on bootstrap iteration for coverage correction in confidence intervals abstract iterated bootstrap procedures may be used to reduce error in many statistical problems we discuss their use in constructing confidence intervals with accurate coverage and show that bootstrap coverage correction produces improvements in coverage accuracy of order n 1 2 in one sided intervals but of order n 1 in two sided intervals explicit formulas are provided for the dominant term in coverage error after iteration in each case these results are used to compare various iterated bootstrap intervals and to assess the effect of bootstrap iteration on other indicators of interval performance such as position of critical points and length of interval we show that for one sided intervals the coverage correction algorithm yields critical points that are second order correct this is not the case for two sided intervals where second order correctness is not crucial in obtaining high order coverage accuracy we also show that the asymptotic mean increase in length between the original and cove

on the removal of skewness by transformation summary we suggest empirical transformations for removing most of the skewness of an asymmetric statistic these transformations may be used as the basis for accurate confidence procedures or hypothesis tests and can be employed in conjunction with either a normal approximation or the bootstrap our approach differs from those of other researchers in that the transformation is monotone and invertible in this paper we suggest elementary but general transformations for eliminating skewness from the distribution of a studentized statistic the transformations are monotone and have simple explicit inversion formulae and so are readily applied to confidence interval problems when the transformations are used in conjunction with the bootstrap they can produce confidence intervals with particularly low levels of coverage error that confidence intervals for symmetric statistics have lower coverage error than their counterparts in the case of asymmetry may be deduced from edgeworth expansions the first term in an expansion of size n 1 2 where n is the sample size describes the error in the usual normal approximation and is due entirely to skewness if it can be eliminated for example by transformation then the normal approxima tion will be in error by only o n 1 thus our method involves transforming one statistic into another where the distribution is virtually symmetric applying the normal approximation or the bootstrap to the new statistic and then regaining the asymmetry of the original problem by inverse transformation at the same time retain ing the high coverage accuracy conferred by applying confidence interval procedures to a symmetric statistic

refining bootstrap simultaneous confidence sets abstract simultaneous confidence sets for a collection of parametric functions may be constructed in several different ways these ways include a the exact pivotal method that underlies tukey s 1953 and scheffe s 1953 simultaneous confidence intervals for linear parametric functions in the normal linear model b the method of asymptotic pivots which is an approximate extension of the pivotal method c the method of bootstrapped roots developed in beran 1988 these three methods share several features each method simultaneously asserts a collection of confidence sets one confidence set for every parametric function of interest each method obtains the uth constituent confidence set by referring a root to a critical value the uth root is a function of the sample and of the uth parametric function each method has the same aim to control the overall level of the simultaneous confidence set and to keep equal the marginal levels of the individual confidence statements that make up the simulta

generalization of optimal hankel norm and balanced model reduction by bilinear mapping abstract in this paper the standard optimal hankel norm model reduction and balanced model reduction are generalized to optimal hankel norm model reduction and balanced model reduction over a disc in the complex plane important properties of the new model reduction concept such as the monotone property of the generalized hankel norm with respect to the domain and the l error bound over a disc d are given applications like the approximation of unstable systems and model reduction for specific working signals are discussed

a discrete model for bootstrap iteration the bootstrap can be validated by considering the sequence of p values obtained by bootstrap iteration rather than asymptotically if this sequence converges to a random variable with the uniform u 0 1 distribution the bootstrap is valid here the model is made discrete and finite characterised by a three dimensional array of probabilities this renders bootstrap iteration to any desired order feasible a unit root test for a process driven by a stationary ma 1 process is known to be unreliable when the ma 1 parameter is near 1 iteration of the bootstrap p value to convergence achieves reliable inference unless the parameter value is very close to 1

miscellanea inference on rare errors using asymptotic expansions and bootstrap calibration summary the number of items in error in an audit population is usually quite small whereas the error distribution is typically highly skewed to the right for applications in statistical auditing where line item sampling is appropriate a new upper confidence limit for the total error amount in an audit population is obtained our method involves an empirical cornish fisher expansion in the first stage in the second stage we employ the bootstrap to calibrate the coverage probability of the resulting interval estimate

