automatic bandwidth choice and confidence intervals in nonparametric regression in the present paper we combine the issues of bandwidth choice and construction of confidence intervals in nonparametric regression main emphasis is put on fully data driven methods we modify the n consistent bandwidth selector of hahrdle hall and marron such that it is appropriate for heteroscedastic data and we show how one can optimally choose the bandwidth g of the pilot estimator m g then we consider classical confidence intervals based on kernel estimators with data driven bandwidths and compare their coverage accuracy we propose a method to put undersmoothing with a data driven bandwidth into practice and show that this procedure outperforms explicit bias correction

directed walks and vertex models renormalization multicriticality and beyond we discuss the exact renormalization of interacting directed random walks drw using dimensional regularization and the exact results for the multicritical points we discuss how these exact results of the drw problem can give exact critical exponents ofthe ferroelectric five vertex model in all dimensions the multicritical drw s lead to a new class of vertex models whose multicritical exponents can be computed exactly a random vertex model and its connection to the drw problem in a random medium is also discussed

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

optimal rates of convergence for deconvolving a density abstract suppose that the sum of two independent random variables x and z is observed where z denotes measurement error and has a known distribution and where the unknown density f of x is to be estimated one application is the estimation of a prior density for a sequence of location parameters a second application arises in the errors in variables problem for nonlinear and generalized linear models when one attempts to model the distribution of the true but unobservable covariates this article shows that if z is normally distributed and f has k bounded derivatives then the fastest attainable convergence rate of any nonparametric estimator of f is only log n k 2 therefore deconvolution with normal errors may not be a practical proposition other error distributions are also treated stefanski carroll 1987a estimators achieve the optimal rates the results given have versions for multiplicative errors where they imply that even optimal rates are exceptionally slow

vicious walks with long range interactions the asymptotic behavior of the survival or reunion probability of vicious walks with short range interactions is generally well studied in many realistic processes however walks interact with a long ranged potential that decays in d dimensions with distance r as r ensuremath d ensuremath ensuremath sigma we employ methods of renormalized field theory to study the effect of such long range interactions we calculate the exponents describing the decay of the survival probability for all values of parameters ensuremath sigma and d to first order in the double expansion in ensuremath epsilon 2 ensuremath d and ensuremath delta 2 ensuremath d ensuremath ensuremath sigma we show that there are several regions in the ensuremath sigma text ensuremath d plane corresponding to different scalings for survival and reunion probabilities furthermore we calculate the leading logarithmic corrections

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

rate of convergence for the wild bootstrap in nonparametric regression this paper concerns the distributions used to construct confidence intervals for the regression function in a nonparametric setup some rates of convergence for the normal limit its plug in approach and the wild bootstrap are obtained conditionally on the explanatory variable x and also unconditionally the bound found for the wild bootstrap approximation is slightly better by a factor n 1 45 than the bounds given by the plug in approach or the clt for the conditional probability on the contrary the unconditional bounds present a different feature the rate obtained when approximating by the clt improves the one given by the plug in approach by a factor of n8 45 while this last one performs better than the wild bootstrap approximation and the corresponding ratio is n1 45 it should be mentioned that these two sequences especially the last one tend to zero at an extremely slow rate

directed abelian algebras and their application to stochastic models with each directed acyclic graph this includes some d dimensional lattices one can associate some abelian algebras that we call directed abelian algebras daas on each site of the graph one attaches a generator of the algebra these algebras depend on several parameters and are semisimple using any daa one can define a family of hamiltonians which give the continuous time evolution of a stochastic process the calculation of the spectra and ground state wave functions stationary state probability distributions is an easy algebraic exercise if one considers d dimensional lattices and chooses hamiltonians linear in the generators in finite size scaling the hamiltonian spectrum is gapless with a critical dynamic exponent z d one possible application of the daa is to sandpile models in the paper we present this application considering one and two dimensional lattices in the one dimensional case when the daa conserves the number of particles the avalanches belong to the random walker universality class critical exponent ensuremath sigma ensuremath tau 3 2 we study the local density of particles inside large avalanches showing a depletion of particles at the source of the avalanche and an enrichment at its end in two dimensions we did extensive monte carlo simulations and found ensuremath sigma ensuremath tau 1 780 ifmmode pm else textpm fi 0 005

