size n although the bootstrap estimates have good properties this bootstrap selection procedure is inconsistent in the sense that the probability of selecting the optimal subset of variables does not converge to 1 as n this inconsistency can be rectified by modifying the sampling method used in drawing bootstrap observations for bootstrapping pairs response explanatory variable it is found that instead of drawing n bootstrap observations a customary bootstrap sampling plan much less bootstrap observations should be sampled the bootstrap selection p

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

on bilinear forms in gaussian random variables and toeplitz matrices we improve a result of szeg on the asymptotic behaviour of the trace of products of toeplitz matrices

anomalous diffusion as modeled by a nonstationary extension of brownian motion if the mean square displacement of a stochastic process is proportional to t ensuremath beta ensuremath beta ensuremath ne 1 then it is said to be anomalous we construct a family of markovian stochastic processes with independent nonstationary increments and arbitrary but a priori specified mean square displacement we label the family as an extended brownian motion and show that they satisfy a langevin equation with time dependent diffusion coefficient if the time derivative of the variance of the process is homogeneous then by computing the fractal dimension it can be shown that the complexity of the family is the same as that of the brownian motion for two particles initially separated by a distance x the finite size lyapunov exponent fsle measures the average rate of exponential separation to a distance ax an analytical expression is developed for the fsles of the extended brownian processes and numerical examples presented the explicit construction of these processes illustrates that contrary to what has been stated in the literature a power law mean square displacement is not necessarily related to a breakdown in the classical central limit theorem clt caused by for example correlation fractional brownian motion or correlated continuous time random walk schemes or infinite variance levy motion the classical clt coupled with nonstationary increments can and often does give rise to power law moments such as the mean square displacement

bootstrap refinements for qml estimators of the garch 1 1 parameters a b s t r a c t this paper reconsiders a block bootstrap procedure for quasi maximum likelihood estimation of garch models based on the resampling of the likelihood function as proposed by goncalves and white 2004 maximum likelihood and the bootstrap for nonlinear dynamic models journal of econometrics 119 199 219 first we provide necessary conditions and sufficient conditions in terms of moments of the innovation process for the existence of the edgeworth expansion of the garch 1 1 estimator up to the k th term second we provide sufficient conditions for higher order refinements for equally tailed and symmetric test statistics in particular the bootstrap estimator based on resampling the likelihood has the same higher order improvements in terms of error in the rejection probabilities as those in andrews 2002 higher order improvements of a computationally attractive k step bootstrap for extremum estimators econometrica 70 119 162

bootstrap technology and applications bootstrap resampling methods have emerged as powerful tools for constructing inferential procedures in modern statistical data analysis although these methods depend on the availability of fast inexpensive computing they offer the potential for highly accurate methods of inference moreover they can even eliminate the need to impose a convenient statistical model that does not have a strong scientific basis in this article we review some bootstrap methods emphasizing applications through illustrations with some real data special attention is given to regression problems with dependent data and choosing tuning parameters for optimal performance

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

a frequency domain bootstrap for ratio statistics in time series analysis the asymptotic properties of the bootstrap in the frequency domain based on studentized periodogram ordinates are studied it is proved that this bootstrap approximation is valid for ratio statistics such as autocorrelations by using edgeworth expansions it is shown that the bootstrap approximation even outperforms the normal approximation the results carry over to whittle estimates in a simulation study the behavior of the bootstrap is studied for empirical correlations and whittle estimates

asymptotic refinements of a misspecification robust bootstrap for generalized method of moments estimators i propose a nonparametric iid bootstrap that achieves asymptotic refinements for t tests and confidence intervals based on gmm estimators even when the model is misspecified in addition my bootstrap does not require recentering the moment function which has been considered as critical for gmm regardless of model misspecification the proposed bootstrap achieves the same sharp magnitude of refinements as the conventional bootstrap methods which establish asymptotic refinements by recentering in the absence of misspecification the key idea is to link the misspecified bootstrap moment condition to the large sample theory of gmm under misspecification of hall and inoue 2003 two examples are provided combining data sets and invalid instrumental variables

