estimation of the distribution function of a standardized statistic for estimating the distribution of a standardized statistic the bootstrap estimate is known to be local asymptotic minimax various computational techniques have been developed to improve on the simulation efficiency of uniform resampling the standard monte carlo approach to approximating the bootstrap estimate two new approaches are proposed which give accurate yet simple approximations to the bootstrap estimate the second of the approaches even improves the convergence rate of the simulation error a simulation study examines the performance of these two approaches in comparison with other modified bootstrap estimates

saddlepoint approximations for the generalized variance and wilks statistic saddlepoint approximations to densities of sums of random variables were introduced by daniels 1954 these methods were further developed by barndorff nielsen cox 1979 to approximate joint and conditional densities for canonical sufficient statistics in exponential family settings approximations for the cumulative distributions of sums were developed by lugannani rice 1980 and robinson 1982 while the approximation of conditional cumulative distribution functions was addressed by skovgaard 1987 the modifications necessary to apply the lugannani rice and skovgaard approximations to conditional distributions of canonical sufficient statistics in exponential family settings have been summarized by davison 1988 reid 1988 provides an overall review of the saddlepoint literature in the present paper the lugannani rice and skovgaard approximations are used to reproduce the distributions of the generalized variance and wilks likelihood ratio statistic in a multivatiate analysis of variance representing the degree of freedom in a gamma regression model as an appropriate linear function the generalized variance distribution arises as the marginal distribution of a canonical sufficient statistic to which the lugannani rice approximation may be applied the more recent approximations of fraser 1990 and barndorff nielsen 1990 are exactly the same for this setting comparison of its accuracy with the approximation of hoel 1937 as well as a normal approximation johnson kotz 1972 p 198 shows the lugannani rice approximation to be highly accurate and uniformly better than its competitors the distribution of wilks likelihood ratio statistic has two distinct characterizations in terms of products of independent beta variates anderson 1958 ch 8 in conjunction with these characterizations we use skovgaard s approximation to construct two distinct

random shearing direction models for isotropic turbulent diffusion recently a rigorous renormalization theory for various scalar statistics has been developed for special modes of random advection diffusion involving random shear layer velocity fields with long range spatiotemporal correlations new random shearing direction models for isotropic turbulent diffusion are introduced here in these models the velocity field has the spatial second order statistics of an arbitrary prescribed stationary incompressible isotropic random field including long range spatial correlations with infrared divergence but the temporal correlations have finite range the explicit theory of renormalization for the mean and second order statistics is developed here with e the spectral parameter for e 4 and measuring the strength of the infrared divergence of the spatial spectrum the scalar mean statistics rigorously exhibit a phase transition from mean field behavior for e 2 to anomalous behavior for e with 2 e 4 as conjectured earlier by avellaneda and the author the universal inertial range renormalization for the second order scalar statistics exhibits a phase transition from a covariance with a gaussian functional form for e with e 2 to an explicit family with a non gaussian covariance for e with 2 e 4 these non gaussian distributions have tails that are broader than gaussian as e varies with 2 e 4 and behave for large values like exp c c x 4 e withc c an explicit constant also here the attractive general principle is formulated and proved that every steady stationary zero mean isotropic incompressible gaussian random velocity field is well approximated by a suitable superposition of random shear layers

saddlepoint expansions in finite population problems summary saddlepoint approximations are derived for the sample mean in the case where samples are obtained randomly without replacement from a finite population asymptotic properties are investigated the great accuracy is demonstrated by numerical examples extensions to more general statistics are discussed