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

on certain applications of the hadamard product abstract in this paper we give an extension of the ozkan and altintas results o ozkan o altintas applications of differential subordination appl math lett 19 2006 728 734 on the inclusion relationships involving various subclasses of analytic and univalent functions defined in terms of linear operators we show that these classes are closed under convolution with convex functions

a weighted bootstrap approach to bootstrap iteration the operation of resampling from a bootstrap resample encountered in applications of the double bootstrap may be viewed as resampling directly from the sample but using probability weights that are proportional to the numbers of times that sample values appear in the resample this suggests an approximate approach to double bootstrap monte carlo simulation where weighted bootstrap methods are used to circumvent much of the labour involved in compounded monte carlo approximation in the case of distribution estimation or equivalently confidence interval calibration the new method may be used to reduce the computational labour moreover the method produces the same order of magnitude of coverage error for confidence intervals or level error for hypothesis tests as a full application of the double bootstrap

some applications of differential subordination and the dziok srivastava convolution operator abstract by making use of the principle of differential subordination and the dziok srivastava convolution operator we introduce and investigate three interesting subclasses of analytic and univalent functions in the open unit disk we obtain their inclusion relationships and also show that these classes are closed under convolution with convex functions the results presented in this paper provide generalizations of several recent works by for example ozkan and altintas 2006 16 trojnar spelina 2008 30 and many other authors

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

nonparametric estimates of standard error the jackknife the bootstrap and other methods summary we discuss several nonparametric methods for attaching a standard error to a point estimate the jackknife the bootstrap half sampling subsampling balanced repeated replications the infinitesimal jackknife influence function techniques and the delta method the last three methods are shown to be identical all the methods derive from the same basic idea which is also the idea underlying the common parametric methods extended numerical comparisons are made for the special case of the correlation coefficient

asymptotic iterated bootstrap confidence intervals an iterated bootstrap confidence interval requires an additive correction to be made to the nominal coverage level of an uncorrected interval such correction is usually performed using a computationally intensive monte carlo simulation involving two nested levels of bootstrap sampling asymptotic expansions of the required correction and the iterated interval endpoints are used to provide two new computationally efficient methods for constructing an approximation to the iterated bootstrap confidence interval the first asymptotic interval replaces the need for a second level of bootstrap sampling with a series of preliminary analytic calculations which are readily automated and from which an approximation to the coverage correction is easily obtained the second interval directly approximates the endpoints of the iterated interval and yields for the first time the possibility of constructing an approximation to an iterated bootstrap confidence interval which does not require any resampling the theoretical properties of the two intervals are considered the computation required for their construction is detailed and has been coded in a fully automatic user friendly fortran program which may be obtained by anonymous ftp a simulation study which illustrates their effectiveness on three examples is presented

neighborhoods of certain p valently analytic functions with negative coefficients in this investigation by making use of the familiar concept of neighborhoods of p valently analytic functions the authors prove coefficient bounds distortion inequalities associated inclusion relations for the n neighborhoods of a family of p valently analytic functions and its certain derivatives which is defined by means of a certain non homogenous differential equation

neighborhood properties of certain classes of multivalently analytic functions associated with the convolution structure abstract in this paper by making use of the familiar concept of neighborhoods of p valently analytic functions we prove coefficient bounds distortion inequalities and associated inclusion relations for the n neighborhoods of a family of p valently analytic functions and their derivatives which is defined by means of a certain general family of non homogenous cauchy euler differential equations

power spectrum reduction by optimal hankel norm approximation of the phase of the outer spectral factor a new approach to stochastic processes power spectrum reduction based on akaike s canonical correlation between the process past and future is developed as a first result the correlation between the canonical past and the canonical future is given its precise functional analytic interpretation as the hankel operator generated lay the all pass function characterizing the phase of the process outer spectral factor the canonical correlation coefficients turn out to he nothing else but the singular values of this hankel operator next the all pass function characterizing the phase of the outer spectral factor is reduced using the optimal hankel norm procedure of adamjan arov and krein the reduction of the all pass phase function is implemented with the safeguards necessary for the magnitude information to be carried all the way through this involves bode s relations between the gain and the phase of an outer function nehari s extension and the winding number of the all pass phase function which is itself shown to be related to the definition of the past and the future finally the net result is a reduced order outer spectral factor that matches the high order spectral factor fairly well

mean squared error properties of generalized ridge estimators abstract mean squared errors are derived and examined for a number of generalized ridge gr regression estimators and the estimators are compared in a number of specific regression models the potential of gr estimators is discussed

inclusion and neighborhood properties for certain classes of multivalently analytic functions of complex order associated with the convolution structure making use of the familiar convolution structure of analytic functions in this paper we introduce and investigate two new subclasses of multivalently analytic functions of complex order among the various results obtained here for each of these function classes we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the class introduced here

on the bootstrap and confidence intervals we derive an explicit formula for the first term in an unconditional edgeworth type expansion of coverage probability for the nonparametric bootstrap technique applied to a very broad class of studentized statistics the class includes sample mean k sample mean sample correlation coefficient maximum likelihood estimators expressible as functions of vector means etc we suggest that the bootstrap is really an empiric one term edgeworth inversion with the bootstrap simulations implicitly estimating the first term in an edgeworth expansion this view of the bootstrap is reinforced by our discussion of the iterated bootstrap which inverts an edgeworth expansion to arbitrary order by simulating simulations

