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

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 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

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

sharp ill posedness results for the kdv and mkdv equations on the torus abstract we establish a new a priori bound for l 2 bounded sequences of solutions to the mkdv equations on the torus this first enables us to construct weak solutions in l 2 for this equation and to check that the solutions constructed by kappeler and topalov in the defocusing case satisfy the equation in some weak sense in a second time we prove that the solution maps associated with the mkdv and the kdv equations are discontinuous for the h s t topology for respectively s 0 and s 1 these last results are sharp

sensitivity analysis of efficiency scores how to bootstrap in nonparametric frontier models efficiency scores of production units are generally measured relative to an estimated pro duction frontier nonparametric estimators dea fdh are based on a finite sample of observed production units the bootstrap is one easy way to analyze the sensitivity of efficiency scores relative to the sampling variations of the estimated frontier the main point in order to validate the bootstrap is to define a reasonable data generating process in this complex framework and to propose a reasonable estimator of it this paper provides a general methodology of bootstrapping in nonparametric frontier models some adapted methods are illustrated in analyzing the bootstrap sampling variations of input efficiency measures of electricity plants

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

extremal depth for functional data and applications abstractwe propose a new notion called extremal depth ed for functional data discuss its properties and compare its performance with existing concepts the proposed notion is based on a measure of extreme outlyingness ed has several desirable properties that are not shared by other notions and is especially well suited for obtaining central regions of functional data and function spaces in particular a the central region achieves the nominal desired simultaneous coverage probability b there is a correspondence between ed based simultaneous central regions and appropriate pointwise central regions and c the method is resistant to certain classes of functional outliers the article examines the performance of ed and compares it with other depth notions its usefulness is demonstrated through applications to constructing central regions functional boxplots outlier detection and simultaneous confidence bands in regression problems supplementary materials for this article are availab

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

bootstrap control in this paper we present a new way to control linear stochastic systems the method is based on statistical bootstrap techniques the optimal future control signal is derived in such a way that unknown noise distribution and uncertainties in parameter estimates are taken into account this is achieved by resampling from existing data when calculating statistical distributions of future process values the bootstrap algorithm takes care of arbitrary loss functions and unknown noise distribution even for small estimation sets the efficient way of utilizing data implies that the method is also well suited for slowly time varying stochastic systems

efficient bias corrected nonparametric spectral estimation summary we consider estimation of a spectral density at a particular frequency a linear model is developed based on the local second order behaviour of kernel spectral estimates and a second stage estimate is derived from this model the estimate is shown to be asymptotically more efficient than the kernel estimate with the optimal but unknown bandwidth whenever the smoothing parameter is chosen so that it oversmooths the kernel estimate a small sample simulation study shows good characteristics of the second stage estimate practical implementation is discussed through examples

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

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

global well posedness of the cauchy problem of the fifth order shallow water equation abstract by using the i method we prove that the cauchy problem of the fifth order shallow water equation t u x 2 t u x 3 u 3 u x u 2 x u x 2 u u x 3 u x 5 u 0 is globally well posed in the sobolev space h s r provided s 5 7 10 4

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

weak convergence of k nn density and regression estimators with varying k and applications on considere le probleme sous une condition de lissite du second ordre dans le cas unidimensionnel

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

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

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

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

a shrinkage estimator for spectral densities summary we propose a shrinkage estimator for spectral densities based on a multilevel normal hierarchical model the first level captures the sampling variability via a likelihood con structed using the asymptotic properties of the periodogram at the second level the spectral density is shrunk towards a parametric time series model to avoid selecting a particular parametric model for the second level a third level is added which induces an estimator that averages over a class of parsimonious time series models the estimator derived from this model the model averaged shrinkage estimator is consistent is shown to be highly competitive with other spectral density estimators via simulations and is computationally inexpensive

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

well posedness of the cauchy problem for a shallow water equation on the circle in this paper we consider the periodic cauchy problem for a fifth order modification of the camassa holm equation we prove local well posedness in appropriate bourgain spaces for initial data in a sobolev space hs t s 1 2 we also prove global well posedness for data in h1 t and of arbitrary size the proofs are based on a priori estimates using fourier analysis techniques microlocalization in phase space an interpolation argument and a fixed point theorem

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

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

change point in stochastic design regression and the bootstrap in this paper we study the consistency of dierent bootstrap procedures for constructing condence intervals cis for the unique jump discontinuity change point in an otherwise smooth regression function in a stochastic design setting this problem exhibits nonstandard asymptotics and we argue that the standard bootstrap procedures in regression fail to provide valid condence intervals for the change point we propose a version of smoothed bootstrap illustrate its remarkable nite sample performance in our simulation study and prove the consistency of the procedure the m out of n bootstrap procedure is also considered and shown to be consistent we also provide sucient conditions for any bootstrap procedure to be consistent in this scenario

