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

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

the essential spectrum of advective equations the geometric optics stability method is extended to a general class of linear advective pde s with pseudodifferential bounded perturbation we give a new short proof of vishik s formula for the essential spectral radius we show that every point in the dynamical spectrum of the corresponding bicharacteristic amplitude system contributes a point into the essential spectrum of the pde generic spectral pictures are obtained in sobolev spaces of sufficiently large smoothness applications to instability are presented

quasi hyperbolic semigroups abstract we introduce the notion of quasi hyperbolic operators and c 0 semigroups examples include the push forward operator associated with a quasi anosov diffeomorphism or flow a quasi hyperbolic operator can be characterised by a simple spectral property or as the restriction of a hyperbolic operator to an invariant subspace there is a corresponding spectral property for the generator of a c 0 semigroup and it characterises quasi hyperbolicity on hilbert spaces but not on other banach spaces we exhibit some weaker properties which are implied by the spectral property

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

subsampling high frequency data the main contribution of this paper is to propose a novel way of conducting inference for an important general class of estimators that includes many estimators of integrated volatility a subsampling scheme is introduced that consistently estimates the asymptotic variance for an estimator thereby facilitating inference and the construction of valid confidence intervals the new method does not rely on the exact form of the asymptotic variance which is useful when the latter is of complicated form the method is applied to the volatility estimator of ait sahalia et al 2011 in the presence of autocorrelated and heteroscedastic market microstructure noise

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

testing that a dependent process is uncorrelated an analysis is presented of a new testing procedure for the null hypothesis that a stochastic process is uncorrelated when the process is possibly dependent unlike with existing procedures the user does not need to choose any arbitrary number to implement the proposed test the asymptotic null distribution of the proposed test statistic is not standard but it is tabulated by means of simulations the test is compared with two alternative test procedures that require selection of user chosen numbers on the basis of asymptotic local power and finite sample behavior although the asymptotic local power of the proposed test is lower than those corresponding to the alternative tests in a monte carlo study i show that in small samples the test typically better controls the type i error and that the loss of power is not substantial

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

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

linear stability in an ideal incompressible fluid we give an explicit construction of approximate eigenfunctions for a linearized euler operator in dimensions two and three with periodic boundary conditions and an estimate from below for its spectral bound in terms of an appropriate lyapunov exponent as a consequence we prove that in dimension 2 the spectral and growth bounds for the corresponding group are equal therefore the linear hydrodynamic stability of a steady state for the euler equations in dimension 2 is equivalent to the fact that the spectrum of the linearized operator is pure imaginary in dimension 3 we prove the estimate from below for the spectral bound that implies the same equality for every example where the relevant lyapunov exponents could be effectively computed for the kinematic dynamo operator describing the evolution of a magnetic field in an ideally conducting incompressible fluid we prove that the growth bound equals the spectral bound in dimensions 2 and 3

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

inference for non stationary time series regression with or without inequality constraints summary we consider statistical inference for time series linear regression where the response and predictor processes may experience general forms of abrupt and smooth non stationary behaviours over time meanwhile the regression parameters may be subject to linear inequality constraints a simple and unified procedure for structural stability checks and parameter inference is proposed in the case where the regression parameters are constrained the methodology proposed is shown to be consistent whether or not the true regression parameters are on the boundary of the restricted parameter space via utilizing an asymptotically invariant geometric property of polyhedral cones

calibrating confidence coefficients abstract two approaches for dealing with the problem of poor coverage probabilities of certain standard confidence intervals are proposed the first is a recommendation that the actual coverage be estimated directly from the data and its value reported in addition to the nominal level this is achieved through a combination of computer simulation and density estimation the asymptotic validity of the procedure is proved for a number of common situations a classical example is the nonparametric estimation of the variance of a population using the normal theory interval here it is shown that the estimated coverage probability consistently estimates the true coverage probability if the population distribution possesses a finite sixth moment the second approach is more traditional it is a procedure for modifying an interval to yield improved coverage properties given a confidence interval its estimated coverage probability obtained in the first approach is used to alter the nominal level of the interval

bootstrapping the box pierce q test a robust test of uncorrelatedness abstract this paper describes a test of the null hypothesis that the first k autocorrelations of a covariance stationary time series are zero in the presence of statistical dependence the test is based on the box pierce q statistic with bootstrap based p values the bootstrap is implemented using a double blocks of blocks procedure with prewhitening the finite sample performance of the bootstrap q test is investigated by simulation in our experiments the performance is satisfactory for samples of n 500 at this sample size the differences between the empirical and nominal rejection probabilities are essentially eliminated

