multiple comparison of several linear regression models research on multiple comparison during the past 50 years or so has focused mainly on the comparison of several population means several years ago spurrier considered the multiple comparison of several simple linear regression lines he constructed simultaneous confidence bands for all of the contrasts of the simple linear regression lines over the entire range when the models have the same design matrices this article extends spurrier s work in several directions first multiple linear regression models are considered and the design matrices are allowed to be different second the predictor variables are either unconstrained or constrained to finite intervals third the types of comparison allowed can be very flexible including pairwise many one and successive two simulation methods are proposed for the calculation of critical constants the methodologies are illustrated with examples

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

coverage probability bias objective bayes and the likelihood principle summary we review objective bayes procedures based on both parametric and predictive coverage probability bias and explore the extent to which such procedures contravene the likelihood principle in the case of a scalar parameter the discussion encompasses choice of objective priors objective posterior probability statements and objective predictive probability statements we conclude with some remarks concerning the future development and implementation of objective priors based on small coverage probability bias

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

confidence intervals for a binomial proportion and asymptotic expansions p z 2n 1 2 p 1 p 1 2 is currently in near universal use we first show that the coverage properties of the wald interval are persistently poor and defy virtually all conventional wisdom we then proceed to a theoretical comparison of the standard interval and four additional alternative intervals by asymptotic expansions of their coverage probabilities and expected lengths the four additional interval methods we study in detail are the score test interval wilson the likelihood ratio test interval a jeffreys prior bayesian interval and an interval suggested by agresti and coull the asymptotic expansions for coverage show that the first three of these alternative methods have coverages that fluctuate about the nominal value while the agresti coull interval has a somewhat larger and more nearly conservative coverage function for the five interval methods we also investigate asymptotically their average coverage relative to distributions for p supported within 0 1 in terms of expected length asymptotic expansions show that the agresti coull interval is always the longest of these the remaining three are rather comparable and are shorter than the wald interval except for p near 0 or 1 these analytical calculations support and complement the findings and the recommendations in brown cai and dasgupta statist sci 2001 16 101 133

a lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets given any family of valid inequalities for the asymmetric traveling salesman polytopep g defined on the complete digraphg we show that all members of are facet defining if the primitive members of usually a small subclass are based on this result we then introduce a general procedure for identifying new classes of facet inducing inequalities forp g by lifting inequalities that are facet inducing forp g whereg is some induced subgraph ofg unlike traditional lifting where the lifted coefficients are calculated one by one and their value depends on the lifting sequence our lifting procedure replaces nodes ofg with cliques ofg and uses closed form expressions for calculating the coefficients of the new arcs which are sequence independent we also introduce a new class of facet inducing inequalities the class of sd source destination inequalities which subsumes as special cases most known families of facet defining inequalities

finger competition dynamics in rotating hele shaw cells we report analytical results for the development of interfacial instabilities in rotating hele shaw cells we execute a mode coupling approach to the problem and examine the morphological features of the fluid fluid interface at the onset of nonlinear effects the impact of normal stresses is accounted for through a modified pressure jump boundary condition a differential equation describing the early nonlinear evolution of the interface is derived being conveniently written in terms of three relevant dimensionless parameters viscosity contrast a surface tension b and gap spacing b we focus our study on the influence of these parameters on finger competition dynamics it is deduced that the link between finger competition and a b and b can be revealed by a mechanism based on the enhanced growth of subharmonic perturbations our results show good agreement with existing experimental and numerical investigations of the problem both in low and high a 0 a 0 lead to enhanced competition of outward inward fingers within this context the role of b and b in determining different finger competition behaviors is also discussed

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

on the monotonization of polyhedra in polyhedral combinatorics one often has to analyze the facial structure of less than full dimensional polyhedra the presence of implicit or explicit equations in the linear system defining such a polyhedron leads to technical difficulties when analyzing its facial structure it is therefore customary to approach the study of such a polytopep through the study of one of its full dimensional relaxations monotonizations known as the submissive and the dominant ofp finding sufficient conditions for an inequality that induces a facet of the submissive or the dominant of a polyhedron to also induce a facet of the polyhedron itself has been posed in the literature as an important research problem our paper goes a long way towards solving this problem we address the problem in the framework of a generalized monotonization of a polyhedronp g mon p that subsumes both the submissive and the dominant and give a sufficient condition for an inequality that defines a facet of g mon p to define a facet ofp for the important cases of the traveling salesman ts polytope in both its symmetric and asymmetric variants and of the linear ordering polytope we give sufficient conditions trivially easy to verify for a facet of the monotone completion to define a facet of the original polytope itself