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

an improved generalized conjugate residual squared algorithm suitable for distributed parallel computing in this paper based on gcrs algorithm in zhang and zhao 2010 and the ideas in gu et al 2007 we present an improved generalized conjugate residual squared igcrs algorithm that is designed for distributed parallel environments the new improved algorithm reduces two global synchronization points to one by changing the computation sequence in the gcrs algorithm in such a way that all inner products per iteration are independent so that communication time required for inner products can be overlapped with useful computation theoretical analysis and numerical comparison of isoefficiency analysis show that the igcrs method has better parallelism and scalability than the gcrs method and the parallel performance can be improved by a factor of about 2 finally some numerical experiments clearly show that the igcrs method can achieve better parallel performance with a higher scalability than the gcrs method and the improvement percentage of communication is up to 52 19 averagely which meets our theoretical analysis

bias corrected confidence bands in nonparametric regression summary bias corrected confidence bands for general nonparametric regression models are considered we use local polynomial fitting to construct the confidence bands and combine the cross validation method and the plug in method to select the bandwidths related asymptotic results are obtained our simulations show that confidence bands constructed by local polynomial fitting have much better coverage than those constructed by using the nadaraya watson estimator the results are also applicable to nonparametric autoregressive time series models

exact results for vicious walker models of domain walls non intersecting or vicious random walker models in one dimension can be interpreted as models of domain walls in two dimensions three problems pertaining to vicious walker models are solved the first is the exact evaluation of the partition function for the random turns model of vicious walkers on a lattice in this model at each tick of the clock a randomly chosen walker must move one step to the left or one step to the right the second problem is the calculation of the mean spacing between walls in terms of the chemical potential for a brownian motion model of continuous domain walls in a strip while the final problem solved is the calculation of the correlation between defects which occur when two domain walls meet and end without crossing the whole system

phase transition and critical behavior in a model of organized criticality we study a model of organized criticality where a single avalanche propagates through an a priori static i e organized sandpile configuration the latter is chosen according to an i i d distribution from a borel probability measure on 0 1 the avalanche dynamics is driven by a standard toppling rule however we simplify the geometry by placing the problem on a directed rooted tree as our main result we characterize which are critical in the sense that they do not admit an infinite avalanche but exhibit a power law decay of avalanche sizes our analysis reveals close connections to directed site percolation both in the characterization of criticality and in the values of the critical exponents

a ridge parameter approach to deconvolution kernel methods for deconvolution have attractive features and prevail in the literature however they have disadvantages which include the fact that they are usually suitable only for cases where the error distribution is infinitely supported and its characteristic function does not ever vanish even in these settings optimal convergence rates are achieved by kernel estimators only when the kernel is chosen to adapt to the unknown smoothness of the target distribution in this paper we suggest alternative ridge methods not involving kernels in any way we show that ridge methods a do not require the assumption that the error distribution characteristic function is nonvanishing b adapt themselves remarkably well to the smoothness of the target density with the result that the degree of smoothness does not need to be directly estimated and c give optimal convergence rates in a broad range of settings

exact fixed point for directed random walks the authors have shown that the renormalization group fixed point for mutually avoiding directed random walks in d 1 dimensions is exactly 2 pi in where in 2 d with no higher order terms in in since the whole calculation can be carried out to all orders exactly the model is also significant from a pedagogical point of view

loop mediated crossover in the three dimensional ising model the authors establish the existence of crossover from a three or any dimensional ising model to the anisotropic five vertex model by suppressing closed loops in the isomorphic dimer problem the crossover exponent is determined exactly using a renormalization group approach

confidence bands in nonparametric regression abstract new bias corrected confidence bands are proposed for nonparametric kernal regression these bands are constructed using only a kernel estimator of the regression curve and its data selected bandwidth they are shown to have asymptotically correct coverage properties and to behave well in a small sample study one consequence of the large sample developments is that bonferroni type bands for the regression curve at the design points also have conservative asymptotic coverage behavior with no bias correction

