higher order properties of the wild bootstrap under misspecification abstract we examine the higher order properties of the wild bootstrap wu 1986 in a linear regression model with stochastic regressors we find that the ability of the wild bootstrap to provide a higher order refinement is contingent upon whether the errors are mean independent of the regressors or merely uncorrelated with them in the latter case the wild bootstrap may fail to match some of the terms in an edgeworth expansion of the full sample test statistic nonetheless we show that the wild bootstrap still has a lower maximal asymptotic risk as an estimator of the true distribution than a normal approximation in shrinking neighborhoods of properly specified models to assess the practical implications of this result we conduct a monte carlo study contrasting the performance of the wild bootstrap with a normal approximation and the traditional nonparametric bootstrap

a bootstrap method for structure detection of narmax models many systems may be described by narmax models using only a few terms however depending on the order of the system the number of candidate terms can become very large selection of a subset of these candidate terms is necessary for an efficient system description this is an unresolved issue in system identification for over parameterized models therefore in this paper we develop a bootstrap structure detection bsd algorithm as a means of determining the structure of highly over parameterized models the performance of this bsd technique was evaluated by using it to estimate the structure of a 1 simple narmax model 2 moderately over parameterized narmax model and 3 highly over parameterized narmax model the results demonstrate that the bsd algorithm is a robust method for detecting the structure of narmax models this method provides accurate estimates of parameter statistics without relying on assumptions made by traditional procedures and yields a parsimonious description of the system

analytic smoothing for equipercentile equating under the common item nonequivalent populations design a cubic spline method for smoothing equipercentile equating relationships under the common item nonequivalent populations design is described statistical techniques based on bootstrap estimation are presented that are designed to aid in choosing an equating method degree of smoothing these include a asymptotic significance tests that compare no equating and linear equating to equipercentile equating b a scheme for estimating total equating error and for dividing total estimated error into systematic and random components the smoothing technique and statistical procedures are explored and illustrated using data from forms of a professional certification test

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

bootstrapping in multiplicative models abstract this paper develops the bootstrap theory to obtain a confidence interval for the constant term of the cobb douglas multiplicative model this also proposes a simple and easily computable estimator of the constant term along with its mean square error

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

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

estimating continuous time processes subject to time deformation an application to postwar u s gnp abstract a class of time series models is presented in which variables evolve on a data based rather than calendar time scale the discrete calendar time model thus obtained exhibits time varying parameters and conditional heteroscedasticity using a procedure based on the kalman filter univariate models are estimated for postwar u s real gross national product gnp and short and long term interest rates the results indicate significant time scale nonlinearities

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

the occupation statistics for variously shaped particles on a rectangular 2 n lattice space utilizing set theoretic arguments matrices of shift operators have been constructed the determinants of which yield recursions that describe exactly the occupational degeneracies for three differently shaped particles trimers l particles t particles distributed on a 2 n rectangular lattice on the basis of these recursions the expectation normalization dispersion and continuous representation of the statistics have been calculated and compared for each kind of particle

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

the occupation statistics for indistinguishable dumbbells on a rectangular lattice space i the method of hock and mcquistan used recently to solve the occupation statistics for indistinguishable dumbbells or dimers on a 2 2 n lattice is extended further to obtain for the l m n lattice general expressions for the normalization expectation and dispersion of the statistics and their limit as n becomes very large in particular an explicit expression of the partition function in the thermodynamic limit x is obtained for any value of the absolute activity x of dimers the developed mathematical formalism is then applied to planar lattices 1 m n with m 1 2 3 and 4 the known results for m 1 and 2 are recovered and some new ones are obtained the recurrence relation for the number a q n of arrangements of q dumbbells on a 1 m n lattice which has 3 and 5 terms when m 1 and 2 respectively is found to have 15 and 65 terms for m 3 and 4 analysis and extrapolation of the results enable one to predict the expectation 1mn on a planar 1 m n lattice to be 63 4 in the limit as both m a