saddlepoint approximations for estimating equations summary the saddlepoint method is used to approximate to the distribution of an estimator defined by an estimating equation two different approaches are available one of which is shown to be equivalent to the technique of field hampel 1982 two recent formulae for the tail probability due to lugannani rice 1980 and to robinson 1982 respectively which are uniformly accurate over the whole range of the estimator are compared numerically with the exact results and those computed by field hampel they are found to be of comparable accuracy while avoiding the use of numerical integration the most accurate is that of lugannani rice hampel 1974 introduced a new technique for approximating to the probability density of an estimator defined by an estimating equation it is an example of what he called small sample asymptotics where high accuracy is achieved for quite small sample sizes n even down to single figures in the original version it gave an approximation to the logarithmic derivative of the density function which was integrated numerically to get the density the distribution function was obtained by a second numerical integration which could then be used to renormalize both the density and the distribution function field hampel 1982 develop the technique in detail and compare its performance with that of other approximation methods as hampel had pointed out his approach is closely related to the saddlepoint method of daniels 1954 which was applied to sample means and ratios of means following the private communication referred to by field hampel 1982 p 31 it was realized that the first numerical integration was unnecessary and that hampel s approach could be shortened to give a direct approximation to the density which is in fact a saddlepoint approximation the purpose of the present paper is to extend the use of the saddlepoint method to estimating equations there are two distinct ways of doing this which lead to different approximations of similar accuracy one appears to be more convenient for approxi mating to tail probabilities by numerical integration and was used to compute the saddlepoint approximations quoted in field hampel s tables 1 and 2 the other gives the density directly in a form equivalent to their equation 4 3 which is then integrated

efficient bootstrap simulation on donne deux idees pour renforcer une simulation la premiere est d equilibrer les echantillons simules la seconde est de faire un usage explicite des approximations qui ne demandent pas de simulation

antithetic resampling for the bootstrap summary an antithetic variates method for the bootstrap is proposed and discussed it is applicable quite generally to bias estimation distribution function estimation and quantile estimation for example it is based on an antithetic permutation of the sample which amounts to ranking values of a certain function of the data once this has been done b uniform resampling operations may be immediately converted into 2b effective resampling operations yielding greater statistical efficiency than 2b totally independent resampling operations we show that antithetic resampling leads to positive nonnegligible gains in performance for the same level of labour when compared with ordinary uniform resampling

theory and multipliers of stable algebras the main theorem is that if a is a c algebra with a countable approximate identity consisting of projections then the unitary group of m a k is contractible this gives a realization via the index map of ko a as components in the set of fredholm operators on ha for a c algebra a let ha ai i ai e a and eai ai converges in norm then ha becomes a right a module under the action ai b aib with the a valued inner product ai b1 ea bi ha is a hilbert a module y ha is the c algebra of operators on ha that have an adjoint k ha is an ideal of y ha called the compact operators on ha set 11 2 definition 4 operators in ha invertible modulo 3f ha are called fredholm and have an index that takes its values in ko a the paper is divided into two parts in part 1 we show that index a ko a is an isomorphism where t denotes the set of path components of the set of fredholm operators in part 2 we show that the unitary group of m a k is contractible when a has a countable approximate identity consisting of projections as 2 ha m a k this in particular implies that the unitary group of s ha is connected when a is unital this latter result is used to prove the isomorphism mentioned above the results of this paper were announced in 14 see remark 2 5 2 this research is based on the author s doctoral work done at dalhousie university under the supervision of professor p a fillmore the author also wishes to thank w j phillips for many useful suggestions part 1 the index of a fredholm operator 1 1 we describe the index map from fredholm operators on hilbert c modules to the corresponding k groups the principal ingredients of this construction have been discussed by several authors kasparov 12 miscenko and fomenko 15 and pimsner popa and voiculescu 17 however as we shall need refinements and variations of these constructions we will give a detailed account of the index map throughout this paper k or y h will denote the c algebra of compact operators on a separable infinite dimensional hilbert space h for any two c algebras a and b a b will denote the completion of the algebraic tensor product aob in the spatial or minimal c norm see e g effros and lance 8 2 received by the editors july 28 1983 and in revised form december 27 1985 1980 mathematics subject classification 1985 revision primary 46l05 46m20 01987 american mathematical society 0002 9947 87 1 00 25 per page

on distributions of ratios a large number of exact inferential procedures in statistics and econometrics involve the sampling distribution of ratios of random variables if the denominator variable is positive then tail probabilities of the ratio can be expressed as those of a suitably defined difference of random variables if in addition the joint characteristic function of numerator and denominator is known then standard fourier inversion techniques can be used to reconstruct the distribution function from it most research in this field has been based on this correspondence but which breaks down when both numerator and denominator are supported on the entire real line the present manuscript derives inversion formulae and saddlepoint approximations that remain valid in this case and reduce to known results when the denominator is almost surely positive applications include the iv est imator of a structural parameter in a just identified equation