dynamics of viscous fingers in rotating hele shaw cells with coriolis effects a growing number of experimental and theoretical works have been addressing various aspects of the viscous fingering formation in rotating hele shaw cells however only a few of them consider the influence of coriolis forces the studies including coriolis effects are mostly restricted to the high viscosity contrast limit and rely on either purely linear stability analyses or intensive numerical simulations we approach the problem analytically and use a modified darcy s law including the exact form of the coriolis effects to execute a mode coupling analysis of the system by imposing no restrictions on the viscosity contrast a dimensionless viscosity difference we go beyond linear stages and examine the onset of nonlinearities our results indicate that when coriolis effects are taken into account an interesting interplay between the reynolds number re and a arises this leads to important changes in the stability and morphological features of the emerging interfacial patterns we contrast our mode coupling approach with previous theoretical models proposed in the literature

a bayesian approach to approximate conditional inference summary sweeting 1995 studies regular bayesian and frequentist approximations within a unified framework in the case of a single parameter and shows that higher order approximations to sampling distributions arise from their bayesian counterparts via an unsmoothing argument in the present paper we extend this programme to include formulae in approximate conditional inference in particular it is shown how bayesian arguments may be used to derive some formulae developed by barndorff nielsen 1980 1983 1986 the development proceeds in terms of likelihood roots

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

corrected confidence intervals after sequential testing with applications to survival analysis corrected confidence intervals are developed for an unknown parameter for data from a sequential experiment the corrections are obtained from asymptotic expansions for the distribution of the signed root transformation and they have a simple form the model does not require independent and identically distributed random variables or smooth stopping boundaries inference about the failure rate of an exponential distribution when data are subject to random censoring censoring by real time and optional stopping is used to illustrate the main results the accuracy of the method is confirmed through simulation studies

symmetric inequalities and their composition for asymmetric travelling salesman polytopes the asymmetric travelling salesmanatspolytope atspv is the convex hull of incidence vectors of all hamiltonian circuits in a complete digraph with node set v this paper studies classes of valid symmetric inequalities ay a0 for ats polytopes with coefficients satisfying aij aji for all pair of cities i and j of particular interest are those inequalities derived from facet defining inequalities for symmetric travelling salesman polytopes stsps we show that known classes of stsp facet defining inequalities such as path wheelbarrow chain and ladder inequalities induce symmetric atsp facet defining inequalities for atspv with v 8 we show that there are precisely four types of stsp facet defining inequalities that do not induce atsp facets we propose a tree composition of symmetric inequalities and use it to generate a large new class of symmetric facet defining inequalities for ats polytopes subsuming all nonspanning clique tree inequalities a hamiltonian path approach is used for most of the polyhedral proofs in the paper

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

double bootstrap for shrinkage estimators computer intensive bootstrap methods are potentially problematic when applied to biased estimators similar to ridge regression because biased estimators may lack a pivot we propose a modification for efron s bias corrected percentiles and suggest an approximate pivoting transformation for ridge regression we indicate practical aspects of beran s double bootstrap designed specifically for situations where there is nonnormality and a lack of a pivot econometric applications to estimating consumption functions a simulation with multicollinear data and new graphical tools are included

facets of the linear ordering polytope letd n be the complete digraph onn nodes and letp lo denote the convex hull of all incidence vectors of arc sets of linear orderings of the nodes ofd n i e these are exactly the acyclic tournaments ofd n we show that various classes of inequalities define facets ofp lo e g the 3 dicycle inequalities the simplek fence inequalities and various mobius ladder inequalities and we discuss the use of these inequalities in cutting plane approaches to the triangulation problem of input output matrices i e the solution of permutation resp linear ordering problems

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

tight approximations of dynamic risk measures this paper compares two frameworks for measuring risk in a multiperiod setting the first corresponds to applying a single coherent risk measure to the cumulative future costs and the second involves applying a composition of one step coherent risk mappings we characterize several necessary and sufficient conditions under which one measurement always dominates the other and introduce a metric to quantify how close the two measures are using this notion we address the question of how tightly a given coherent measure can be approximated by lower or upper bounding compositional measures we exhibit an interesting asymmetry between the two cases the tightest upper bound can be exactly characterized and corresponds to a popular construction in the literature whereas the tightest lower bound is not readily available we show that testing domination and computing the approximation factors are generally np hard even when the risk measures are comonotonic and law invariant however we characterize conditio

