modelling of covariance structures in generalised estimating equations for longitudinal data summary when used for modelling longitudinal data generalised estimating equations specify a working structure for the within subject covariance matrices aiming to produce efficient parameter estimators however misspecification of the working covariance structure may lead to a large loss of efficiency of the estimators of the mean parameters in this paper we propose an approach for joint modelling of the mean and covariance structures of longitudinal data within the framework of generalised estimating equations the resulting estimators for the mean and covariance parameters are shown to be consistent and asymptotically normally distributed real data analysis and simulation studies show that the proposed approach yields efficient estimators for both the mean and covariance parameters

monotonic quantile regression with bernstein polynomials for stochastic simulation quantile regression is an important tool to determine the quality level of service product and operation systems via stochastic simulation it is frequently known that the quantiles of the output distribution are monotonic functions of certain inputs to the simulation model because there is typically high variability in estimation of tail quantiles it can be valuable to incorporate this information in quantile modeling however the existing literature on monotone quantile regression with multiple inputs is sparse in this article we propose a class of monotonic regression models which consists of functional analysis of variance fanova decomposition components modeled with bernstein polynomial bases for estimating quantiles as a function of multiple inputs the polynomial degrees of the bases for the model and the fanova components included in the model are selected by a greedy algorithm real examples demonstrate the advantages of incorporating the monotonicity assumption in quantile regression a

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

bootstrap confidence bands for regression curves and their derivatives confidence bands for regression curves and their first p derivatives are obtained via local pth order polynomial estimation the method allows for multiparameter local likelihood estimation as well as other unbiased estimating equations as an alternative to the confidence bands obtained by asymptotic distribution theory we also study smoothed bootstrap confidence bands simulations illustrate the finite sample properties of the methodology

bootstrap prediction intervals for autoregression abstract the nonparametric bootstrap is applied to the problem of prediction in autoregression let y t t 0 1 2 be a stationary autoregressive process of known order p ar p given a realization of the series up to time t y 1 y 2 y t a 100 prediction interval for y t k is desired standard forecasting techniques which assume that the error sequence of the process y t is gaussian rely on the fact that the conditional distribution of y t k given the data is gaussian as well as a nonparametric alternative the bootstrap provides an estimate of the conditional distribution of y t k the method is similar to other applications of the bootstrap for linear models because the residuals are resampled the proposed methodology represents a different approach since an alternative representation for ar p series is used allowing for bootstrap replicates generated backward in time it follows that the resulting replicates all have the same conditionally fixed values at the end

quadratic inference functions for varying coefficient models with longitudinal data nonparametric smoothing methods are used to model longitudinal data but the challenge remains to incorporate correlation into nonparametric estimation procedures in this article we propose an efficient estimation procedure for varying coefficient models for longitudinal data the proposed procedure can easily take into account correlation within subjects and deal directly with both continuous and discrete response longitudinal data under the framework of generalized linear models the proposed approach yields a more efficient estimator than the generalized estimation equation approach when the working correlation is misspecified for varying coefficient models it is often of interest to test whether coefficient functions are time varying or time invariant we propose a unified and efficient nonparametric hypothesis testing procedure and further demonstrate that the resulting test statistics have an asymptotic chi squared distribution in addition the goodness of fit test is applied to test whether the model assumption is satisfied the corresponding test is also useful for choosing basis functions and the number of knots for regression spline models in conjunction with the model selection criterion we evaluate the finite sample performance of the proposed procedures with monte carlo simulation studies the proposed methodology is illustrated by the analysis of an acquired immune deficiency syndrome aids data set

bootstrap choice of bandwidth for density estimation abstract a bootstrap based choice of bandwidth for kernel density estimation is introduced the method works by estimating the integrated mean squared error imse for any given bandwidth and then minimizing over all bandwidths a straightforward application of the bootstrap method to estimate the imse fails because it does not capture the bias component a smoothed bootstrap method based on an initial density estimate is described that solves this problem it is possible to construct pointwise and simultaneous confidence intervals for the density the simulation study compares cross validation and the bootstrap method over a wide range of densities a long tailed a short tailed an asymmetric and a bimodal among others the bootstrap method uniformly outperforms cross validation the accuracy of the constructed confidence bands improves as the sample size increases