improvements of jackknife confidence limit methods summary edgeworth expansion is applied to studentized parameter estimates when the standard error has been computed by a jackknife method adjustments to the usual jackknife confidence limit formulae are obtained this approach is contrasted with a bootstrap approach in numerical illustrations for estimation of a ratio jackknife methods are nonparametric methods for estimating the bias and standard error of an estimate t approximate confidence limits for the estimand can be found by using a large sample normal approximation for t somewhat curiously little is known about possible improvements to the normal approximation in this context although improvements based on edgeworth expansions are familiar in other problems in this paper we show that edgeworth expansion methods can be applied in the jackknife context numerical results for a particular application raise the possibility that the resulting improvements of jackknife methods can be matched by a suitable use of bootstrap methods efron 1982

factorizations of invertible operators and k theory of c algebras let a be a unital c algebra we describe k skeleton factorizations of all invertible operators on a hilbert c module a in particular on h 12 with the fredholm index as an invariant we then outline the isomorphisms k 0 a 2k p 0 2k gl r p a and k 1 a 2k 1 p 0 2k 1 gl r p a for k 0 where p 0 denotes the class of all compact perturbations of a projection p in the infinite grassmann space gr 022 a and gl r p a stands for the group of all those invertible operators on h a essentially commuting with p

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

conditional bootstrap methods in the mean shift model summary bootstrap methods are not inherently conditional but they can be made so by appropri ate stratification of the simulated samples which bootstrap produces we show how stratification can work in a bootstrap analysis of mean shift in nile river flow data the results are compared with both parametric and semiparametric likelihood analyses the paper ends with some general remarks on conditional bootstraps bootstrap methods efron 1982 are simulation methods for assessing frequency properties of statistical analyses the simulation model being that which is fitted to the actual data often the simulation involves monte carlo rather than theoretical evaluation unfortunately the bootstrap as usually described is inherently unconditional rubin s 1981 discussion bears on this point however if the bootstrap simulation results can be properly stratified then it is possible to estimate relevant conditional probabilities stratification can sometimes be effected in the model itself as when fixing the values of sample sizes or explanatory variables otherwise one must run the unconditional bootstrap and stratify its output in the present paper we illustrate the latter situation in the analysis of a mean shift model section 2 describes the application and its analysis by parametric methods sections 3 and 4 produce alternative bootstrap analyses the first of which uses the stratification of an unconditional monte carlo simulation the second bootstrap analysis involves estimation of a likelihood function the final section contains some brief general remarks about conditional bootstraps

saddlepoint approximations in resampling methods summary saddlepoint approximations are shown to be easy to use and accurate in a variety of simple bootstrap and randomization applications examples include mean estimation ratio estimation two sample comparisons and autoregressive estimation

a new algorithm with plane waves and wavelets for random velocity fields with many spatial scales abstract a new monte carlo algorithm for constructing and sampling stationary isotropic gaussian random fields with power law energy spectrum infrared divergence and fractal self similar scaling is developed here the theoretical basis for this algorithm involves the fact that such a random field is well approximated by a superposition of random one dimensional plane waves involving a fixed finite number of directions in general each one dimensional plane wave is the sum of a random shear layer and a random acoustical wave these one dimensional random plane waves are then simulated by a wavelet monte carlo method for a single space variable developed recently by the authors the computational results reported in this paper demonstrate remarkable low variance and economical representation of such gaussian random fields through this new algorithm in particular the velocity structure function for an imcorepressible isotropic gaussian random field in two space dimensions with the kolmogoroff spectrum can be simulated accurately over 12 decades with only 100 realizations of the algorithm with the scaling exponent accurate to 1 1 and the constant prefactor accurate to 6 in fact the exponent of the velocity structure function can be computed over 12 decades within 3 3 with only 10 realizations furthermore only 46 592 active computational elements are utilized in each realization to achieve these results for 12 decades of scaling behavior