continental scale relationship between bankfull width and drainage area for single thread alluvial channels we explore the bankfull width wbf versus drainage area ada relationship across a range of climatic and geologic environments and ask 1 is the relationship between ln wbf and ln ada best described by a linear function and 2 can a reliable relationship be developed for predicting wbf with ada as the only independent variable the principal data set for this study was compiled from regional curve studies and other reports that represent 1018 sites 1 m wbf 110 m and 0 50 km2 ada 22 000 km2 in the continental united states two additional data sets were used for validation after dividing the data into small medium and large size basins which respectfully correspond to ada 4 95 km2 4 95 km2 ada 337 km2 and ada 337 km2 regression lines from each data set were compared using one way analysis of covariance ancova a second ancova was performed to determine if mean annual precipitation p is an extraneous factor in the wbf versus ada relationship the ancova results reveal that using ada alone does not yield a reliable wbf versus ada relationship that is applicable across a wide range of environments and that p is a significant extraneous factor in the relationship considering data for very small basins ada 0 49 km2 and very large basins ada 1 0 105 km2 we conclude that a two segment linear model is the most probable form of the ln wbf versus ln ada relationship this study provides useful information for building complex multivariate models for predicting wbf

transfer matrices for first second and third neighbor interactions on two dimensional hexagonal cell lattices abstract we discuss the use of the shift operator matrix som method for the construction of transfer matrices appropriate to systems of simple indistinguishable first second and third neighbor interacting particles distributed on planar m n hexagonal cell lattices the matrix elements may be identified in such a way as to distinguish the contributions of particles and first second and third neighbor pairs according to their position relative to the lattice edges and of pairs according to their orientation by a simple transformation the transfer matrix for an m n planar lattice can be mapped into the transfer matrix for a cylindrical lattice of the same dimensions the matrix elements of the resulting transfer matrix may be identified in such a way that the orientation circumferential axial of the various kinds of neighbor pairs may be distinguished

bootstrapping in controlled calibration experiments we consider the determination of an unknown quantity for example the concentration of a particular chemical in a given sample or samples using controlled calibration here several samples are prepared with concentrations chosen to cover a required range and these are used to establish the relationship between concentration and the measured response to an assay method this relationship is then used to estimate the concentration in the unknown samples from their measured responses confidence intervals for the estimated concentrations can usually be calculated by inverting a prediction interval but in some situations this method becomes intractable we explore the use of the bootstrap as an alternative in linear nonlinear and multivariate controlled calibration using both simulation and real datasets from the field of immunoassay we also discuss the alternatives afforded by replication of the design points the bootstrap is found to be comparable to the standard method in simple situations and is ea

a multidimensional extension of the combinatorics function technique i linear and homogeneous partial difference equations recently the solution of multidimensional linear and homogeneous recurrence relations or partial difference equations pde was obtained via a multidimensional extension of the combinatorics function technique developed by antippa and phares combinatorics functions of the first and second kind are representations of restricted paths connecting two points in an n dimensional space these functions are shown to give the solution of the most general linear and inhomogeneous pde the consistency of the pde with the initial value conditions is also discussed applications of the method are given elsewhere

bootstrap of residual processes in regression to smooth or not to smooth summaryin this paper we consider regression models with centred errors independent of the covariates given independent and identically distributed data and given an estimator of the regression function which can be parametric or nonparametric in nature we estimate the distribution of the error term by the empirical distribution of estimated residuals to approximate the distribution of this estimator koul lahiri 1994 and neumeyer 2009 proposed bootstrap procedures based on smoothing the residuals before drawing bootstrap samples so far it has been an open question as to whether a classical nonsmooth residual bootstrap is asymptotically valid in this context here we solve this open problem and show that the nonsmooth residual bootstrap is consistent we illustrate the theoretical result by means of simulations which demonstrate the accuracy of this bootstrap procedure for various models testing procedures and sample sizes

bootstrapping realized multivariate volatility measures we study bootstrap methods for statistics that are a function of multivariate high frequency returns such as realized regression coefficients and realized covariances and correlations for these measures of covariation the monte carlo simulation results of barndorff nielsen and shephard 2004 show that finite sample distortions associated with their feasible asymptotic theory approach may arise if sampling is not too frequent this motivates our use of the bootstrap as an alternative tool of inference for covariation measures we consider an i i d bootstrap applied to the vector of returns we show that the finite sample performance of the bootstrap is superior to the existing first order asymptotic theory nevertheless and contrary to the existing results in the bootstrap literature for regression models subject to heteroskedasticity in the error term the edgeworth expansion for the i i d bootstrap that we develop here shows that this method is not second order accurate we argue that this is due to the fact that the conditional mean parameters of realized regression models are heterogeneous under stochastic volatility

bootstrap confidence intervals for a class of parametric problems summary we consider the following class of problems having observed a multivariate normal data vector y with unknown mean vector q covariance matrix the identity find an approximate confidence interval for 0 t q a real valued function of q a simple geometric construction is given which leads to highly accurate solutions this construction shows that the standard approximation based on maximum likelihood theory 04 cz00 can be quite misleading when 0 is nonlinear in q we discuss bootstrap based confidence intervals which remove most of the error in the standard approximation at the expense of considerably more calculation the bootstrap intervals are invariant under transformation of both y and i and so they automatically produce accurate solutions in problems which can be transformed to multivariate normality without requiring knowledge of the normalizing transformation