on bootstrap resampling and iteration summary we propose a single unifying approach to bootstrap resampling applicable to a very wide range of statistical problems it enables attention to be focused sharply on one or more characteristics which are of major importance in any particular problem such as coverage error or length for confidence intervals or bias for point estimation our approach leads easily and directly to a very general form of bootstrap iteration unifying and generalizing present disparate accounts of this subject it also provides simple solutions to relatively complex problems such as a suggestion by lehmann 1986 for conditionally short confidence intervals we set out a single unifying principle guiding the operation of bootstrap resampling applicable to a very wide range of statistical problems including bias reduction shrinkage hypothesis testing and confidence interval construction our principle differs from other approaches in that it focuses attention directly on a measure of quality or accuracy expressed in the form of an equation whose solution is sought a very general form of bootstrap iteration is an immediate consequence of iterating the empirical solution to this equation so as to improve accuracy when employed for bias reduction iteration of the resampling principle yields a competitor to the generalized jackknife enabling bias to be reduced to arbitrarily low levels when applied to confidence intervals it produces the techniques of hall 1986 and beran 1987 the resampling principle leads easily to solutions of new complex problems such as empirical versions of confidence intervals proposed by lehmann 1986 lehmann argued that an ideal confidence interval is one which is short when it covers the true parameter value but not necessarily otherwise the resampling principle suggests a simple empirical means of constructing such intervals section 2 describes the general principle and 3 shows how it leads naturally to bootstrap iteration there we show that in many problems of practical interest such as bias reduction and coverage error reduction in two sided confidence intervals each iteration reduces error by the factor n 1 where n is sample size in the case of confidence intervals our result sharpens one of beran 1987 who showed that coverage error is reduced by the factor n 2 in two sided intervals the main exception to our n 1 rule is coverage error of one sided intervals where error is reduced by the factor n a at each iteration our approach to bootstrap iteration serves to unify not just the philosophy of iteration for different statistical problems but also different techniques of iteration for the same

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

fast and accurate approximate double bootstrap confidence intervals we propose two new methods for the construction of approximate iterated bootstrap confidence intervals our methods follow a saddlepoint approach but employ approximate rather than exact solutions to the nonlinear saddlepoint equations that arise in applying that technique the main advantages of the new techniques are that they yield confidence intervals with high coverage accuracy in small to moderately sized samples and that they achieve this accuracy with very substantial reductions in computation time compared to other methods

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

prepivoting to reduce level error of confidence sets summary approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function s sampling distribution we call this function the root of the confidence set either asymptotic theory or bootstrap methods can be used to estimate the desired quantile when the root is not a pivot in the sense of classical statistics the actual level of the approximate confidence set may differ substantially from the intended level prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function prepivoting can be iterated bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root the first prepivoting is nearly equivalent to studentizing when that operation is appropriate further iterations of prepivoting make higher order corrections automatically

certain subclasses of meromorphically multivalent functions associated with generalized hypergeometric function in this paper we introduce and study two subclasses s p q s a 1 a b l and s p q s a 1 a b l of meromorphic p valent functions of order l email protected email protected

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

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

neighborhoods of certain p valent analytic functions with complex order in this paper the authors prove several coefficient bounds distortion inequalities of the class s g n p l b b of p valent analytic functions of complex order by making use of the familiar concept of neighborhoods of p valent analytic functions they obtained several inclusion relations associated with the n p n d neighborhood of this class and its certain derivatives which is defined by means of a certain non homogeneous cauchy euler differential equations

neighborhoods of a certain family of multivalent functions with negative coefficients abstract by making use of the familiar concept of neighborhoods of analytic and p valent functions the authors prove coefficient bounds and distortion inequalities and associated inclusion relations for the n neighborhoods of a family of multivalent functions with negative coefficients which is defined by means of a certain nonhomogeneous cauchy euler differential equation relevant connections of the various function classes investigated in this paper with those considered by earlier workers on the subject are also mentioned

neighborhoods for certain subclasses of multivalently analytic functions defined by using a differential operator in the present investigation by making use of the familiar concept of neighborhoods of analytic and multivalent functions we derive coefficient bounds and distortion inequalities associated inclusion relations for the n d neighborhoods of subclasses of analytic and multivalent functions with negative coefficients which are defined by means of a certain nonhomogenous differential equation several special cases of the main results are mentioned