inference on distribution functions under measurement error abstract this paper is concerned with inference on the cumulative distribution function cdf f x in the classical measurement error model x x we consider the case where the density of the measurement error is unknown and estimated by repeated measurements and show validity of a bootstrap approximation for the distribution of the deviation in the sup norm between the deconvolution cdf estimator and f x we allow the density of to be ordinary or super smooth we also provide several theoretical results on the bootstrap and asymptotic gumbel approximations of the sup norm deviation for the case where the density of is known our approximation results are applicable to various contexts such as confidence bands for f x and its quantiles and for performing various cdf based tests such as goodness of fit tests for parametric models of x two sample homogeneity tests and tests for stochastic dominance simulation and real data examples illustrate satisfactory performance of the proposed methods

a matrix crs iterative method for solving a class of coupled sylvester transpose matrix equations abstract in this paper we apply kronecker product and vectorization operator to extend the conjugate residual squared crs method for solving a class of coupled sylvester transpose matrix equations some numerical examples are given to compare the accuracy and efficiency of the new matrix iterative method with other methods presented in the literature numerical results validate that the proposed method can be much more efficient than some existing methods

effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density tion and so bias must be corrected by other means when the bootstrap is used to construct confidence intervals for a probability density bias may either be estimated explicitly or allowed for by undersmoothing the curve estimator which of these two approaches is to be preferred in the present paper we address this question from the viewpoint of coverage accuracy assuming a given number of derivatives of the unknown density we conclude that the simpler undersmoothing method is more efficacious undersmoothing also has advantages from the standpoint of minimizing interval width we derive formulae for bandwidths which are optimal in terms of coverage accuracy and also give formulae for the coverage error which results from using those bandwidths 1 introduction the bootstrap can be a notoriously poor estimator of bias in problems of nonparametric curve estimation e g 5 and this fact should be taken into account when using bootstrap methods to construct confidence intervals bias is usually significant when a curve estimator is constructed so as to minimize mean squared error but decreases as the amount of statistical smoothing is reduced therefore one way of alleviating bias is to smooth the curve estimator less than would be appropriate for point estimation as opposed to interval estimation a statistician must either estimate bias explicitly or allow for bias by undersmoothing the curve estimator which of these two methods is to be preferred in this paper we address this problem in the case of nonparametric density estimation and from the viewpoint of coverage error we use the percentile t approach to confidence interval construction and show that in a sense which we shall make precise two paragraphs below the undersmoothing method produces confidence intervals with greater coverage accuracy than those obtained by explicit bias correction l l f x nh h for the density estimator based on kernel k and bandwidth h let b and o2

exact solution of the totally asymmetric oslo model recently it has been found pruessner g and jensen h j 2003 phys rev lett 91 244303 that a totally asymmetric variant of the oslo model christensen k et al 1996 phys rev lett 77 107 represents the entire universality class of the oslo model with anisotropy the totally asymmetric model can be solved without scaling assumptions by finding recursively the eigenvectors of the markov matrix which can be suitably modified to produce the moment generating function of the relevant observable this method should be applicable to many other stochastic processes

additivity property and emergence of power laws in nonequilibrium steady states we show that an equilibriumlike additivity property can remarkably lead to power law distributions observed frequently in a wide class of out of equilibrium systems the additivity property can determine the full scaling form of the distribution functions and the associated exponents the asymptotic behavior of these distributions is solely governed by branch cut singularity in the variance of subsystem mass to substantiate these claims we explicitly calculate using the additivity property subsystem mass distributions in a wide class of previously studied mass aggregation models as well as in their variants these results could help in the thermodynamic characterization of nonequilibrium critical phenomena

monte carlo methods for nonparametric regression with heteroscedastic measurement error nonparametric regression is a fundamental problem in statistics but challenging when the independent variable is measured with error among the first approaches was an extension of deconvoluting kernel density estimators for homescedastic measurement error the main contribution of this article is to propose a new simulation based nonparametric regression estimator for the heteroscedastic measurement error case similar to some earlier proposals our estimator is built on principles underlying deconvoluting kernel density estimators however the proposed estimation procedure uses monte carlo methods for estimating nonlinear functions of a normal mean which is different than any previous estimator we show that the estimator has desirable operating characteristics in both large and small samples and apply the method to a study of benzene exposure in chinese factory workers

directed polymers in a random medium abstract the problem of a polymer in a random medium can be formulated as follows consider a regular lattice with on each bond ij a random energy ij the energy e w of a self avoiding walk w on this lattice is the sum of the energies of all the bonds visited by the walk then the problem is to compute the partition function z w exp e w t several recent results will be presented 1 the mean field version of this problem can be solved exactly and one observes a low temperature phase similar to the spin glass phase 2 for finite dimensional lattices one can derive bounds on the transition temperature in high enough dimension 3 for hierarchical lattices the problem can be reduced to finding stable laws when one combines random variables in a non linear way for example z z 1 z 2 z 3 z 4