convergence of fourier wavelet models for gaussian random processes mean square convergence and convergence in probability of fourier wavelet models fwm of stationary gaussian random processes in the metric of banach space of continuously differentiable functions and in sobolev space are studied sufficient conditions for the convergence formulated in the frame of spectral functions are given it is shown that the given rates of convergence of fwm in the mean square obtained in the nikolskib besov classes cannot be improved

comparative analysis of multiscale gaussian random field simulation algorithms abstract we analyze and compare the efficiency and accuracy of two simulation methods for homogeneous random fields with multiscale resolution we consider in particular the fourier wavelet method and three variants of the randomization method a without any stratified sampling of wavenumber space b with stratified sampling of wavenumbers with equal energy subdivision c with stratified sampling with a logarithmically uniform subdivision we focus primarily on fractal gaussian random fields with kolmogorov type spectra previous work has shown that variants a and b of the randomization method are only able to generate a self similar structure function over three to four decades with reasonable computational effort by contrast variant c along with the fourier wavelet method is able to reproduce accurate self similar scaling of the structure function over a number of decades increasing linearly with computational effort for our examples we will show that nine decades can be reproduced we provide some conceptual and numerical comparison of the various cost contributions to each random field simulation method we find that when evaluating ensemble averaged quantities like the correlation and structure functions as well as some multi point statistical characteristics the randomization method can provide good accuracy with less cost than the fourier wavelet method the cost of the randomization method relative to the fourier wavelet method however appears to increase with the complexity of the random field statistics which are to be calculated accurately moreover the fourier wavelet method has better ergodic properties and hence becomes more efficient for the computation of spatial rather than ensemble averages which may be important in simulating the solutions to partial differential equations with random field coefficients

more efficient bootstrap computations abstract this article concerns computational methods for the bootstrap that are more efficient than the straightforward monte carlo methods usually used the bootstrap is considered in its simplest form in a one sample nonparametric problem where the goal is to estimate the bias or variance of some statistic by bootstrap sampling or to set approximate confidence intervals for a parameter of interest in terms of various percentiles of the bootstrap distribution the methods of this article can in favorable situations reduce the necessary number of bootstrap replications manyfold moreover simple diagnostics are available to see whether or not any particular case is accessible to these methods

an application of φ subgaussian technique to fourier analysis abstract we present an application to random fourier series for subgaussian random variables in particular we generalize previously known notions of dependence for random variables by introducing the concept of f manageable random variables and consider fourier series of f manageable subgaussian random variables gaussian series are then a particular case of the series considered in the paper conditions for the uniform convergence of such series in probability are established and a rate of convergence is investigated moreover the results are new even for the case of independent random variables

general saddlepoint approximations of marginal densities and tail probabilities saddlepoint approximations of marginal densities and tail probabilities of general nonlinear statistics are derived these are based on the expansion of the statistic up to the second order their accuracy is shown in a variety of examples including logit and probit models and rank estimators for regression

deformation of a projection in the multiplier algebra and projection lifting from the corona algebra of a non simple c algebra let x be a unit interval or a unit circle and let b be a p unital purely infinite simple c algebra such that its multiplier algebra m b has real rank zero then we determine necessary and sufficient conditions for a projection in the corona algebra of c x b to be liftable to a projection in the multiplier algebra this generalizes a result proved by l brown and the author brown and lee 2012 3 the main technical tools are divided into two parts the first part is borrowed from the author s result lee 2011 14 theorem 3 3 the second part is a proposition showing that we can produce a sub projection with an arbitrary rank which is prescribed as k theoretical data of a projection or a co projection in the multiplier algebra of c x b under a suitable infinite rank and co rank condition

saddlepoint approximations for marginal and conditional probabilities of transformed variables the lugannani and rice formula for tail areas in the univariate case has recently been extended to tail areas for marginal distributions and for conditional distributions in certain multivariate settings however the results on relative order of errors are given only formally or under the strong continuity assumptions necessary to obtain density approximations this paper attempts to give a unified treatment of these results for smooth transformations of multivariate means under weaker conditions appropriate to indirect edgeworth approximations for probabilities

