markov decision evolutionary games we present a class of evolutionary games involving large populations that have many pairwise interactions between randomly selected players the fitness of a player depends not only on the actions chosen in the interaction but also on the individual state of the players players have a finite life time during which they participate in several local interactions and take actions the actions taken by a player determine not only the immediate fitness but also the transition probabilities to its next individual state we define and characterize the evolutionary stable strategies for these games and propose a method to compute them we illustrate the model and results through a networking problem

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

a comparison of certain bootstrap confidence intervals in the cox model abstract we study bootstrap confidence intervals for three types of parameters in cox s proportional hazards model the regression parameter the survival function at fixed time points and the median survival time at fixed values of a covariate several types of bootstrap confidence intervals are studied and the type of interval is determined by two factors one factor is the method of drawing the bootstrap sample we consider three such methods 1 ordinary resampling from the empirical cumulative distribution function 2 resampling conditional on the covariates and 3 resampling conditional on the covariates and the censoring pattern another factor is the method of forming the confidence interval from a given sample the methods considered are the percentile hybrid and bootstrap t all the methods of forming confidence intervals are compared to each other and to the standard asymptotic method via a monte carlo study the data sets for this monte carlo study are simulated conditionally on the c

simulation based finite and large sample tests in multivariate regressions abstract in the context of multivariate linear regression mlr models it is well known that commonly employed asymptotic test criteria are seriously biased towards overrejection in this paper we propose a general method for constructing exact tests of possibly nonlinear hypotheses on the coefficients of mlr systems for the case of uniform linear hypotheses we present exact distributional invariance results concerning several standard test criteria these include wilks likelihood ratio lr criterion as well as trace and maximum root criteria the normality assumption is not necessary for most of the results to hold implications for inference are two fold first invariance to nuisance parameters entails that the technique of monte carlo tests can be applied on all these statistics to obtain exact tests of uniform linear hypotheses second the invariance property of the latter statistic is exploited to derive general nuisance parameter free bounds on the distribution of the lr statistic for arbitrary hypotheses even though it may be difficult to compute these bounds analytically they can easily be simulated hence yielding exact bounds monte carlo tests illustrative simulation experiments show that the bounds perform well our findings illustrate the value of the bounds as a tool to be used in conjunction with more traditional simulation based test methods e g the parametric bootstrap which may be applied when the bounds are not conclusive

on sensitivity of gauss christoffel quadrature in numerical computations the question how much does a function change under perturbations of its arguments is of central importance in this work we investigate sensitivity of gauss christoffel quadrature with respect to small perturbations of the distribution function in numerical quadrature a definite integral is approximated by a finite sum of functional values evaluated at given quadrature nodes and multiplied by given weights consider a sufficiently smooth integrated function uncorrelated with the perturbation of the distribution function then it seems natural that given the same number of function evaluations the difference between the quadrature approximations is of the same order as the difference between the original and perturbed approximated integrals that is perhaps one of the reasons why to our knowledge the sensitivity question has not been formulated and addressed in the literature though several other sensitivity problems motivated in particular by computation of the quadrature nodes and weights from moments have been thoroughly studied by many authors we survey existing particular results and show that even a small perturbation of a distribution function can cause large differences in gauss christoffel quadrature estimates we then discuss conditions under which the gauss christoffel quadrature is insensitive under perturbation of the distribution function present illustrative examples and relate our observations to known conjectures on some sensitivity problems

bootstrap confidence bands for shrinkage estimators empirical application of shrinkage estimators has been limited by the inability to estimate confidence intervals we investigate confidence band estimators based on edgeworth expansions and bootstrapping our monte carlo results suggest that both efron s bias correction method with acceleration and the simple percentile bootstrap methods generate reasonable confidence bands approximations based on edgeworth expansions performed poorly we then use the percentile method with a single bootstrap to generate bands for predictions of gnp growth rates from a leading indicators model our study shows that simple percentile bootstrap confidence bands perform well enough to support empirical applications of shrinkage estimators