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

a parallel version of qmrcgstab method for large linear systems in distributed parallel environments in this paper a parallel qmrcgstab method pqmrcgstab method for solving large sparse linear systems with unsymmetrical coefficient matrices is proposed for distributed parallel environments the method reduces four global synchronization points to one by reconstructing qmrcgstab method it combines the elements of numerical stability with the characters of design of parallel algorithms the cost is only a little increased computation performance analysis shows that pqmrcgstab method has better parallelism and scalability than qmrcgstab method numerical experiments show the effectiveness of our method

confidence intervals for nonparametric curve estimates numerous nonparametric regression methods exist that yield consistent estimators of function curves often one is also interested in constructing confidence intervals for the unknown function when a function estimate is based on a single global smoothing parameter the resulting confidence intervals may hold their desired confidence level 1 on average but because bias in nonparametric estimation is not uniform they do not hold the desired level uniformly at all design points most research in this area has focused on mean squared error properties of the estimator for example mise itself a global measure in addition measures like mise are one step removed from the practical issue of coverage probability recent work that focuses on coverage probability has considered only coverage in an average sense ignoring the important issue of uniformity of coverage across the design space to deal with this problem a new estimator is developed which uses a local cross validation criterion lcv to determi

on deconvolution with repeated measurements in a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known although in many applications it is unrealistic to make this assumption the problem is often insoluble without it however if additional data are available then it is possible to estimate consistently the unknown error density data are seldom available directly on the transformation but repeated or replicated measurements increasingly are becoming available such data consist of intrinsic values that are measured several times with errors that are generally independent working in this setting we treat the nonparametric deconvolution problems of density estimation with observation errors and regression with errors in variables we show that even if the number of repeated measurements is quite small it is possible for modified kernel estimators to achieve the same level of performance they would if the error distribution were known indeed density and regression estimators can be constructed from replicated data so that they have the same first order properties as conventional estimators in the known error case without any replication but with sample size equal to the sum of the numbers of replicates practical methods for constructing estimators with these properties are suggested involving empirical rules for smoothing parameter choice

an improved bi conjugate residual algorithm suitable for distributed parallel computing an improved bi conjugate residual bicr method ibicr method in brief for solving large sparse linear systems with unsymmetrical coefficient matrices was proposed for distributed parallel environments the method reduced two global synchronization points to one by reconstructing bicr method and all inner products per iteration were independent and communication time required for inner product can be overlapped efficiently with computation time of vector updates it combines the elements of numerical stability with the characters of design of parallel algorithms the cost is only a little increased computation performance and isoefficiency analysis shows that ibicr method has better parallelism and scalability than bicr method numerical experiments show that the parallel performance can be improved by a factor of about 2 we compared also ibicr with ibicg and bicr with bicg methods the results show that bicr and ibicr methods convergent at the same number of iteration and they convergent faster than bicg and ibicg methods respectively furthermore ibicr method conquers the vibration of residual norm of ibicg method

moments of vicious walkers and mobius graph expansions a system of brownian motions in one dimension all started from the origin and conditioned never to collide with each other in a given finite time interval 0 t is studied the spatial distribution of such vicious walkers can be described by using the repulsive eigenvalue statistics of random hermitian matrices and it was shown that the present vicious walker model exhibits a transition from the gaussian unitary ensemble gue statistics to the gaussian orthogonal ensemble goe statistics as the time t goes on from 0 to t in the present paper we characterize this gue to goe transition by presenting the graphical expansion formula for the moments of positions of vicious walkers in the gue limit stackrel ensuremath rightarrow t 0 only the ribbon graphs contribute and the problem is reduced to the classification of orientable surfaces by genus following the time evolution of the vicious walkers however the graphs with twisted ribbons called m obius graphs increase their contribution to our expansion formula and we have to deal with the topology of nonorientable surfaces application of the recent exact result of dynamical correlation functions yields closed expressions for the coefficients in the m obius expansion using the stirling numbers of the first kind