examples of abelian surfaces with everywhere good reduction we describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction these examples provide evidence for the eichler shimura conjecture for hilbert modular forms over a real quadratic field several of the examples also support a conjecture of brumer and kramer on abelian varieties associated to siegel modular forms with paramodular level structures

content corrected tolerance limits based on the bootstrap this article considers k factor tolerance limits in which the p content is data dependent the proposed content corrected method is based on a bootstrap estimate of the given content and is shown to be robust in the sense of preserving the confidence coefficient for a variety of distributions for small or moderate sample sizes to all large samples we develop the theoretical basis for the asymptotic results and prove both asymptotic normality and weak consistency of the corresponding bootstrap statistic the method is shown to be asymptotically accurate with respect to the confidence coefficient for a wide class of distributions several simulations and two examples show the advantage of this method when the normality of the data cannot be assumed

the exact bootstrap mean and variance of an l estimator exact analytic expressions for the bootstrap mean and variance of any l estimator are obtained thus eliminating the error due to bootstrap resampling the expressions follow from the direct calculation of the bootstrap mean vector and covariance matrix of the whole set of order statistics by using these expressions recommendations can be made about the appropriateness of bootstrap estimation under given conditions

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

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

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

model diagnostic tests for selecting informative correlation structure in correlated data in the generalized method of moments approach to longitudinal data analysis unbiased estimating functions can be constructed to incorporate both the marginal mean and the correlation structure of the data increasing the number of parameters in the correlation structure corresponds to increasing the number of estimating functions thus building a correlation model is equivalent to selecting estimating functions this paper proposes a chi squared test to choose informative unbiased estimating functions we show that this methodology is useful for identifying which source of correlation it is important to incorporate when there are multiple possible sources of correlation this method can also be applied to determine the optimal working correlation for the generalized estimating equation approach

constrained smoothing of histograms and scatterplots with simulated annealing histogram and scatterplot models are often required for statistical inference in the field of petroleum engineering stochastic simulation algorithms require among other statistics a model for the histogram of the petrophysical attribute porosity permeability being simulated often this model is taken to be the deciustered distribution of the sample data when there are many data say greater than l 000 this histogram may be reasonably informed most often however the sample histogram shows multiple sawtoothlike spikes that are not representative of the entire population the sample histogram must be smoothed a simulated annealing based procedure is proposed for smoothing one variable univariate histograms and two variable scatterplots bivariate histograms the smoothed histograms are constrained to the sample mean variance specified quantiles and a measure of smoothness in the bivariate case the distribution must be consistent with both marginal histograms and can be additionally cons

strong approximation of density estimators from weakly dependent observations by density estimators from independent observations we derive an approximation of a density estimator based on weakly dependent random vectors by a density estimator built from independent random vectors we construct on a sufficiently rich probability space such a pairing of the random variables of both experiments that the set of observations x 1 x n from the time series model is nearly the same as the set of observations y 1 y n from the i i d model with a high probability all sets of the form x 1 x n delta y 1 y n boolean and a 1 b 1 x x a d b d contain no more than o n 1 2 pi b i a i 1 log n elements respectively although this does not imply very much for parametric problems it has important implications in nonparametric statistics it yields a strong approximation of a kernel estimator of the stationary density by a kernel density estimator in the i i d model moreover it is shown that such a strong approximation is also valid for the standard bootstrap and the smoothed bootstrap using these results we derive simultaneous confidence bands as well as supremum type nonparametric tests based on reasoning for the i i d model

building adaptive estimating equations when inverse of covariance estimation is difficult summary to construct an optimal estimating function by weighting a set of score functions we must either know or estimate consistently the covariance matrix for the individual scores in problems with high dimensional correlated data the estimated covariance matrix could be unreliable the smallest eigenvalues of the covariance matrix will be the most important for weighting the estimating equations but in high dimensions these will be poorly determined generalized estimating equations introduced the idea of a working correlation to minimize such problems however it can be difficult to specify the working correlation model correctly we develop an adaptive estimating equation method which requires no working correlation assumptions this methodology relies on finding a reliable approximation to the inverse of the variance matrix in the quasi likelihood equations we apply a multivariate generalization of the conjugate gradient method to find estimating equations that preserve the information well at fixed low dimensions this approach is particularly useful when the estimator of the covariance matrix is singular or close to singular or impossible to invert owing to its large size