relative efficiency of gaussian stochastic process sampling procedures various methods for sampling stationary gaussian stochastic processes are investigated and compared with an emphasis on applications to processes with power law energy spectra several approaches are considered including a riemann summation using left endpoints the use of random wave numbers to sample a the spectrum in proportion to the energy it contains and a combination of the two the fourier wavelet method of elliott et al is investigated and compared with other methods all of which are evaluated in terms of their ability to sample the stochastic process over a large number of decades for a given computational cost the fourier wavelet method has accuracy which increases linearly with the computational complexity while the accuracy of the other methods grows logarithmically for the kolmogorov spectrum a hybrid quadrature method is as efficient as the fourier wavelet method if no more than eight decades of accuracy are required the effectiveness of this hybrid method wanes when one samples fields whose energy spectrum decays more rapidly near the origin the fourier wavelet method has roughly the same behavior independently of the exponent of the power law the fourier wavelet method returns samples which are gaussian over the range of values where the structure function is well approximated by contrast multi point gaussianity may be lost at the smaller length scales when one uses methods with random wave numbers

model selection in estimating equations summary model selection is a necessary step in many practical regression analyses but for methods based on estimating equations such as the quasi likelihood and generalized estimating equation gee approaches there seem to be few well studied model selection techniques in this article we propose a new model selection criterion that minimizes the expected predictive bias epb of estimating equations a bootstrap smoothed cross validation bcv estimate of epb is presented and its performance is assessed via simulation for overdispersed generalized linear models for illustration the method is applied to a real data set taken from a study of the development of ewe embryos

approximate bayesian inference with the weighted likelihood bootstrap we introduce the weighted likelihood bootstrap wlb as a way to simulate approximately from a posterior distribution this method is often easy to implement requiring only an algorithm for calculating the maximum likelihood estimator such as iteratively reweighted least squares in the generic weighting scheme the wlb is first order correct under quite general conditions inaccuracies can be removed by using the wlb as a source of samples in the sampling importance resampling sir algorithm which also allows incorporation of particular prior information the sir adjusted wlb can be a competitive alternative to other integration methods in certain models asymptotic expansions elucidate the second order properties of the wlb which is a generalization of rubin s bayesian bootstrap the calculation of approximate bayes factors for model comparison is also considered we note that given a sample simulated from the posterior distribution the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values a modification of this estimator that avoids instability is also noted these methods provide simple ways of calculating approximate bayes factors and posterior model probabilities for a very wide class of models

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

empirical saddlepoint approximations for multivariate m estimators summary in this paper we investigate the use of the empirical distribution function in place of the underlying distribution function f to construct an empirical saddlepoint approximation to the density fn of a general multivariate m estimator we obtain an explicit form for the error term in the approximation investigate the effect of renormalizing the estimator carry out some numerical comparisons and discuss the regression problem the sampling distribution of a statistic is a basic tool for evaluating the properties of the statistic and for constructing inference procedures based on the statistic the density of the sampling distribution may be used to compute moments to develop approximations or to compute the distribution function and hence various tail prob abilities which can be used to construct tests or confidence intervals when the exact density is intractable it may be possible to approximate it by a normal density an edgeworth expansion or a saddlepoint approximation daniels 1954 the saddle point approximation is derived from an asymptotic expansion but is often very accurate even in quite small samples to highlight this property hampel 1973 described these and related techniques as small sample asymptotic approximations as can be seen from equation 3 later the main property of this approximation and its major advantage over edgeworth expansions is that it is always non negative and the relative error is uniformly of order n 1 for recent reviews see reid 1988 barndorff nielsen and cox 1989 and field and ronchetti 1990 the construction of these approximations typically requires knowledge of the underlying distribution however just as we can estimate the variance of a normal approxima tion or estimate the terms in an edgeworth expansion we can estimate the saddle point approximation in this paper we examine the properties of an estimate of the saddlepoint approximation our investigation is motivated by three considerations firstly the estimated approximation may be useful for comparing various estimators on a given data set if the sampling distribution is non normal at least in principle we need to compare the whole distribution and comparisons based on the estimated asymptotic variance alone may be misleading secondly the estimated approximation may be useful for evaluating the quality of simpler approximations including the normal approxima tion finally it may be useful for developing inference procedures methods like