non parametric confidence bands in deconvolution density estimation summary uniform confidence bands for densities f via non parametric kernel estimates were first constructed by bickel and rosenblatt in this paper this is extended to confidence bands in the deconvolution problem g f for an ordinary smooth error density under certain regularity conditions we obtain asymptotic uniform confidence bands based on the asymptotic distribution of the maximal deviation l distance between a deconvolution kernel estimator and f further consistency of the simple non parametric bootstrap is proved for our theoretical developments the bias is simply corrected by choosing an undersmoothing bandwidth for practical purposes we propose a new data driven bandwidth selector that is based on heuristic arguments which aims at minimizing the l distance between and f although not constructed explicitly to undersmooth the estimator a simulation study reveals that the bandwidth selector suggested performs well in finite samples in terms of both area and coverage probability of the resulting confidence bands finally the methodology is applied to measurements of the metallicity of local f and g dwarf stars our results confirm the g dwarf problem i e the lack of metal poor g dwarfs relative to predictions from closed box models of stellar formation

scaling of fluctuation for directed polymers with random interaction using a finite size scaling form for reunion probability we show numerically the existence of a binding unbinding transition for directed polymers with random interaction the cases studied are a1 two chains in 1 1 dimensions a2 two chains in 2 1 dimensions and b three chains in 1 1 dimensions a similar finite size scaling form for fluctuation establishes a disorder induced transition with identical exponents for cases a2 and b the length scale exponents in all the three cases are in agreement with previous exact renormalization group results

on convergence rates of suprema it is shown that the convergence rate of suprema of stationary gaussian and related processes such as processes defined by the empirical distribution function is logarithmically slow even if the rates are to be uniform over as few as three points it is proved that the bootstrap approximation provides a substantial improvement

using simex for smoothing parameter choice in errors in variables problems simex methods are attractive for solving curve estimation problems in errors in variables regression using parametric or semiparametric techniques however nonparametric approaches are generally of quite a different type being based on for example kernels local linear modeling ridging orthogonal series or splines all of these techniques involve the challenging and not well studied issue of empirical smoothing parameter choice we show that simex can be used effectively for selecting smoothing parameters when applying nonparametric methods to errors in variable regression in particular we suggest an approach based on multiple error inflated or remeasured data sets and extrapolation

bootstrap wild bootstrap and asymptotic normality we show for an i i d sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works an asymptotic approach is used where everything may depend onn the result is extended to the case of independent but not necessarily identically distributed random variables furthermore it is shown that wild bootstrap works under the same conditions as bootstrap

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

sandpile models and random walkers on finite lattices abelian sandpile models both deterministic such as the bak tang wiesenfeld btw model p bak c tang and k wiesenfeld phys rev lett 59 381 1987 and stochastic such as the manna model s s manna j phys a 24 l363 1991 are studied on finite square lattices with open boundaries the avalanche size distribution p l n is calculated for a range of system sizes l the first few moments of this distribution are evaluated numerically and their dependence on the system size is examined the sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries it is shown that the conservation law provides an interesting connection between the sandpile models and random walk models using this connection it is shown that the average avalanche sizes l for the btw and manna models are equal to each other and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size l this is in spite of the fact that the sandpile models with deterministic btw and stochastic manna toppling rules exhibit different critical exponents indicating that they belong to different universality classes

uniform confidence bands in deconvolution with unknown error distribution abstract this paper develops a method to construct uniform confidence bands in deconvolution when the error distribution is unknown simulation studies demonstrate the performance of the multiplier bootstrap confidence band in the finite sample we apply our method to the outer continental shelf ocs auction data and draw confidence bands for the density of common values of mineral rights on oil and gas tracts we also present an application of our main theoretical result specifically to additive fixed effect panel data models and we draw confidence bands for the density of the total factor productivity in a manufacturing industry in chile