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

bootstrap of residual processes in regression to smooth or not to smooth summaryin this paper we consider regression models with centred errors independent of the covariates given independent and identically distributed data and given an estimator of the regression function which can be parametric or nonparametric in nature we estimate the distribution of the error term by the empirical distribution of estimated residuals to approximate the distribution of this estimator koul lahiri 1994 and neumeyer 2009 proposed bootstrap procedures based on smoothing the residuals before drawing bootstrap samples so far it has been an open question as to whether a classical nonsmooth residual bootstrap is asymptotically valid in this context here we solve this open problem and show that the nonsmooth residual bootstrap is consistent we illustrate the theoretical result by means of simulations which demonstrate the accuracy of this bootstrap procedure for various models testing procedures and sample sizes

quadratic mathbb q curves units and hecke l values we show that if k is a quadratic field and if there exists a quadratic mathbb q curve e k of prime degree n satisfying weak conditions then any unit u of o k satisfies a congruence u r equiv 1 pmod n where r mathrm g c d n 1 12 if k is imaginary quadratic we prove a congruence modulo a divisor of n between an algebraic hecke character tilde psi and roughly speaking the elliptic curve we show that this divisor then occurs in a critical value l tilde psi 2 by constructing a non zero element in a selmer group and applying a theorem of kato

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

testing whether independent treatment groups have equal medians the paper suggests new methods for comparing the medians corresponding to independent treatment groups the procedures are based on the harrell davis estimator in conjunction with a slight modification and extension of the bootstrap calibration technique suggested by loh alternatives to the harrell davis estimator are briefly discussed for the special case of two treatment groups the proposed procedure always had more power than the fligner rust solution as well as the procedure examined by wilcox and charlin included is an illustration using real data that comparing medians rather than means can yield a substantially different conclusion as to whether two distributions differ in terms of some measure of central location

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

a smoothed em approach to indirect estimation problems with particular reference to stereology and emission tomography there are many practical problems where the observed data are not drawn directly from the density g of real interest but rather from another distribution derived from g by the application of an integral operator the estimation of g then entails both statistical and numerical difficulties a natural statistical approach is by maximum likelihood conveniently implemented using the em algorithm but this provides unsatisfactory reconstructions of g in this paper we modify the maximum likelihood em approach by introducing a simple smoothing step at each em iteration in our experience this algorithm converges in relatively few iterations to good estimates of g that do not depend on the choice of starting configuration some theoretical background is given that relates this smoothed em algorithm to a maximum penalized likelihood approach two applications are considered in detail the first is the classical stereology problem of determining particle size distributions from data collected on a plane section through a composite medium the second concerns the recovery of the structure of a section of the human body from external observations obtained by positron emission tomography for this problem we also suggest several technical improvements on existing methodology

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

a note on bootstrapping the correlation coefficient summary smoothed bootstrap estimation of the sampling standard deviation of the variance stabilized correlation coefficient is reconsidered an approximation to the mean squared error of the bootstrap estimator is obtained and an empirical procedure for choosing the degree of smoothing in the bootstrap estimation is presented performance of the procedure is examined in a simulation study silverman young 1987 discuss use of the smoothed bootstrap for estimation of the sampling standard deviation of the variance stabilized correlation coefficient in bivariate samples and examine from a theoretical viewpoint circumstances when some smoothing is advantageous in the bootstrap estimation in the present paper the analytical tools used in that investigation are applied to estimate the mean squared error of the bootstrap estimator and to define a procedure for empirical choice of the degree of smoothing the procedure is illustrated in a small simulation study

non parametric estimates of overlap kernel densities provide accurate non parametric estimates of the overlapping coefficient or the proportion of similar responses psr in two populations non parametric estimates avoid strong assumptions on the shape of the populations such as normality or equal variance and possess sampling variation approaching that of parametric estimates we obtain accurate standard error estimates by bootstrap resampling we illustrate the practical use of these methods in two examples and use simulations to explore the properties of the estimators under various sampling situations