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

on the number of bootstrap simulations required to construct a confidence interval abstract the purpose of this document is to make two points about the effect of the number of bootstrap simulations b on percentile t bootstrap confidence intervals the first point concerns coverage probability the second distance of the simulated critical point form the true critical point derived with b infinity in both cases the author has in mind applications to smooth statistics such as the studentized mean of a sample drawn from a continuous distribution he indicates the change that have to be made if the distribution of the statistic is not smooth additional keywords exponentive functions

bootstrapping survival times in stochastic systems by using saddlepoint approximations summary the single bootstrap is implemented by using a saddlepoint approximation to determine estimates for the survival and hazard functions of first passage times in complicated semi markov processes the double bootstrap is also implemented by resampling saddlepoint inversions and provides bca confidence bands for these functions confidence intervals for the mean and variance of first passage times are easily computed a new characterization of the asymptotic hazard rate for survival times is presented and leads to an indirect method for constructing its bootstrap confidence interval

the weighted log rank class of permutation tests p values and confidence intervals using saddlepoint methods test statistics from the weighted log rank class are commonly used to compare treatment with control when there is right censoring this paper uses saddlepoint methods to determine mid p values from the null permutation distributions of tests from the weighted log rank class analytical saddlepoint computations replace the permutation simulations and provide mid p values that are virtually exact for all practical purposes the speed of these saddlepoint computations makes it practicable to invert the weighted log rank tests to determine nominal 95 confidence intervals for the treatment effect with right censored data such analytical inversions lead to permutation confidence intervals that are easily computed and virtually identical to the exact intervals that would normally require massive amounts of simulation

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

balanced design of bootstrap simulations summary davison et al 1986 have shown that finite bootstrap simulations can be improved by forcing balance in the aggregate of simulated data sets their methods yield first order balance which principally affects bootstrap estimation of bias here we extend the methodology to second order balance which principally affects bootstrap estimation of variance the particular techniques involve latin square and balanced incomplete block designs numerical examples are given to illustrate both the positive and the negative features of the balanced simulations

distribution estimation using concomitants of order statistics with application to monte carlo simulation for the bootstrap we show that a simulation method suggestted by efron for approximating bootstrap distributions is closely related to tecniques based on concomitants of order statistics and develop its asymptotic properties from that viewpoint we prove that the method produces monte carlo approximations with variance and mean squared error decreasing like b 1 n 1 2 where b denotes the number of simulations and n equals the sample size therefore efron s method can be a competitor with techniques such as balanced resampling importance resampling and antithetic resampling where variance and mean squared error decrease like b 1 with no significant contribution from sample size

a fourier wavelet monte carlo method for fractal random fields a new hierarchical method for the monte carlo simulation of random fields called the fourier wavelet method is developed and applied to isotropic gaussian random fields with power law spectral density functions this technique is based upon the orthogonal decomposition of the fourier stochastic integral representation of the field using wavelets the meyer wavelet is used here because its rapid decay properties allow for a very compact representation of the field the fourier wavelet method is shown to be straightforward to implement given the nature of the necessary precomputations and the run time calculations and yields comparable results with scaling behavior over as many decades as the physical space multiwavelet methods developed recently by two of the authors however the fourier wavelet method developed here is more flexible and in particular applies to anisotropic spectra generated through solutions of differential equations simulation results using this new technique and the well known nonhierarchical simulation technique the randomization method are given and compared for both a simple shear layer model problem as well as a two dimensional isotropic gaussian random field the fourier wavelet method results are more accurate for statistical quantities depending on moments higher than order 2 in addition to showing a quite smooth decay to zero on the scales smaller than the scaling regime when compared with the randomization method results the only situation in which the nonhierarchical randomization method is more computationally efficient occurs when no more than four decades of scaling behavior are needed and the statistical quantities of interest depend only on second moments

homotopy classification of homogeneous projections in the corona algebra of c x b for a graph x abstract in this paper we consider certain projections in the corona algebra of c x b for x a directed graph and b a stable c algebra of which the multiplier algebra has real rank zero assuming a kind of homogeneity on the projections we characterize when two such projections are homotopy equivalent as a result for a simple graph we show that such projections are always homotopic