a matrix game solution of the single controller stochastic game we consider a stochastic game on finitely many states with limiting average payoff further we assume that the law of motion depends on the actions of one player only say the minimizer we show that these games have the following properties a the value of the stochastic game for each state s is the same as the value of the matrix game whose rows and columns are the pure stationary strategies of the players and whose entries are the corresponding payoffs b these matrix games have a common optimal strategy for the maximizer which in turn yields his optimal stationary strategy in the stochastic game c if the value of a stochastic subgame obtained by deleting a column or a row in a particular state coincides with the value of the original game at that state then it coincides at all states d if all actions of the minimizer at each state are essential for optimal play then in the transient states under optimal play the minimizer has only one action in such a case the original game can be solve

numerical stability of descent methods for solving linear equations in this paper we perform a round off error analysis of descent methods for solving a liner systemax b wherea is supposed to be symmetric and positive definite this leads to a general result on the attainable accuracy of the computed sequence x i when the method is performed in floating point arithmetic the general theory is applied to the gauss southwell method and the gradient method both methods appear to be well behaved which means that these methods compute an approximationx i to the exact solutiona 1 b which is the exact solution of a slightly perturbed linear system i e a a x i b a of order a where is the relative machine precision and denotes the spectral norm

the effect of monte carlo approximation on coverage error of double bootstrap confidence intervals a double bootstrap confidence interval must usually be approximated by a monte carlo simulation consisting of two nested levels of bootstrap sampling we provide an analysis of the coverage accuracy of the interval which takes account of both the inherent bootstrap and monte carlo errors the analysis shows that by a suitable choice of the number of resamples drawn at the inner level of bootstrap sampling we can reduce the order of coverage error we consider also the effects of performing a finite monte carlo simulation on the mean length and variability of length of two sided intervals an adaptive procedure is presented for the choice of the number of inner level resamples the effectiveness of the procedure is illustrated through a small simulation study

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

on the rate of convergence of the preconditioned conjugate gradient method we derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum we present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function

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

analysis of the symmetric lanczos algorithm with reorthogonalization methods we present an error analysis of the symmetric lanczos algorithm in finite precision arithmetic the loss of orthogonality among the computed lanczos vectors is explained with the help of a recurrence formula a backward error analysis shows that semiorthogonality among the lanczos vectors is enough to guarantee the accuracy of the computed quantities up to machine precision the results of this analysis are then extended to the more general case of the lanczos algorithm with a semiorthogonalization strategy based on the recurrence formula a new reorthogonalization method called partial reorthogonalization is introduced we show that both partial reorthogonalization and selective orthogonalization as introduced by parlett and scott 15 are semiorthogonalization strategies finally we discuss the application of our results to the solution of linear systems of equations and to the eigenvalue problem

the numerical stability analysis of pipelined conjugate gradient methods historical context and methodology algebraic solvers based on preconditioned krylov subspace methods are among the most powerful tools for large scale numerical computations in applied mathematics sciences technology as well as i

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

monotone convergence of the extended krylov subspace method for laplace stieltjes functions of hermitian positive definite matrices abstract the extended krylov subspace method is known to be very efficient in many cases in which one wants to approximate the action of a matrix function f a on a vector b in particular when f belongs to the class of laplace stieltjes functions we prove that the euclidean norm of the error decreases strictly monotonically in this situation when a is hermitian positive definite similar results are known for the polynomial lanczos method for f a b and we demonstrate how the techniques of proof used in the polynomial krylov case can be transferred to the extended krylov case

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

on the convergence rate of the conjugate gradients in presence of rounding errors we investigate here rounding error effects on the convergence rate of the conjugate gradients more precisely we analyse on both theoretical and experimental basis how finite precision arithmetic affects known bounds on iteration numbers when the spectrum of the system matrix presents small or large isolated eigenvalues