kernel smoothing to improve bootstrap confidence intervals summary some studies of the bootstrap have assessed the effect of smoothing the estimated distribution that is resampled a process usually known as the smoothed bootstrap generally the smoothed distribution for resampling is a kernel estimate and is often rescaled to retain certain characteristics of the empirical distribution typically the effect of such smoothing has been measured in terms of the mean squared error of bootstrap point estimates the reports of these previous investigations have not been encouraging about the efficacy of smoothing in this paper the effect of resampling a kernel smoothed distribution is evaluated through expansions for the coverage of bootstrap percentile confidence intervals it is shown that under the smooth function model proper bandwidth selection can accomplish a first order correction for the one sided percentile method with the objective of reducing the coverage error the appropriate bandwidth for one sided intervals converges at a rate of n 4 rather than the familiar n 115 for kernel density estimation applications of this same approach to bootstrap t and two sided intervals yield optimal bandwidths of order n 112 these bandwidths depend on moments of the smooth function model and not on derivatives of the underlying density of the data the relationship of this smoothing method to both the accelerated bias correction and the bootstrap t methods provides some insight into the connections between three quite distinct approximate confidence intervals

a comparison of related density based minimum divergence estimators this paper compares the minimum divergence estimator of basu et al 1998 to a competing minimum divergence estimator which turns out to be equivalent to a method proposed from a different perspective by windham 1995 both methods can be applied for any parametric model and contain maximum likelihood as a special case efficiencies are compared under model conditions and robustness properties are studied overall the two methods are found to perform quite similarly some relatively small advantages of the former method over the latter are identified

tools for the symbolic computation of asymptotic expansions summary this paper describes a collection of procedures for the systematic computation of asymptotic expansions that are common in statistical theory and practice expansions of functions of sums of independent and identically distributed random variables the procedures permit the expansion of maximum likelihood estimates the associated deviance or drop in likelihood and more general functions of random variables with distributions involving one or more parameters the procedures are illustrated with examples involving general and specific laws much of statistical theory and practice is based on asymptotic expansions many programs are available to assist in the numerical evaluation of such expansions but there is a need for computational tools to assist in their derivation and symbolic evaluation heller 1991 shows how symbolic calculation may be used in a wide variety of statistical problems kendall 1988 1990 gives procedures for the symbolic computation of expressions in the analysis of the diffusion of euclidean shape silverman and young 1987 use computer algebra to evaluate criteria on which the decision to smooth a bootstrap distribution is based young and daniels 1990 apply symbolic computation to evaluate expressions in the assessment of bootstrap bias venables 1985 utilizes symbolic computation heavily to obtain expansions of maximum marginal likelihood estimates most notably fisher s a statistic barndorff nielsen and blasild 1986 describe procedures for the numerical calculation of bartlett factors in cases where cumulants of the likelihood function may be specified most of these references involve the evaluation of complicated formulae in particular cases and not the derivation of the formulae themselves here we give general procedures for both the derivation of formulae and their evaluation in specific cases the derivation of asymptotic expansions is typically a simple but laborious task consider for example the calculation of the expectation of the likelihood ratio test statistic for a one parameter family to order 1 n this may be accomplished in general

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

alternative smoothed bootstraps bootstrap estimation of simple population functionals is considered the smoothed bootstrap is compared with an alternative estimation procedure based on resampling ideas of kendall and kendall specific examples of the estimation problem are considered showing that the competing estimators may display different qualitative behaviours

modeling persistent trends in distributions abstractwe present a nonparametric framework to model a short sequence of probability distributions that vary both due to underlying effects of sequential progression and confounding noise to distinguish between these two types of variation and estimate the sequential progression effects our approach leverages an assumption that these effects follow a persistent trend this work is motivated by the recent rise of single cell rna sequencing experiments over a brief time course which aim to identify genes relevant to the progression of a particular biological process across diverse cell populations while classical statistical tools focus on scalar response regression or order agnostic differences between distributions it is desirable in this setting to consider both the full distributions as well as the structure imposed by their ordering we introduce a new regression model for ordinal covariates where responses are univariate distributions and the underlying relationship reflects consistent changes i

the bootstrap to smooth or not to smooth summary the bootstrap and smoothed bootstrap are considered as alternative methods of estimating properties of unknown distributions such as the sampling error of parameter estimates criteria are developed for determining whether it is advantageous to use the smoothed bootstrap rather than the standard bootstrap key steps in the argument leading to these criteria include the study of the estimation of linear functionals of distributions and the approximation of general functionals by linear functionals consideration of an example the estimation of the standard error in the variance stabilized sample correlation coefficient elucidates previously published simulation results and also illustrates the use of computer algebraic manipulation as a useful technique in asymptotic statistics finally the various approximations used are vindicated by a simulation study