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

fluctuation effects and multiscaling of the reaction diffusion front for a b to oe we consider the properties of the diffusion controlled reaction a b to oe in the steady state where fixed currents of a and b particles are maintained at opposite edges of the system using renormalization group methods we explicitly calculate the asymptotic forms of the reaction front and particle densities as expansions in jd 1 mod x mod d 1 1 where j are the equal applied currents and d the equal diffusion constants for the asymptotic densities of the minority species we find in addition to the expected exponential decay fluctuation induced power law tails which for d 2 we find similar power laws with mu d 3 but with non universal coefficients logarithmic corrections occur in d 2 these results imply that in the time dependent case with segregated initial conditions the moments integral mod x mod qr x t dx fail to satisfy simple scaling for q mu 1 finally it is shown that the fluctuation induced wandering of the position of the reaction front centre may be neglected for large enough systems

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

a design adaptive local polynomial estimator for the errors in variables problem local polynomial estimators are popular techniques for nonparametric regression estimation and have received great attention in the literature their simplest version the local constant estimator can be easily extended to the errors in variables context by exploiting its similarity with the deconvolution kernel density estimator the generalization of the higher order versions of the estimator however is not straightforward and has remained an open problem for the last 15 years we propose an innovative local polynomial estimator of any order in the errors in variables context derive its design adaptive asymptotic properties and study its finite sample performance on simulated examples we provide not only a solution to a long standing open problem but also provide methodological contributions to error in variable regression including local polynomial estimation of derivative functions

estimation of distributions moments and quantiles in deconvolution problems when using the bootstrap in the presence of measurement error we must first estimate the target distribution function we cannot directly resample since we do not have a sample from the target these and other considerations motivate the development of estimators of distributions and of related quantities such as moments and quantiles in errors in variables settings we show that such estimators have curious and unexpected properties for example if the distributions of the variable of interest w say and of the observation error are both centered at zero then the rate of convergence of an estimator of the distribution function of w can be slower at the origin than away from the origin this is an intrinsic characteristic of the problem not a quirk of particular estimators the property holds true for optimal estimators

bootstrapping in nonparametric regression local adaptive smoothing and confidence bands abstract the operation of the bootstrap in the context of nonparametric regression is considered bootstrap samples are taken from estimated residuals to study the distribution of a suitably recentered kernel estimator the application of this principle to the problem of local adaptive choice of bandwidth and to the construction of confidence bands is investigated and compared with a direct method based on asymptotic means and variances the technique of the bootstrap is to replace any occurrence of the unknown distribution in the definition of the statistical function of interest by the empirical distribution function of the observed errors in a regression context these errors are not directly observed although their role can be played by the residuals from the fitted model in this article the fitted model is a kernel nonparametric regression estimator since nonparametric smoothing is involved an additional difficulty is created by the bias incurred in smoothing this bias however can be estimated

bootstrap simultaneous error bars for nonparametric regression simultaneous error bars are constructed for nonparametric kernel estimates of regression functions the method is based on the bootstrap where resampling is done from a suitably estimated residual distribution the error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints applications to an economic problem are given and comparison to both pointwise and bonferroni type bars is presented through a simulation study

simulation extrapolation estimation in parametric measurement error models we describe a simulation based method of inference for parametric measurement error models in which the measurement error variance is known or at least well estimated the method entails adding add

estimation of conditional prevalence from group testing data with missing covariates we consider estimating the conditional prevalence of a disease from data pooled according to the group testing mechanism consistent estimators have been proposed in the literature but they rely o

families of vicious walkers we consider a generalization of the vicious walker problem in which n random walkers in rd are grouped into p families using field theoretic renormalization group methods we calculate the asymptotic behaviour of the probability that no pairs of walkers from different families have met up to time t for d 2 this is constant but for d 2 it decays as a power t which we compute to e2 in an expansion in e 2 d the second order term depends on the ratios of the diffusivities of the different families in two dimensions we find a logarithmic decay ln t and compute exactly

nonparametric regression with errors in variables the effect of errors in variables in nonparametric regression estimation is examined to account for errors in covariates deconvolution is involved in the construction of a new class of kernel estimators it is shown that optimal local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution in fact there are two types of rates of convergence according to whether the error is ordinary smooth or super smooth it is also shown that these results hold uniformly over a class of joint distributions of the response and the covariate which is rich enough for many practical applications furthermore to achieve optimality we show that the convergence rates of all possible estimators have a lower bound possessed by the kernel estimators