double bootstrap methods that use a single double bootstrap simulation we show that when the double bootstrap is used to improve performance of bootstrap methods for bias correction techniques based on using a single double bootstrap sample for each single bootstrap sample can be particularly effective in particular they produce third order accuracy for much less computational expense than is required by conventional double bootstrap methods however this improved level of performance is not available for the single double bootstrap methods that have been suggested to construct confidence intervals or distribution estimators

fractional calculus and certain starlike functions with negative coefficients abstract a certain subclass t n p of starlike functions in the unit disk is introduced the object of the present paper is to derive several interesting properties of functions belonging to the class t n p various distortion inequalities for fractional calculus of functions in the class t n p are also given

neighborhoods of certain multivalent functions with negative coefficients by means of a differential operator we introduce and investigate two new subclasses of p valently analytic functions of complex order the various results obtained here for each of these function classes include coefficient inequalities and the consequent inclusion relationships involving the neighborhoods of the p valently analytic functions

prepivoting by weighted bootstrap iteration summary prepivoting by conventional bootstrap iteration is known to yield a progressively more accurate pivot in certain problems and has important application in the construction of confidence limits and estimation of null distributions we investigate the theoretical effects of weighted bootstrap iteration on prepivoting and show that each weighted bootstrap iteration with weights chosen carefully but empirically is asymptotically equivalent to two consecutive conventional bootstrap iterations in terms of reducing the order of error prepivoting can therefore be carried out much more efficiently if based on weighted boot

empirical bootstrap bias correction and estimation of prediction mean square error in small area estimation we develop a method for bias correction which models the error of the target estimator as a function of the corresponding estimator obtained from bootstrap samples and the original estimators and bootstrap estimators of the parameters governing the model fitted to the sample data this is achieved by considering a number of plausible parameter values generating a pseudo original sample for each parameter and bootstrap samples for each such sample and then searching for an appropriate functional relationship under certain conditions the procedure also permits estimation of the mean square error of the bias corrected estimator the method is applied for estimating the prediction mean square error in small area estimation of proportions under a generalized mixed model empirical comparisons with jackknife and bootstrap methods are presented

some families of linear operators associated with certain subclasses of multivalent functions making use of a certain linear operator which is defined here by means of the hadamard product or convolution we introduce two novel subclasses p a c a b p l andp a c a b p l of the class a p of normalized p valent analytic functions in the open unit disk the main objective of the present paper is to investigate the various important properties and characteristics of each of these subclasses furthermore several properties involving neighborhoods of functions in these subclasses are investigated we also derive many results for the modified hadamard products of functions belonging to the class p a c a b p l finally some applications of fractional calculus operators are considered

bootstrap methods for finite populations abstract we show that the familiar bootstrap plug in rule of efron has a natural analog in finite population settings in our method a characteristic of the population is estimated by the average value of the characteristic over a class of empirical populations constructed from the sample our method extends that of gross to situations in which the stratum sizes are not integer multiples of their respective sample sizes moreover we show that our method can be used to generate second order correct confidence intervals for smooth functions of population means a property that has not been established for other resampling methods suggested in the literature a second resampling method is proposed that also leads to second order correct confidence intervals and is less computationally intensive than our bootstrap but a simulation study reveals that the second method can be quite unstable in some situations whereas our bootstrap performs very well

a note on the jackknife the bootstrap and the delta method estimators of bias and variance three methods for the nonparametric reduction of bias and estimation of variance are in common use the jackknife the delta method i e the method of statistical differentials and the bootstrap in recent work efron 1981 a b 1982 has compared these three methods numerically for the case of the sample correlation coefficient for present purposes we define the procedures as follows consider a statistic tn thought of as an estimator of an unknown parameter 0 we wish to study the possibility of bias reduction by estimating the bias and subtracting off the estimate and variance estimation for g tj considered as an estimator of g 0 throughout we assume that we observe x1 xn independent and identically distributed with common cumulative distribution function f let tn denote the abovementioned statistic computed based on x1 xn we assume tn to be symmetric in its arguments furthermore let t 1 denote the above mentioned statistic computed based upon the sample of size n 1 consisting of x1 xi1 xi 1 xn the jackknife estimator of the bias of g tn is then given as b n n 1 1 g t 1 g tn n and the jackknife variance estimator is just

classes of analytic functions associated with the generalized hypergeometric function using the generalized hypergeometric function we introduce and study a class of analytic functions with negative coefficients coefficients estimates distortion theorems extreme points and the radii of convexity and starlikeness for this class are given relevant connections of these results with those in several earlier investigations are indicated