small time edgeworth type expansions for weakly convergent nonhomogeneous markov chains we consider triangular arrays of markov chains that converge weakly to a diffusion process second order edgeworth type expansions for transition densities are proved the paper differs from recent results in two respects we allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero small time asymptotics are motivated by statistical applications and by resulting approximations for the joint density of diffusion values at an increasing grid of points

sur le theoreme de berry esseen pour les suites faiblement dependantes nous etendons la methode de demonstration du theoreme de berry esseen proposee par bergstrom aux suites de variables aleatoires faiblement dependantes en particulier nous montrons que pour les suites stationnaires de variables aleatoires reelles bornees la vitesse de convergence dans le theoreme limite central en distance de levy est de l ordre den 1 2 des que la suite p p 0 des coefficients de melange uniforme satisfait la condition p 0 p p

on blocking rules for the bootstrap with dependent data summary we address the issue of optimal block choice in applications of the block bootstrap to dependent data it is shown that optimal block size depends significantly on context being equal to n 3 n 4 and nll5 in the cases of variance or bias estimation estimation of a onesided distribution function and estimation of a two sided distribution function respectively a clear intuitive explanation of this phenomenon is given together with outlines of theoretical arguments in specific cases it is shown that these orders of magnitude of block sizes can be used to produce a simple practical rule for selecting block size empirically that technique is explored numerically

testing for structural change in conditional models in the past decade we have seen the development of a new set of tests for structural change of unknown timing in regression models most notably the supf statistic of andrews 1993 econometrica 61 825 856 the expf and avef statistics of andrewsploberger 1994 econometrica 62 1383 1414 and the statistic of nyblom 1989 journal of american statistical association 84 223 230 the distribution theory used for these tests is primarily asymptotic and has been derived under the maintained assumption that the regressors are stationary this excludes structural change in the marginal distribution of the regressors as a result these tests technically cannot discriminate between structural change in the conditional and marginal distributions this paper attempts to remedy this de ciency by deriving the large sample distributions of the test statistics allowing for structural change in the marginal distribution of the regressors we nd that the asymptotic distributions of the supf expf avef and statistics are not invariant to structural change in the regressors to solve the size problem we introduce a xed regressor bootstrap which achieves the rst order asymptotic distribution and appears to possess reasonable size properties in small samples our bootstrap theory allows for arbitrary structural change in the regressors including structural shifts polynomial trends and exogenous stochastic trends it allows for lagged dependent variables and heteroskedastic error processes 2000 elsevier science s a all rights reserved

testing for granger causality in large mixed frequency vars abstract we analyze granger causality testing in a mixed frequency var where the difference in sampling frequencies of the variables is large implying parameter proliferation problems in case we attempt to estimate the model unrestrictedly we propose several tests based on reduced rank restrictions including bootstrap versions thereof to account for factor estimation uncertainty and improve the finite sample properties of the tests and a bayesian var extended to mixed frequencies we compare these methods to a test based on an aggregated model the m a x test ghysels et al 2016a and an unrestricted var based test ghysels et al 2016b using monte carlo simulations an empirical application illustrates the techniques

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

asymptotic operating characteristics of an optimal change point detection in hidden markov models let 0 1 1 be observations from the hidden markov model with probability distribution p 0 and let 1 be observations from the hidden markov model with probability distribution p 1 the parameters 0 and 1 are given while the change point is unknown the problem is to raise an alarm as soon as possible after the distribution changes from p 0 to p 1 but to avoid false alarms specifically we seek a stopping rule n which allows us to observe the s sequentially such that e n is large and subject to this constraint sup k e k n k n k is as small as possible here e k denotes expectation under the change point k and e denotes expectation under the hypothesis of no change whatever in this paper we investigate the performance of the shiryayev roberts pollak srp rule for change point detection in the dynamic system of hidden markov models by making use of markov chain representation for the likelihood function the structure of asymptotically minimax policy and of the bayes rule and sequential hypothesis testing theory for markov random walks we show that the srp procedure is asymptotically minimax in the sense of pollak ann statist 13 1985 206 227 next we present a second order asymptotic approximation for the expected stopping time of such a stopping scheme when w 1 motivated by the sequential analysis in hidden markov models a nonlinear renewal theory for markov random walks is also given