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

the graphical relaxation a new framework for the symmetric traveling salesman polytope a present trend in the study of thesymmetric traveling salesman polytope stsp n is to use as a relaxation of the polytope thegraphical relaxation gtsp n rather than the traditionalmonotone relaxation which seems to have attained its limits in this paper we show the very close relationship between stsp n and gtsp n in particular we prove that every non trivial facet of stsp n is the intersection ofn 1 facets of gtsp n n of which are defined by the degree inequalities this fact permits us to define a standard form for the facet defining inequalities for stsp n that we calltight triangular and to devise a proof technique that can be used to show that many known facet defining inequalities for gtsp n define also facets of stsp n in addition we give conditions that permit to obtain facet defining inequalities by composition of facet defining inequalities for stsp n and general lifting theorems to derive facet defining inequalities for stsp n k from inequalities defining facets of stsp n

the symmetric traveling salesman polytope new facets from the graphical relaxation the path the wheelbarrow and the bicycle inequalities have been shown by cornuejols fonlupt and naddef to be facet defining for the graphical relaxation of stsp n the polytope of the symmetric traveling salesman problem on an n node complete graph we show that these inequalities and some generalizations of them define facets also for stsp n in conclusion we characterize a large family of facet defining inequalities for stsp n that include as special cases most of the inequalities currently known to have this property as the comb the clique tree and the chain inequalities most of the results given here come from a strong relationship of stsp n with its graphical relaxation that we have pointed out in another paper where the basic proof techniques are also described

an o brien fleming sequential trial for comparing three treatments we consider a sequential procedure for comparing three treatments with the goal of ultimately selecting the best treatment this procedure starts with a sequential test to detect an overall treatment difference and eliminates the apparently inferior treatment if this test rejects the equality of the treatments it then proceeds with a sequential test of the remaining two treatments we base these sequential tests on the stopping boundaries popularized by o brien and fleming our procedure is similar in structure to that used by siegmund in conjunction with modified repeated significance tests we compare the performances of the two procedures via a simulation experiment we derive analytic approximations for an error probability the power and the expected sample size of our procedure which we compare to simulated values furthermore we propose a modification of the procedure for the comparison of a standard treatment with experimental treatments

nonparametric comparison of mean directions or mean axes samples of directional or axial measurements arise in geophysical biological and econometric contexts we represent the rotational difference between two mean directions or two mean axes as a direction or axis we then construct nonparametric simultaneous confidence sets for all pairwise rotational differences among the mean directions or mean axes of s samples by specialization this methodology yields nonparametric simultaneous tests for pairwise equality of directional means or axes

facets of the asymmetric traveling salesman polytope in this paper we consider the asymmetric traveling salesman polytope pn defined as the convex hull of the incidence vectors of tours in a complete digraph with n vertices and its monotonization pi n several classes of valid inequalities for both there polytopes have been introduced in the literature which are not known to define facets of pn or pi n we describe a general technique which can often be used to prove that a given inequality defines a facet of pn the method is applied to prove that the so called dk dk c3 comb and c2 inequalities define facets of pn thus solving open problems as for the monotone polytope a necessary and sufficient condition for a facet defining inequality of pn to define a facet of pi n as well is introduced which applies to the dk dk c3 comb and c2 inequalities in addition a simple procedure for transforming any facet defining inequality of pn into an equivalent one which also defines a facet of pi n is given

asymptotics beyond all orders asymptotology and borel summation asymptotics superasymptotics hyperasymptotics m berry the geometric model of crystal growth numerical analysis of the geometric model of crystal growth j m hammersley dendritic crystal growth an experimental assessment of continuum models of dendritic growth j p gollub directional solidification of solids directional growth of dilute mixtures and lamellar eutectics k kassner et al flow in a hele shaw cell viscous displacement in a hele shaw cell s tanveer the rapidly forced pendulum exponentially small estimate for separatrix splittings j scheurle et al ordinary differential equations singular perturbations of solitons y pomeau solitary water waves in the presence of small surface tension solitary waves with ripples beyond all orders j t beale problems in optics potpourri 23 additional articles index

a note on a class of solitary like solutions of the tzitzeica equation generated by a similarity reduction abstract in this paper new class of solutions of the tzitzeica equation are derived by using the classical lie symmetry analysis the important aspect of this paper however is the fact that the analysis results in a new class of solitary like solutions the so called cusp solitary solutions this special type of solutions are not found in the current literature and represents a necessary contribution for the whole solution manifold the studied equation was originally found in the field of geometry otherwise the tzitzeica equation takes place in many branches of non linear sciences therefore explicit class of solutions connected by a physical meaning are of great importance the analysis is restricted to the case of traveling waves represented by a similarity variable describing any wave propagation a complete characterization of the group properties is given the classical lie point symmetries are derived and algebraic properties are determined similarity solutions and transformations are given in a most general form and have been derived for the first time in terms of jacobian elliptic functions it is worth to mention that the application of known powerful algebraic methods e g special function transform methods are not appropriate to study the solution manifold hence the present paper is therefore suitable to create a deeper insight into the solution manifold with respect to the traveling wave picture