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

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

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

the lanczos algorithm with partial reorthogonalization the lanczos algorithm is becoming accepted as a powerful tool for finding the cigenvalues and for solving linear systems of equations any practical implementation of the algorithm suffers however from roundoff errors which usually cause the lanczos vectors to lose their mutual orthogonality in order to maintain some level of orthogonality full reorthogonalization fro and selective orthogonalization so have been used in the past as a remedy here partial reorthogonalization pro is proposed as a new method for maintain ing semiorthogonality among the lanczos vectors pro is based on a simple recurrence which allows us to monitor the loss of orthogonality among the lanczos vectors directly wvithout computing the inner products based on the information from the recurrence reorthogonalizations occur only when necessary thus substantial savings are made as compared to fro in some numerical examples we apply the lanczos algorithm with pro to the solution of large symmetric systems of linear equations and show that it is a robust and efficient algorithm for maintaining semiorthogonality among the lanczos vectors the results obtained compare favorably with the conjugate gradient method

on the real convergence rate of the conjugate gradient method abstract we present a parametrized class of matrices for which the rate of convergence of the conjugate gradient method varies greatly with the parameter and does not appreciably depend on the algorithm implementation a small change in the eigenvalue distribution can lead to a large change in the sensitivity of cg to rounding errors a theorem is proved which gives a necessary and sufficient condition for ordering exact arithmetic cg processes for systems with different spectra according to the energy norm of the error theorems 4 1 and 4 2 continue paige s and greenbaum s work

on the level error after bartlett adjustment of the likelihood ratio statistic summary it is well known that bartlett adjustment reduces level error of the likelihood ratio statistic from order n l to order n3 2 in the present note we show that level error of the adjusted statistic is actually order n2

a review of bootstrap confidence intervals on passe en revue plusieurs methodes distinctes basees sur le bootstrap pour construire des intervalles de confiance

game interactions and dynamics on networked populations a new mathematical formulation of evolutionary game dynamics on networked populations is proposed the model extends the standard replicator equation to a finite set of players organized on an arbitrary network of connections graph classical results of multipopulation evolutionary game theory are used in combination with graph theory to obtain the mathematical model specifically the players located at the vertices of the graph are interpreted as subpopulations of a multipopulation dynamical game the members of each subpopulation are replicators engaged at each time instant into 2 player games with the members of other connected subpopulations this idea allows us to write an extended equation describing the game dynamics of a finite set of players connected by a graph the obtained equation does not require any assumption on the game payoff matrices nor graph topology stability of steady states nash equilibria and the relationship of the proposed model to the standard replicator equation are discussed the dynamical behavior of the model over different graphs is also investigated by means of extended simulations

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

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

convergence of linear threshold decision making dynamics in finite heterogeneous populations abstract linear threshold models have been studied extensively in structured populations however less attention is paid to perception differences among the individuals of the population to focus on this effect we exclude structure and consider a well mixed population of heterogeneous agents each associated with a threshold in the form of a fixed ratio within zero and one that can be unique to this agent the agents are initialized with a choice of strategy a or b and at each time step one agent becomes active to update if the ratio of agents playing a is higher resp lower than her threshold she updates to a resp b we show that for any given initial condition after a finite number of time steps the population reaches an equilibrium where no agent s threshold is violated however the equilibrium is not necessarily uniquely determined by the initial condition but depends on the agents activation sequence we find all those possible equilibria that the dynamics may reach from a given initial condition and show that in contrast to the case of homogeneous populations heterogeneity in the agents thresholds gives rise to several equilibria where both a playing and b playing agents coexist perception heterogeneity can hence preserve decision diversity even in the absence of population structure we also investigate the asymptotic stability of the equilibria and show how to calculate the contagion probability for a population with two thresholds

solving some large scale semidefinite programs via the conjugate residual method most current implementations of interior point methods for semidefinite programming use a direct method to solve the schur complement equation sce m delta y h in computing the search direction when the number of constraints is large the problem of having insufficient memory to store m can be avoided if an iterative method is used instead numerical experiments have shown that the conjugate residual cr method typically takes a huge number of steps to generate a high accuracy solution on the other hand it is difficult to incorporate traditional preconditioners into the sce except for block diagonal preconditioners we decompose the sce into a 2 2 block system by decomposing delta y similarly for h into two orthogonal components with one lying in a certain subspace that is determined from the structure of m numerical experiments on semidefinite programming problems arising from the lovasz theta function of graphs and maxcut problems show that high accuracy solutions can be obtained with a moderate number of cr steps using the proposed equation