small sample confidence intervals abstract in this article we present a technique for constructing one or two sided confidence intervals which are second order correct in terms of coverage for either parametric or nonparametric problems the construction is valid in the presence of nuisance parameters the situation we consider is this there are p parameters and we want a confidence interval for some function of them possibly one of the parameters itself the p parameters are estimated by m estimates which means they are obtained as a solution of a system of equations maximum likelihood estimates are included as a special case the essential intermediate result given in equation 3 4 says that the estimated parameter of interest can be written as a mean up to order o p 1 n the representation 3 4 is attained when is a smooth function of the parameters and is the solution of a well behaved system of equations we avoid the use of pivots and strive to obtain accurate coverage confidence intervals are construc

small sample asymptotic distributions of m estimators of location summary asymptotic formulae for the distribution of m estimators i e maximum likelihood type estimators of location including the arithmetic mean are derived which numerical studies show to give relative errors for densities and tail areas of the order of magnitude of 1 down to sample sizes 3 and 4 even in the extreme tails the paper is the continuation of earlier work by the second author and is also closely related to daniels s work on the saddlepoint approximation the method consists in expanding the derivative of the logarithm of the unstandardized density of the estimator in powers of 1 n at each point using recentring by means of conjugate distributions this method yields a unified point of view for the comparison of other asymptotic methods namely saddlepoint method edgeworth expansion and large deviations approach which are also compared numerically

nonlinear approximation of random functions given an orthonormal basis and a certain class x of vectors in a hilbert space h consider the following nonlinear approximation process approach a vector x in x by keeping only its n largest coordinates and let n go to infinity in this paper we study the accuracy of this process in the case where h l 2 i and we use either the trigonometric system or a wavelet basis to expand this space the class of function that we are interested in is described by a stochastic process we focus on the case of piecewise stationary processes that describe functions which are smooth except at isolated points we show that the nonlinear wavelet approximation is optimal in terms of mean square error and that this optimality is lost either by using the trigonometric system or by using any type of linear approximation method i e keeping the n first coordinates the main motivation of this work is the search for a suitable mathematical model to study the compression of images and of certain types of signals

variance stabilization for a scalar parameter summary we present a variance stabilizing transformation for inference about a scalar parameter that is estimated by a function of a multivariate m estimator the transformation proposed is automatic computationally simple and can be applied quite generally though it is based on an intuitive notion and entirely empirical the transformation is shown to have an appropriate justification in providing variance stabilization when viewed from both parametric and nonparametric perspectives further the transformation repairs deficiencies of existing methods for variance stabilization the transformation proposed is illustrated in a range of examples and its effectiveness to yield confidence limits having low coverage error is demonstrated in a numerical example

importance sampling for bootstrap confidence intervals abstract the use of importance sampling methods to substantially reduce the amount of resampling necessary for the construction of nonparametric bootstrap confidence intervals is investigated the classical importance sampling method of hammersley and handscomb 1964 is modified to apply to the estimation of quantiles based on this method a resampling procedure is introduced that recenters the bootstrap distribution of a robust estimator of location to produce estimated confidence limits that are much more accurate for a given amount of resampling than those obtained by the usual bootstrap method efron 1982 the required recentering is accomplished with a suitable exponential tilting similar to that used in another connection by field and hampel 1982 this importance sampling procedure is used to produce bootstrap confidence intervals for location based on a class of estimators that includes symmetric m estimators these interval estimates are asymptotically optimal in a certain sense simulati

an empirical study of the ratio estimator and estimators of its variance abstract this paper reports results from an empirical study of the ratio estimator for a finite population total from each of six real populations 1 000 simple random samples 1 000 restricted random samples and three nonrandom samples of size 32 are drawn performance of the ratio estimator and of five estimators of its variance is compared with theoretical results generated using a prediction superpopulation models and b probability sampling distributions the results presented graphically show that theory based on prediction models can reveal relationships that are essential in making inferences but that are concealed in probability sampling analyses

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12