very weak expansions for sequential confidence levels on obtient des developpements asymptotiques pour une classe de moyennes de probabilites de recouvrement pour des bornes de confiance sequentielles

a framework for bayesian and likelihood approximations in statistics summary regular bayesian and frequentist approximations in statistics are studied within a unified framework in particular it is shown how some higher order likelihood based approximations arise from their bayesian counterparts via an unsmoothing argument this approach serves to shed new light on these formulae and to clarify relationships between bayesian and frequentist inferences for example bayesian interpretations of higher order approximations can give insights into the acceptability or otherwise of these approximations from the point of view of relevance to the actual data observed furthermore this approach is a very natural one for the study of more general nonregular problems models for dependent data and approximate conditional inference for ease of exposition the development is in terms of a single real parameter the main analytic development proceeds in terms of signed roots of log density ratios

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

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

balanced simultaneous confidence sets abstract suppose that t tu is a family of parametric functions indexed by the variable u for each u an approximate confidence set cn u for tu may be obtained by referring a function of tu and the sample to an estimated quantile dn u of that function s sampling distribution this approach is sometimes called the pivotal method for constructing a confidence set even when it is not based on a true pivot a simultaneous confidence set cn for t is then obtained by simultaneously asserting the individual confidence sets cn u the problem is to choose the critical values for the cn u in such a way that the overall coverage probability of cn is correct and the coverage probabilities of the individual confidence sets cn u are equal the second property is termed balance it means that the simultaneous confidence set cn treats each constituent confidence statement cn u fairly aside from a few special cases the problem just described is too difficult for analytical approaches whe

on unions and dominants of polytopes a well known result on unions of polyhedra in the same space gives an extended formulation in a higher dimensional space whose projection is the convex hull of the union here in contrast we study the unions of polytopes in different spaces giving a complete description of the convex hull without additional variables when the underlying polytopes are monotone the facets are described explicitly generalizing results of hong and hooker on cardinality rules and an efficient separation algorithm is proposed these results are based on an explicit representation of the dominant of a polytope and an algorithm for the separation problem for the dominant for non monotone polytopes both the dominant and the union are characterized we also give results on the unions of polymatroids both on disjoint ground sets and on the same ground set generalizing results of conforti and laurent

integrable expansions for posterior distributions for a two parameter exponential family asymptotic expansions of posterior distributions are derived for a two dimensional exponential family which includes normal gamma inverse gamma and inverse gaussian distributions reparameterization allows us to use a data dependent transformation convert the likelihood function to the two dimensional standard normal density and apply a version of stein s identity to assess the posterior distributions applications are given to characterize optimal noninformative priors in the sense of stein to suggest the form of a high order correction to the distribution function of a sequential likelihood ratio statistic and to provide confidence intervals for one parameter in the presence of other nuisance parameters

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

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

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

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

systematic weakly nonlinear analysis of radial viscous fingering we present a weakly nonlinear analysis of the interface dynamics in a radial hele shaw cell driven by both injection and rotation we extend the systematic expansion introduced in e alvarez lacalle et al phys rev e 64 016302 2001 to the radial geometry and compute explicitly the first nonlinear contributions we also find the necessary and sufficient condition for the uniform convergence of the nonlinear expansion within this region of convergence the analytical predictions at low orders are compared satisfactorily to exact solutions and numerical integration of the problem this is particularly remarkable in configurations with no counterpart in the channel geometry for which the interplay between injection and rotation allows that condition to be satisfied at all times in the case of the purely centrifugal forcing we demonstrate that nonlinear couplings make the interface more unstable for lower viscosity contrast between the fluids

balanced confidence regions based on tukey s depth and the bootstrap summary we propose and study the bootstrap confidence regions for multivariate parameters based on tukey s depth the bootstrap is based on the normalized or studentized statistic formed from an independent and identically distributed random sample obtained from some unknown distribution in rq the bootstrap points are deleted on the basis of tukey s depth until the desired confidence level is reached the proposed confidence regions are shown to be second order balanced in the context discussed by beran we also study the asymptotic consistency of tukey s depth based bootstrap confidence regions the applicability of the method proposed is demonstrated in a simulation study