on the associated graded rings of parameter ideals in buchsbaum rings is an invariant 1 a of a not depending on the particular choice of a parameter ideal q of a where i a q and e a denote the length of the amodule a q and the multiplicity of a relative to q respectively this is equivalent to saying that every system a a ad of parameters for a is a weak a sequence i e the equality a ai a i a ai m holds for all 0 i d 1 21 thus every cohen macaulay local ring a is clearly buchsbaum with i a 0 and vice versa in this sense the notion of buchsbaum rings is an extension of that of cohen macaulay rings and is originated in an answer of vogel 29 to a problem of buchsbaum 11 notice that the concept of buchsbaum rings may be canonically generalized to that of buchsbaum modules cf e g 11 221 now let us fix a parameter ideal q of a and let g denote the associated graded ring g a ona q qn of q recall that the ring g is cohenmacaulay if a is so actually g is a polynomial ring over a q in this case then what happens when a is buchsbaum this is a motive problem in the present research and our conclusion is briefly stated

the theory of d sequences and powers of ideals in this paper we study the question of finding depth r i given an ideal i in a commutative ring this problem is difficult in general even for simple examples depth r z can be difficult to compute brodmann has shown that for any commutative noetherian ring depth r z become stable for large n finding this value however is quite hard this problem originally arose through consideration of the following example suppose k is a field and x xij is an n

nonlinear instability in two dimensional ideal fluids the case of a dominant eigenvalue it is proved that any steady 2 dimensional ideal fluid flow is nonlinearly unstable with respect to l2 growth in the velocity provided there exists an eigenvalue for the linearised euler equation with re here is the maximal lyapunov exponent of the steady flow

subsampling and model selection in time series analysis in this paper the subsampling method of carlstein 1986 is used to estimate the risk of prediction for time series data we extend carlstein s result by proving strong consistency of the subsampling estimator and we propose a procedure for selecting a time series model empirically from a set of possibly nonnested and misspecified models by using estimated risk of prediction as a selection criterion when this procedure is applied to the selection of the order of an autoregressive model it is shown to be a consistent order selector if an appropriate subsample size is chosen we propose a practical model selection procedure with a common subsample size chosen by hall jing s 1996 procedure

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

on the buchsbaum property of associated graded rings abstract let i be an m primary ideal in a buchsbaum local ring a m in this paper we investigate the buchsbaum property of the associated graded ring of i when the equality i 2 q i holds for some minimal reduction q of i however the buchsbaum property does not always follow even if i 2 q i so we give certain conditions for associated graded rings to be buchsbaum in this situation

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

on symmetric bootstrap confidence intervals on montre que dans des circonstances generales les intervalles de confiance du bootstrap de percentile t symetrique ont une erreur de recouvrement o n 2 ou n denote la taille d echantillon

the spectrum of the kinematic dynamo operator for an ideally conducting fluid the spectrum of the kinematic dynamo operator for an ideally conducting fluid and the spectrum of the corresponding group acting in the space of continuous divergence free vector fields on a compact riemannian manifold are described we prove that the spectrum of the kinematic dynamo operator is exactly one vertical strip whose boundaries can be determined in terms of the lyapunov oseledets exponents with respect to all ergodic measures for the eulerian flow also we prove that the spectrum of the corresponding group is obtained from the spectrum of its generator by exponentiation in particular the growth bound for the group coincides with the spectral bound for the generator

theoretical measures of relative performance of classifiers for high dimensional data with small sample sizes summary we suggest a technique related to the concept of detection boundary that was developed by ingster and by donoho and jin for comparing the theoretical performance of classifiers constructed from small training samples of very large vectors the resulting classification boundaries are obtained for a variety of distance based methods including the support vector machine distance weighted discrimination and kth nearest neighbour classifiers for thresholded forms of those methods and for techniques based on donoho and jin s higher criticism approach to signal detection assessed in these terms standard distance based methods are shown to be capable only of detecting differences between populations when those differences can be estimated consistently however the thresholded forms of distance based classifiers can do better and in particular can correctly classify data even when differences between distributions are only detectable not estimable other methods including higher criticism classifiers can on occasion perform better still but they tend to be more limited in scope requiring substantially more information about the marginal distributions moreover as tail weight becomes heavier the classification boundaries of methods designed for particular distribution types can converge to and achieve the boundary for thresholded nearest neighbour approaches for example although higher criticism has a lower classification boundary and in this sense performs better in the case of normal data the boundaries are identical for exponentially distributed data when both sample sizes equal 1