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

using bootstrap likelihood ratios in finite mixture models statistical inference using the likelihood ratio statistic for the number of components in a mixture model is complicated when the true number of components is less than that of the proposed model since this represents a non regular problem the true parameter is on the boundary of the parameter space and in some cases the true parameter is in a non identifiable subset of the parameter space the maximum likelihood estimator is shown to converge to the subset characterized by the same density function and connection is made to the bootstrap method proposed by aitkin and co workers and mclachlan for testing the number of components in a finite mixture and deriving confidence regions in a finite mixture

bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator for rather general parametric models a simple connection is established between the bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant c of the formula c i i 1 2l for the conditional distribution of a maximum likelihood estimator as applied to the full model and the model of the hypothesis tested this leads to a relatively simple demonstration that division of the likelihood ratio statistic by a suitable constant or estimated factor improves the chi squared approximation to its distribution various expressions for these quantities are discussed in particular for the case of a one dimensional parameter an approximation to the constants involved is derived which does not require integration over the sample space

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

theoretical comparison of bootstrap confidence intervals we develop a unified framework within which many commonly used bootstrap critical points and confidence intervals may be discussed and compared in all seven different bootstrap methods are examined each being usable in both parametric and nonparametric contexts emphasis is on the way in which the methods cope with first and second order departures from normality percentile t and accelerated bias correction emerge as the most promising of existing techniques certain other methods are shown to lead to serious errors in coverage and position of critical point an alternative approach based on shortest bootstrap confidence intervals is developed we also make several more technical contributions in particular we confirm efron s conjecture that accelerated bias correction is second order correct in a variety of multivariate circumstances and give a simple interpretation of the acceleration constant

accuracy and effectiveness of preconditioned conjugate gradient algorithms for large and ill conditioned problems abstract different implementations of the preconditioned conjugate gradient method are presented for solving large and very ill conditioned algebraic systems of equations resulting from the finite element method a special implementation based on mixed precision calculations proved to be robust in performance and reliable in accuracy as well as storage effective additionally a new type of iterative refinement scheme is introduced to improve the accuracy of direct methods based on conjugate gradient iterations it is found to be a significant improvement on the standard iterative refinement approach

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

prepivoting test statistics a bootstrap view of asymptotic refinements abstract approximate tests for a composite null hypothesis about a parameter may be obtained by referring a test statistic to an estimated critical value either asymptotic theory or bootstrap methods can be used to estimate the desired quantile the simple asymptotic test a refers the test statistic to a quantile of its asymptotic null distribution after unknown parameters have been estimated the bootstrap approach used here is based on the concept of prepivoting prepivoting is the transformation of a test statistic by the cdf of its bootstrap null distribution the simple bootstrap test b refers the prepivoted test statistic to a quantile of the uniform 0 1 distribution under regularity conditions the bootstrap test b has a smaller asymptotic order of error in level than does the asymptotic test a provided that the asymptotic null distribution of the test statistic does not depend on unknown parameters in the contrary case both a and b have the same order of level error certain class

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

sequential iterated bootstrap confidence intervals a version of the sequential probability ratio test for testing simultaneously a set of nested hypotheses is developed this procedure is then applied to define a sequential procedure of sampling at the inner level of the two nested levels of resampling required by monte carlo construction of an iterated bootstrap percentile method confidence interval the sequential resampling scheme reduces very significantly the computational demands of construction of the iterated bootstrap confidence interval the scheme may be applied simply and without any adaptation to construct a confidence interval for any parameter of interest the performance of the sequential iterated bootstrap confidence interval is illustrated on two examples involving the ratio of two population means and a population variance