an exact recursion for the composite nearest neighbor degeneracy for a 2 n lattice space a set theoretic argument is utilized to develop a recursion relation that yields exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space the associated generating functions as well as the expectation of the resulting statistics are also treated

shift operator matrices for the nearest neighbor problem in two dimensional hexagonal cell lattices abstract we show how to construct shift operator matrices som s the eigenvalues of which yield the partition function for a system of simple indistinguishable nearest neighbor interacting particles distributed on planar m n hexagonal cell lattice spaces by an appropriate transformation the som for an m planar lattice having two free edges is mapped into the som for a cylindrical lattice having no free edges

asymptotic behavior of the empiric distribution of m estimated residuals from a regression model with many parameters on etudie le comportement asymptotique des residus d un modele de regression lorsque les m estimateurs sont utilises et que le nombre de parametres peut croitre avec la taille de l echantillon

the occupational statistics for indistinguishable trimers on a 3 n lattice space a 20 term recursion relation is derived that describes exactly the occupational degeneracy for trimers distributed on a 3 n lattice space the associated generating functions expectation normalization dispersion and continuous representation are also developed and discussed

bootstrapping neural networks knowledge about the distribution of a statistical estimator is important for various purposes such as the construction of confidence intervals for model parameters or the determination of critical values of tests a widely used method to estimate this distribution is the so called boot strap which is based on an imitation of the probabilistic structure of the data generating process on the basis of the information provided by a given set of random observations in this article we investigate this classical method in the context of artificial neural networks used for estimating a mapping from input to output space we establish consistency results for bootstrap estimates of the distribution of parameter estimates

bootstrapping robust estimates of regression we introduce a new computer intensive method to estimate the distribution of robust regression estimates the basic idea behind our method is to bootstrap a reweighted representation of the estimates to obtain a bootstrap method that is asymptotically correct we include the auxiliary scale estimate in our reweighted representation of the estimates our method is computationally simple because for each bootstrap sample we only have to solve a linear system of equations the weights we use are decreasing functions of the absolute value of the residuals and hence outlying observations receive small weights this results in a bootstrap method that is resistant to the presence of outliers in the data the breakdown points of the quantile estimates derived with this method are higher than those obtained with the bootstrap we illustrate our method on two datasets and we report the results of a monte carlo experiment on confidence intervals for the parameters of the linear model

edgeworth expansions for bootstrapping regression models on etudie la performance asymptotique de la methode bootstrap dans des modeles de regression lineaire des expansions d edgeworth montrent que la methode bootstrap est asymptotiquement toujours au moins aussi bonne et dans certains cas meilleure que l approximation normale classique les performances de l approximation normale et du bootstrap dependent du taux de croissance dans les elements de la matrice plan

least squares theory based on general distributional assumptions with an application to the incomplete observations problem the linear regression modely x e is reanalyzed taking the modest position that x is an approximation of the best predictor ofy we derive the asymptotic distribution ofb andr 2 under mild assumptions the method of derivation yields an easy answer to the estimation of from a data set which contains incomplete observations where the incompleteness is random

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

spurious periodicity in inappropriately detrended time series econometric analysis of time series data is frequently preceded by regression on time to remove a trent component in the date the resulting residuals are then treated as a stationary series to which procedures requiring stationarity such as spectral analysis can be applied the objective is often to investigate the dynamics of transitory movements in the systems for example in econometric models of the business cycle when the data does consist of a deterministic function of time plus a stationary error then regression residuals will clearly be unbiased estimates of the stationary component however if the data is generated by possibly repeated summation of a satisfactory and inevitable process then the series cannot be expressed as a deterministic function of time plus a stationary deviation even though a least squares trend line and the associated residuals can always be calculated for any given finite sample in a recent paper chan hayya and ord 1977 hereafter cho were able to show that a residuals from linear regression of a realization of a random walk the summation of a purely random series on time have autocovariances which for given lag are a function of time and thereafter that the residuals are not stationary further cho established that the expected sample autocovariance function the expected autocovariances for given lag averaged over the time interval of the sample is a function of sample size as well as lag and therefore an artifact of the detrending procedure this function is characterized by cho in their figure 1 as being effectively linear in lag although the exact function is a fifth degree polynomial with the rate of decay from unity at the origin depending inversely on sample size

