on inverse problems with unknown operators consider a problem of recovery of a smooth function signal image f spl isin spl fscr spl isin l sub 2 0 1 sup d passed through an unknown filter and then contaminated by a noise a typical model discussed in the paper is described by a stochastic differential equation dy sub f sup spl epsi t hf t dt spl epsi dw t t spl isin 0 1 sup d spl epsi 0 where h is a linear operator modeling the filter and w is a brownian motion sheet modeling a noise the aim is to recover f with asymptotically as spl epsi spl rarr 0 minimax mean integrated squared error traditionally the problem is studied under the assumption that the operator h is known then the ill posedness of the problem is the main concern in this paper a more complicated and more realistic case is considered where the operator is unknown instead a training set of n pairs e sub l y e sub l sup spl sigma l 1 2 n where e sub l is an orthonormal system in l sub 2 and y e sub l sup spl sigma denote the solutions of stochastic differential equations of the above type with f e sub l and spl epsi spl sigma is available an optimal in a minimax sense over considered operators and signals data driven recovery of the signal is suggested the influence of spl epsi spl sigma and n on the recovery is thoroughly studied in particular we discuss an interesting case of a larger noise during the training and present formulas for threshold levels for n beyond which no improvement in recovery of input signals occurs we also discuss the case where h is an unknown perturbation of a known operator we describe a class of perturbations for which the accuracy of recovery of the signal is asymptotically the same up to a constant as in the case of precisely known operator

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

nonasymptotic behavior of a common eulerian approximation for transport in random velocity fields lagrangian methods of analyzing solute transport in random velocity fields deal with the motion of indivisible solute particles through such fields whereas eulerian methods treat the solute concentration c as the dependent variable in a stochastic mass balance equation both approaches lead to nonlinear stochastic differential equations which are difficult to solve analytically without approximation a particular eulerian approximation which has become popular in recent years derives from the premise that in mildly fluctuating velocity fields terms involving the products of fluctuating quantities can be disregarded in comparison to terms which involve only one such quantity this leads to a non fickian low order approximation for the ensemble mean concentration c where the dispersive flux is not proportional to c but is given instead by a convolution integral the spatial moments of c based on this approximation are shown to be in conflict with those obtained from a first order lagrangian analysis we explore this apparent contradiction and conclude that the reason lies in the nonasymptotic nature of the eulerian approximation whereby terms neglected are of the same order as terms retained

on the ni x integral function and its application to the airy s non homogeneous equation in this article we discuss a recently introduced function ni x to which we will refer as the nield kuznetsov function this function is attractive in the solution of inhomogeneous airy s equation we derive and document some elementary properties of this function and outline its application to airy s equation subject to initial conditions we introduce another function ki x that arises in connection with ni x when solving airy s equation with a variable forcing function in appendix a we derive a number of properties of both ni x and ki x their integral representation ascending and asymptotic series representations we develop iterative formulae for computing all derivatives of these functions and formulae for computing the values of the derivatives at x 0 an interesting finding is the type of differential equations ni x satisfies in particular it poses itself as a solution to langer s comparison equation

a fourier approach to nonparametric deconvolution of a density estimate we consider the problem of constructing a nonparametric estimate of a probability density function h from independent random samples of observations from densities a and f when a represents the convolution of h and f our approach is based on truncated fourier inversion in which the truncation point plays the role of a smoothing parameter we derive the asymptotic mean integrated squared error of the estimate and use this formula to suggest a simple practical method for choosing the truncation point from the data strikingly when the smoothing parameter is chosen in this way then in many circumstances the estimator behaves to first order as though the true f were known

estimation of the density of regression errors estimation of the density of regression errors is a fundamental issue in regression analysis and it is typically explored via a parametric approach this article uses a nonparametric approach with the mean integrated squared error mise criterion it solves a long standing problem formulated two decades ago by mark pinsker about estimation of a nonparametric error density in a nonparametric regression setting with the accuracy of an oracle that knows the underlying regression errors the solution implies that under a mild assumption on the differentiability of the design density and regression function the mise of a data driven error density estimator attains minimax rates and sharp constants known for the case of directly observed regression errors the result holds for error densities with finite and infinite supports some extensions of this result for more general heteroscedastic models with possibly dependent errors and predictors are also obtained in the latter case the marginal error density is estimated in all considered cases a blockwise shrinking efromovich pinsker density estimate based on plugged in residuals is used the obtained results imply a theoretical justification of a customary practice in applied regression analysis to consider residuals as proxies for underlying regression errors numerical and real examples are presented and discussed and the s plus software is available