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

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

interface dynamics in hele shaw flows with centrifugal forces preventing cusp singularities with rotation a class of exact solutions of hele shaw flows without surface tension in a rotating cell is reported we show that the interplay between injection and rotation modifies the scenario of formation of finite time cusp singularities for a subclass of solutions we show that for any given initial condition there exists a critical rotation rate above which cusp formation is suppressed we also find an exact sufficient condition to avoid cusps simultaneously for all initial conditions within the above subclass

clique tree inequalities and the symmetric travelling salesman problem the linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful cf crowder and padberg crowder h p m w padberg 1980 solving large scale symmetric travelling salesman problems to optimality management sci 26 495 509 grotschel grotschel m 1980a on the symmetric travelling salesman problem solution of a 120 city problem math programming stud 12 61 77 padberg and hong padberg m w s hong 1980 on the symmetric travelling salesman problem a computational study math programming stud 12 78 107 one of the reasons for this success is certainly the fact that instead of ordinary cutting planes gomory cuts etc problem specific cutting planes could be used which define facets of the underlying integer programming polytopes in this paper we shall define a new class of inequalities clique tree inequalities valid for the travelling salesman polytope which properly contains many of the known classes of inequalities like subtour elimination constraints 2 matching constraints comb inequalities and we show that all these new inequalities induce facets of the travelling salesman polytope since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem

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

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

on the acyclic subgraph polytope the acyclic subgraph problem can be formulated as follows given a digraph with arc weights find a set of arcs containing no directed cycle and having maximum total weight we investigate this problem from a polyhedral point of view and determine several classes of facets for the associated acyclic subgraph polytope we also show that the separation problem for the facet defining dicycle inequalities can be solved in polynomial time this implies that the acyclic subgraph problem can be solved in polynomial time for weakly acyclic digraphs this generalizes a result of lucchesi for planar digraphs

bounds on causal effects in three arm trials with non compliance summary the paper considers the analysis of three arm randomized trials with non compliance in these trials the average causal effects of treatments within principal strata of compliance behaviour are of interest for better understanding the effect of the treatment unfortunately even with the usual assumptions the average causal effects of treatments within principal strata are not point identified however the observable data do provide useful information on the bounds of the identification regions of the parameters of interest under two sets of assumptions we derive sharp bounds for the causal effects within principal strata for binary outcomes and construct confidence intervals to cover the identification regions the methods are illustrated by an analysis of data from a randomized study of treatments for alcohol dependence

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

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

improving coverage accuracy of nonparametric prediction intervals summary methods are suggested for improving the coverage accuracy of intervals for predicting future values of a random variable drawn from a sampled distribution it is shown that properties of solutions to such problems may be quite unexpected for example the bootstrap and the jackknife perform very poorly when used to calibrate coverage although the jackknife estimator of the true coverage is virtually unbiased a version of the smoothed bootstrap can be employed for successful calibration however interpolation among adjacent order statistics can also be an effective way of calibrating although even there the results are unexpected in particular whereas the coverage error can be reduced from o n 1 to orders o n 2 and o n 3 where n denotes the sample size by interpolating among two and three order statistics respectively the next two orders of reduction require interpolation among five and eight order statistics respectively

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

dominants and submissives of matching polyhedra letp be the convex hull of perfect matchings of a graphg v e the dominant ofp is y re y x for somex p a theorem of fulkerson implies that ifg is bipartite then the dominant ofp can be described by linear inequalities having 0 1 valued coefficients however this is far from true in general here it is proved that for every positive integern there exists a graph for which the dominant has an essential valid inequality whose coefficient set includes the firstn positive integers a similar result holds for the submissive ofp y re 0 y x for somex p

the fixed outdegree 1 arborescence polytope a 1 arborescence is a spanning arborescence rooted at node 1 plus one arc incident into node 1 the 1 arborescence polytope the convex hull of incidence vectors of 1 arborescences is a well known relaxation of the asymmetric traveling salesman ats polytope in this paper we study the fixed outdegree 1 arborescence fda polytope defined as the convex hull of incidence vectors of 1 arborescences having outdegree 1 at a prescribed subset f of nodes the fda polytope becomes the 1 arborescence polytope when f and it becomes the ats polytope when f is the entire node set we derive several classes of valid inequalities for the fda polytope and show that they are facet defining this is done by using a theorem that establishes sufficient conditions for a facet defining inequality for the ats polytope to also be facet defining for the fda polytope we then introduce the class of fd inequalities and show that they define facets of both the ats polytope and the fda polytope finally we define a canonical form that facilitates rigorous comparisons between inequalities and use it to show that the new facets are distinct from each other and from other known facets

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