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 cauchy problem for a higher order shallow water type equation on the circle abstract in this paper we investigate the cauchy problem for a higher order shallow water type equation u t u t x x x 2 j 1 u x 2 j 3 u 3 u u x 2 u x u x x u u x x x 0 where x t r 2 and j n firstly we prove that the cauchy problem for the shallow water type equation is locally well posed in h s t with s j 2 2 for arbitrary initial data by using the i method we prove that the cauchy problem for the shallow water type equation is globally well posed in h s t with 2 j 1 j 2 2 j 1 s 1 our results improve the results of a a himonas g misiolek himonas and misiolek 1998 11 2000 12

global well posedness for a fifth order shallow water equation in sobolev spaces abstract the cauchy problem of a fifth order shallow water equation t u x 2 t u x 3 u 3 u x u 2 x u x 2 u u x 3 u x 5 u 0 is shown to be globally well posed in sobolev spaces h s r for s 6 10 17 4 the proof relies on the i method developed by colliander keel staffilani takaoka and tao for this equation lacks scaling invariance we reconsider the local result and pay special attention to the relationship between the lifespan of the local solution and the initial data we prove the almost conservation law and combine it with the local result to obtain the global well posedness

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

well posedness for a higher order modified camassa holm equation abstract we show that the cauchy problem for a higher order modification of the camassa holm equation is locally well posed for initial data in the sobolev space h s r for s s where 1 4 s 1 2 and the value of s depends on the order of equation employing harmonic analysis methods we derive the corresponding bilinear estimate and then use a contraction mapping argument to prove existence and uniqueness of solutions

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

on periodic kp i type equations we analyze bilinear estimates related to periodic kp i type equations we prove that the cauchy problem for the fifth order kp i equation posed on is globally well posed in the energy space despite the loss of a smoothing relation which is essential to recuperate the derivative loss used in the context of the kdv and kp ii type equations we are able to gain a moothing thanks to some specific properties of the set of fourier modes where the smoothing relation fails counterexamples in the context of kp i type equations posed on are given these examples indicate that contrary to the kp ii case it would be difficult to obtain the local well posedness for the periodic kp i type equations by an iteration scheme using bourgain spaces

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

global wellposedness of kdv in h 1 t r by the inverse method we show that the korteweg de vries equation kdv tv x t x3v x t 6v x t xv x t x t t r is globally in time wellposed in the sobolev space of distributions h t r for any 1

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 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

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

kernel and nearest neighbor estimation of a conditional quantile let x 1 z 1 x 2 z 2 x n z n be iid as x z and for 0 p 1 let p x denote the conditional p quantile of z given x x i e p z p x x x p kernel and near est neighbor estimators of p x are proposed in order to study the asymptotics of these estimates bahadur type representations of the sample conditional quantiles are obtained these representations are used to examine the important issue of choosing the smoothing parameter by a local approach for a fixed x based on weak convergence of these estimators the results lead to asymptotic linear models which motivate certain estimators

a bilinear estimate with applications to the kdv equation u x 0 u0 x where u0 h r our principal aim here is to lower the best index s for which one has local well posedness in h r i e existence uniqueness persistence and continuous dependence on the data for a finite time interval whose size depends on u0 hs equation in 1 1 was derived by korteweg and de vries 21 as a model for long wave propagating in a channel a large amount of work has been devoted to the existence problem for the ivp 1 1 for instance see 9 10 the inverse scattering method applies to this problem and under appropriate decay assumptions on the data several existence results have been established see 5 6 14 28 33 another approach inherited from hyperbolic problems relies on the energy estimates and in particular shows that 1 1 is locally well posed in h r for s 3 2 see 2 3 12 29 30 31 using these results and conservation laws global in time well posedness in h r s 2 was established see 3 12 30 also global in time weak solutions in the energy space h r were constructed in 34 in 13 and 22 a local smoothing effect for solutions of 1 1 was discovered this combined with the conservation laws was used in 13 and 22 to construct global in time weak solutions with data in h r and even in l r in 16 we introduced oscillatory integral techniques to establish local well posedness of 1 1 in h r s 3 4 and hence global in time well posedness in h r s 1 in 16 we showed how to obtain the above mentioned result by picard iteration in an appropriate function space in 4 j bourgain introduced new function spaces adapted to the linear operator t 3 x for which there are good bilinear estimates for the nonlinear term x u 2 using these spaces bourgain was able to establish local well posedness of 1 1 in h r l r and hence by a conservation

the cauchy problem for higher order modified camassa holm equations on the circle abstract in this paper we investigate the cauchy problem for the shallow water type equation u t x 2 n 1 u 1 2 x u 2 x 1 x 2 1 u 2 1 2 u x 2 0 with low regularity data in the periodic settings firstly we proved that the bilinear estimate related to the nonlinear term of the equation in space w s defined in page 5 is invalid with s n 2 1 then the locally well posed of the cauchy problem for the periodic shallow water type equation is obtained in h s t with s n 3 2 n 2 for arbitrary initial data thus our result improves the result of himonas and misiolek commun partial differ equ 23 1998 123 139 where they have proved that the problem is locally well posed for small initial data in h s t with s n 2 1 n n with the aid of the standard fourier restriction norm method