on the probable error of the ratio of two gamma means summary the two sample student s t interval can be derived by parametric bootstrap coverage calibration of the naive plug in interval that fails to account for the estimation of the common variance parameter the same technique results in an essentially exact confidence interval for the ratio of two means based on independent samples from gamma distributions the gamma problem is more difficult computationally because an exact pivot is not available however the computational burden of bootstrap calibration can be reduced to a routine level using the fact that the ratio of sample means is proportional to an exact f variate that is independent of the conditional maximum likelihood estimator of the shape parameter the density of the conditional estimator can be approximated with extreme accuracy using the p formula and a rejection sampler can be used to simulate from this approximate density this greatly simplifies bootstrap calibration in this context an alternative interval is obtained by inverting an exact test described by jensen 1986 simulation studies suggest that the two intervals have similar performance in terms of length and coverage in practice we recommend the bootstrap calibration method because it is much simpler to implement and is more versatile

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

nonparametric likelihood ratio confidence intervals we consider construction of two sided nonparametric confidence intervals in a smooth function model setting a nonparametric likelihood approach based on stein s least favourable family is proposed as an alternative to empirical likelihood the approach enjoys the same asymptotic properties as empirical likelihood but is analytically and computationally less cumbersome the simplicity of the method allows us to propose and analyse asymptotic and bootstrapping techniques as a means of reducing coverage error to levels comparable with those obtained by more computationally intensive techniques such as the iterated bootstrap a simulation study confirms that coverage error may be substantially reduced by simple analytic adjustment of the nonparametric likelihood interval and that bootstrapping the distribution of the nonparametric likelihood ratio results in very desirable coverage accuracy

bootstrap recycling a monte carlo alternative to the nested bootstrap abstract a monte carlo algorithm is described that can be used in place of the nested bootstrap it is particularly advantageous when there is a premium on the number of bootstrap samples either because samples are hard to generate or because expensive computations are applied to each sample this recycling algorithm is useful because it enables inference procedures like prepivoting and bootstrap iteration in models where nested bootstrapping is computationally impractical implementation of the recycling algorithm is quite straightforward as a replacement of the double bootstrap for example bootstrap recycling involves two stages of sampling as does the double bootstrap the first stage of both algorithms is the same simulate from the fitted model in the second stage of recycling one batch of samples is simulated from one measure a measure dominating all the first stage fits these samples are recycled with each first stage sample to yield estimated adjustments to the original inference procedur

bias correction with jackknife bootstrap and taylor series we analyze bias correction methods using jackknife bootstrap and taylor series we focus on the binomial model and consider the problem of bias correction for estimating f p where f in c 0 1 is arbitrary we characterize the supremum norm of the bias of general jackknife and bootstrap estimators for any continuous functions and demonstrate the in delete d jackknife different values of d may lead to drastically different behaviors in jackknife we show that in the binomial model iterating the bootstrap bias correction infinitely many times may lead to divergence of bias and variance and demonstrate that the bias properties of the bootstrap bias corrected estimator after r 1 rounds are of the same order as that of the r jackknife estimator if a bounded coefficients condition is satisfied

monte carlo approximation and the iterated bootstrap summary we discuss optimal choice of the numbers of resamples in the two stages of the iterated bootstrap when that technique is used to calibrate bootstrap confidence intervals if the numbers of resamples in the first and second stages are denoted by b and c respectively we show that it is optimal to take c to be approximately a constant multiple of b the value of the constant is derived and shown to depend only on the nominal coverage level of a confidence interval however it assumes different values in the cases of one and twosided intervals

confidence bands in non parametric errors in variables regression summary errors in variables regression is important in many areas of science and social science e g in economics where it is often a feature of hedonic models in environmental science where air quality indices are measured with error in biology where the vegetative mass of plants is frequently obscured by mismeasurement and in nutrition where reported fat intake is typically subject to substantial error to date in non parametric contexts the great majority of work has focused on methods for estimating the mean as a function with relatively little attention being paid to techniques for empirical assessment of the accuracy of the estimator we develop methodologies for constructing confidence bands our contributions include techniques for tuning parameter choice aimed at minimizing the coverage error of confidence bands

on the bootstrap and likelihood based confidence regions summary we describe a method for constructing likelihood based confidence regions for a vector parameter using the bootstrap and nonparametric density estimation the technique is illustrated by application to a numerical example and its theoretical properties are elucidated it is argued that likelihood based regions should not be approximated by ellipses if we are to have any hope of capturing first order departures from normality bootstrap algorithms for constructing simultaneous confidence intervals for the components of a vector parameter are also presented advantages of the percentile t method over the ordinary percentile method are demonstrated in a multivariate setting