on bootstrap confidence intervals in nonparametric regression priate in this paper we point out some of the theoretical advantages of pivoting they include a reduction in the error of the bootstrap distribution function estimate from n 1 2 to n lh 1 2 where h denotes bandwidth and a reduction in coverage error of confidence intervals from either n l 1 2h 1 2 or n 1 2h1 2 depending on which nonpivotal method is used to n 1 several comparisons are drawn with the case of nonparametric density estimation where a pivotal approach also reduces errors associated with confidence intervals but where the orders of magnitude of the respective errors are quite different from their counterparts for nonparametric regression

on the optimal rates of convergence for nonparametric deconvolution problems deconvolution problems arise in a variety of situations in statistics an interesting problem is to estimate the density f of a random variable x based on n i i d observations from y x varepsilon where varepsilon is a measurement error with a known distribution in this paper the effect of errors in variables of nonparametric deconvolution is examined insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions the smoother the harder in fact there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth it is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators

uniform confidence bands for nonparametric errors in variables regression abstract this paper develops a method to construct uniform confidence bands for a nonparametric regression function where a predictor variable is subject to a measurement error we allow for the distribution of the measurement error to be unknown but assume the availability of validation data or repeated measurements on the latent predictor variable the proposed confidence band builds on the deconvolution kernel estimation and a novel application of the multiplier bootstrap method we establish asymptotic validity of the proposed confidence band to our knowledge this is the first paper to derive asymptotically valid uniform confidence bands for nonparametric errors in variables regression

exactly solved model of self organized critical phenomena we define a variant of the model of bak tang and wiesenfeld of self organized critial behavior by introducing a preferred direction we characterize the critical state and by establishing equivalence to a voter model determine the critical exponents exactly in arbitrary dimension d the upper critical dimension for this model is three in two dimensions the model is equivalent to an earlier solved special case of directed percolation

reunion and survival of interacting walkers the reunion and survival probabilities of p random walkers in d dimensions with a mutual repulsive interaction are formulated via appropriate partition functions of directed polymers the exponents that describe the decay of these probabilities with length are obtained through renormalization group theory to o mathrm ensuremath epsilon 2 where ensuremath epsilon 2 d the distribution function and the probability of n out of p walkers meeting are also discussed to first order the distribution function is a gaussian one modified by an anomalous exponent of the length of the polymer n the procedure is generalized to multicritical many body interactions for these multicritical cases the exponents are obtained to second order in the relevant ensuremath epsilon s at the upper critical dimension of the interaction there is a logarithmic correction other than the gaussian exponent an interesting consequence is a logarithmic correction for one dimensional walkers with a three body repulsive interaction

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

algebraic aspects of abelian sandpile models the abelian sandpile models feature a finite abelian group g generated by the operators corresponding to particle addition at various sites we study the canonical decomposition of g as a product of cyclic groups g z d1 z d2 z d3 z dg where g is the least number of generators of g and di is a multiple of di 1 the structure of g is determined in terms of the toppling matrix delta we construct scalar functions linear in the height variables of the pile that are invariant under toppling at any site these invariants provide convenient coordinates to label the recurrent configurations of the sandpile for an l l square lattice we show that g l in this case we observe that the system has non trivial symmetries transcending the obvious symmetries of the square namely those coming from the action of the cyclotomic galois group gall of the 2 l 1 th roots of unity which operates on the set of eigenvalues of h these eigenvalues are algebraic integers the product of which is the order mod g mod with the help of gall we are able to group the eigenvalues into certain subsets the products of which are separately integers and thus obtain an explicit factorization of mod g mod we also use gall to define other simpler sets of toppling invariants

bootstrap procedures under some non i i d models it is shown in this article that the classical i i d bootstrap remains a valid procedure for estimating the sampling distributions of certain symmetric estimators of location as long as the random observations are independently drawn from distributions with essentially a common location this may be viewed as a robust property of the classical i i d bootstrap also included is a study of the second order properties of a different bootstrap procedure proposed by wu in the context of heteroscedasticity in regression