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

the statistics of interacting dumbbells on a 2 n lattice space a set theoretic argument is utilizied to determine a 31 term recursion that describes exactly the composite nearest neighbor degeneracy for indistinguishable dumbbell particles distributed on a rectangular 2 n lattice space the associated generating functions the expectation of the lattice coverage and the density of occupied nearest neighbor pairs are also determined

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

a suboptimal bootstrap method for structure detection of non linear output error models with application to human ankle dynamics identification of non linear systems involves estimating unknown parameters and structure detection selection of a subset of candidate terms that best describe the observed output this is a necessary procedure to compute an efficient system description which may afford greater insight into the functionality of the system or a simpler controller design for non linear systems simple output additive noise can generate multiplicative terms between the input output and noise the terms associated with noise need to be modelled to obtain unbiased parameter estimates significantly increasing the number of candidate terms to be estimated and considered in special cases it may be possible to use an output error oe model structure and the instrumental variable iv estimator to obtain unbiased parameters without modelling the noise this significantly reduces the dimensionality of the structure computation problem therefore in this paper we propose a suboptimal bootstrap structure detection sobsd algo

the occupation statistics for indistinguishable dumbbells on a 2 2 n lattice space a 20 term recursion relation is derived that describes exactly the occupation statistics for indistinguishable dumbbells which occupy two nearest neighbor sites distributed on a 2 2 n lattice space on the basis of this recursion the normalization expectation dispersion and continuous representation of the statistics are also developed when the lattice space is completely filled the recursion relationship reduces to four terms permitting a calculation of the orientational degeneracy of a completely filled 2 2 n lattice space

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

choosing the resampling scheme when bootstrapping a case study in reliability abstract often when dealing with complex data structures there is no unique way to bootstrap if the data can be viewed as u 1 u n iid from some distribution p then one can bootstrap by resampling the u s alternately one can resample in a more model based way that is by making use of the structure of the model p a typical example of this is linear regression in which the data is y i x i i 1 n one can resample the pairs y i x i or one can resample the residuals from a fitted model this phenomenon arises over quite a wide spectrum of problems and in many cases the different methods of bootstrapping can give substantially different results it seems hopeless to come up with a general theory that compares the different ways of bootstrapping in this article we study in some detail a certain model that arises in reliability theory in which there are two natural ways to bootstrap this model is described as follows available for testing is a sample of n iid systems each having t

the bootstrap and identification of prognostic factors via cox s proportional hazards regression model this paper describes the use of the bootstrap 1 2 a new computer based statistical methodology to help validate a regression model resulting from the fitting of cox s proportional hazards model3 to a set of censored survival data as an example we define a prognostic model for outcome in childhood acute lymphocytic leukemia with the cox model and use of a training set of 224 patients to validate the accuracy of the model we use a bootstrap resampling technique to mimic the population under study in two stages first we select the important prognostic factors via a stepwise regression procedure with 100 bootstrap samples secondly we estimate the corresponding regression parameters for these important factors with 400 bootstrap samples the bootstrap result suggests that the model constructed from the training set is reasonable

bootstrap methods using prior information summary bayesian analysis is subject to the same kinds of misspecification problems which motivate the robustness and nonparametric literature we present a method of incorporating prior information which performs well without direct knowledge of the error distribution this is accomplished by replacing the likelihood in bayes s formula by a bootstrap estimate of the sampling density of a robust estimator the flexibility of the method is illustrated by examples and its performance relative to standard bayesian techniques is evaluated in a monte carlo study

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

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

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 effect of estimating weights in weighted least squares abstract in weighted least squares it is typical that the weights are unknown and must be estimated most packages provide standard errors assuming that the weights are known this is fine for sufficiently large sample sizes but what about for small to moderate sample sizes the investigation of this article into the effect of estimating weights proceeds under the assumption typical in practice that one has a parametric model for the variance function in this context generalized least squares consists of a an initial estimate of the regression parameter b a method for estimating the variance function and c the number of iterations in reweighted least squares by means of expansions for the covariance it is shown that each of a c can matter in problems of small to moderate size a few conclusions may be of practical interest first estimated standard errors assuming that the weights are known can be too small in practice the investigation indicates that a simple bootstrap operation resul

a note on the use of local maxima to predict turning points in related series abstract this article deals with the issue of to what extent local maxima of one time series can be used to predict the turning points of other series of interest it is argued that the theory of high level crossings is the appropriate framework to handle such issues the article then defines precisely the notion of a turning point and introduces a test for investigating the predictive value of the u s leading indicator series