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

a general resampling scheme for triangular arrays of alpha mixing random variables with application to the problem of spectral density estimation in 1989 kunsch introduced a modified bootstrap and jackknife for a statistic which is used to estimate a parameter of the m dimensional joint distribution of stationary and alpha mixing observations the modification amounts to resampling whole blocks of consecutive observations or deleting whole blocks one at a time liu and singh independently proposed in 1988 the same technique for observations that are m dependent however many time series statistics notably estimators of the spectral density function involve parameters of the whole infinite dimensional joint distribution and hence do not fit in this framework in this report we generalize the moving blocks resampling scheme of kunsch and liu and singh a still modified version of the nonparametric bootstrap and jackknife is seen to be valid for general linear statistics that are asymptotically normal and consistent for a parameter of the whole joint distribution we then apply this result to the problem of estimation of the spectral density

on bootstrapping kernel spectral estimates an approach to bootstrapping kernel spectral density estimates is described which is based on resampling from the periodogram of the original data we show that it is asymptotically valid under suitable conditions and we illustrate its performance for a medium sized time series sample with a small simulation study

blockwise bootstrapped empirical process for stationary sequences we apply the bootstrap for general stationary observations proposed by kunsch to the empirical process for p dimensional random vectors it is known that the empirical process in the multivariate case converges weakly to a certain gaussian process we show that the bootstrapped empirical process converges weakly to the same gaussian process almost surely assuming that the block length l for constructing bootstrap replicates satisfies l n o n 1 2 varepsilon 0 varepsilon frac 1 2 and l n rightarrow infty an example where the multivariate setup arises are the robust gm estimates in an autoregressive model we prove the asymptotic validity of the bootstrap approximation by showing that the functional associated with the gm estimates is frechet differentiable

on stability for symmetric hyperbolic systems ii in a very wide spectrum of applications the problem of stability plays a central role there is a vast literature available in the field but still the problem is solved only in special cases the incompleteness of the literature is especially noticeable when the governing equations are partial differential equations the normal procedure in a stability research for such equations is to linearize the equations of perturbation and confine the study to the simpler linear equations without any mathematical justification even though the linearization has been mathematically justified only in a few very special cases it is a fact that usually the linear stability criteria are in excellent agreement with experimental observations in this paper we shall also confine our study to linear partial differential equations the conventional approach to the problem of stability for such equations is the so called normal mode method which under certain restrictions transfers the problem to an eigenvalue problem for a partial differential operator with boundary conditions in general this eigenvalue problem is very complicated it is therefore customary to restrict the study to the special cases where the problem essentially is transferred to an eigenvalue problem for an ordinary differential operator even in these special cases the eigenvalue problem is not completely resolved an alternative approach which has been given some attention in recent years is liapunov s direct method which in principle can be applied to all kinds of stability problems including nonlinear problems the results obtained by this method so far however are rather limited the difficulties in applying the method consist of constructing functionals with specific properties a new approach to the problem of stability for linear hyperbolic equations was given by eckhoff i on a somewhat speculative and conjectural basis the approach is based on the generalized progressing wave expansion method described by friedlander 2 and ludwig 3 and it has been applied to problems of stability in fluid mechanics by eckhoff and storesletten 4 51 the purpose of this paper is to give a rigorous

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

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

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

large sample confidence regions based on subsamples under minimal assumptions in this article the construction of confidence regions by approximating the sampling distribution of some statistic is studied the true sampling distribution is estimated by an appropriate normalization of the values of the statistic computed over subsamples of the data in the i i d context the method has been studied by wu in regular situations where the statistic is asymptotically normal the goal of the present work is to prove the method yields asymptotically valid confidence regions under minimal conditions essentially all that is required is that the statistic suitably normalized possesses a limit distribution under the true model unlike the bootstrap the convergence to the limit distribution need not be uniform in any sense the method is readily adapted to parameters of stationary time series or more generally homogeneous random fields for example an immediate application is the construction of a confidence interval for the spectral density function of a homogeneous random field

cocycles and mane sequences with an application to ideal fluids abstract exponential dichotomy of a strongly continuous cocycle is proved to be equivalent to existence of a mane sequence either for or for its adjoint as a consequence we extend some of the classical results to general banach bundles the dynamical spectrum of a product of two cocycles one of which is scalar is investigated and applied to describe the essential spectrum of the euler equation in an arbitrary spacial dimension