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

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

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

sieve bootstrap for smoothing in nonstationary time series we propose a sieve bootstrap procedure for time series with a deterministic trend the sieve for constructing the bootstrap is based on nonparametric trend estimation and autoregressive approximation for some noise process the bootstrap scheme itself does i i d resampling of estimated innovations from fitted autoregressive models we show the validity and indicate second order correctness of such sieve bootstrap approximations for the limiting distribution of nonparametric linear smoothers the resampling can then be used to construct nonparametric confidence intervals for the underlying trend in particular we show asymptotic validity for constructing confidence bands which are simultaneous within a neighborhood of size of the order of the smoothing bandwidth our resampling procedure yields satisfactory results in a simulation study for finite sample sizes we also apply it to the longest series of total ozone measurements from arosa switzerland and find a significant decreasing trend

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

on edgeworth expansions for dependency neighborhoods chain structures and stein s method let w be the sum of dependent random variables and h x be a function this paper provides an edgeworth expansion of an arbitrary length for e h w in terms of certain characteristics of dependency and of the smoothness of h and or the distribution of w the core of the class of dependency structures for which these characteristics are meaningful is the local dependency but in fact the class is essentially wider the remainder is estimated in terms of lyapunov s ratios the proof is based on a stein s method

efficient likelihood estimation in state space models motivated by studying asymptotic properties of the maximum likelihood estimator mle in stochastic volatility sv models in this paper we investigate likelihood estimation in state space models we first prove under some regularity conditions there is a consistent sequence of roots of the likelihood equation that is asymptotically normal with the inverse of the fisher information as its variance with an extra assumption that the likelihood equation has a unique root for each n then there is a consistent sequence of estimators of the unknown parameters if in addition the supremum of the log likelihood function is integrable the mle exists and is strongly consistent edge worth expansion of the approximate solution of likelihood equation is also established several examples including markov switching models arma models g arch models and stochastic volatility sv models are given for illustration

asymptotics of solute dispersion in periodic porous media the concentration c bf x t of a solute in a saturated porous medium is governed by a second order parabolic equation partial c partial t u 0 bf b cdot nabla c tfrac 1 2 sum d ij partial 2 c partial x i partial x j in the case that bf b is periodic and divergence free and d ij are constants and d ij positive definite the concentration is asymptotically gaussian for large times this article analyzes the dependence of the dispersion matrix bf k of the limiting gaussian distribution on the velocity parameter u 0 and the period a it is shown that each coefficient k ii is asymptotically quadratic in au 0 if b i bar bar b j has a nonzero component in the null space of bf b cdot nabla and asymptotically constant in au 0 if b i bar bar b j belongs to the range of bf b cdot nabla it is shown in a more general context that bf k depends only on au 0 an asymptotic expansion of the cramer edgeworth typ

a bootstrap theory for weakly integrated processes abstract this paper develops a bootstrap theory for models including autoregressive time series with roots approaching to unity as the sample size increases in particular we consider the processes with roots converging to unity with rates slower than n 1 we call such processes weakly integrated processes it is established that the bootstrap relying on the estimated autoregressive model is generally consistent for the weakly integrated processes both the sample and bootstrap statistics of the weakly integrated processes are shown to yield the same normal asymptotics moreover for the asymptotically pivotal statistics of the weakly integrated processes the bootstrap is expected to provide an asymptotic refinement and give better approximations for the finite sample distributions than the first order asymptotic theory for the weakly integrated processes the magnitudes of potential refinements by the bootstrap are shown to be proportional to the rate at which the root of the underlying process converges to unity the order of boostrap refinement can be as large as o n 1 2 for any 0 our theory helps to explain the actual improvements observed by many practitioners which are made by the use of the bootstrap in analyzing the models with roots close to unity