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

higher order effects on flow and transport in randomly heterogeneous porous media advective solute transport in nonuniform geologic media is generally nonlocal and non fickian neuman 1993 cushman and ginn 1993 in statistically homogeneous log conductivity fields under uniform mean flow the transport is expected to become asymptotically local and fickian at late time during the earlier preasymptotic regime macrodispersivity a measure of the rate at which a plume spreads is expected to vary with solute residence time a first order linear in the natural log hydraulic conductivity variance 2 analysis of this variation has been performed by dagan 1984 1987 1988 he found that when local dispersion is neglected the longitudinal macrodispersivity increases monotonically from zero toward a constant asymptote however the transverse macrodispersivity first increases from zero to a peak value then decreases monotonically toward zero the first order asymptotic analyses by winter 1982 gelhar and axness 1983 and winter et al 1984 also yield zero transverse macrodispersivity when local dispersion is disregarded

deconvolution of a distribution function abstract we consider the estimation of a distribution function when observations from this distribution are contaminated by measurement error the unknown distribution is modeled as a mixture of a finite number of known distributions model parameters can be estimated and confidence intervals constructed using well known likelihood theory we show that it is also possible to apply this approach to estimation of a unimodal distribution an application is presented using data from a dietary survey simulation results are given to indicate the performance of the estimators and the confidence interval procedures

deconvolution density estimation on so n this paper develops nonparametric deconvolution density estimation over so n the group of n x n orthogonal matrices of determinant 1 the methodology is to use the group and manifold structures to adapt the euclidean deconvolution techniques to this lie group environment this is achieved by employing the theory of group representations explicit to so n general consistency results are obtained with specific rates of convergence achieved under sufficient smoothness conditions application to empirical bayes prior estimation and inference is also discussed

an eulerian scheme for the second order approximation of subsurface transport moments the moments of a conservative tracer cloud migrating in a mean uniform flow field are estimated using an operator approximation scheme results are presented for the second third and fourth central moments in the mean flow direction it is assumed that the spatially variable flow field and therefore the tracer migration problem itself is amenable to a probabilistic description the effects of local dispersion on cloud migration are neglected in this study variation in the flow field is assumed to be the result of spatial variation in the hydraulic conductivity spatial variation in porosity is assumed negligible the operator approximation scheme as implemented in this study is second order correct which requires a second order correct approximation of the velocity field correlation structure because estimation of the velocity correlation structure is decidedly the most difficult aspect of second order analysis an ad hoc extension of the imperfectly stratified approximation developed earlier is implemented for this purpose the first order approximation resulting from the operator expansion scheme is equivalent to small perturbation eulerian results presented earlier naff 1990 1992 the infinite order approximation resulting from this scheme is equivalent to the exponential operator results obtained by van kampen 1976

adaptively local one dimensional subproblems with application to a deconvolution problem in this paper a method for finding global minimax lower bounds is introduced the idea is to adjust automatically the direction of a local one dimensional subproblem at each location to the nearly hardest one and to use locally the difficulty of the one dimensional subproblem this method has the advantages of being easily implemented and understood the lower bound is then applied to nonparametric deconvolution to obtain the optimal rates of convergence for estimating a whole function other applications are also addressed

a note on nonparametric estimation of linear functionals precise asymptotic descriptions of the minimax affine risks and bias variance tradeoffs for estimating linear functionals are given for a broad class of moduli the results are complemented by illustrative examples including one where it is possible to construct an estimator which is fully adaptive over a range of parameter spaces

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

mixtures of exponential distributions arbitrary nonparametric mixtures of exponential and weibull fixed shape distributions are considered as possible models for a lifetime distribution a characterization of such distributions is given by the well known characterization of laplace transforms the maximum likelihood estimate of the mixing distribution is investigated and found to be supported on a finite number of points it is shown to be unique and weakly convergent to the true mixing measure with probability one a practical algorithm for computing the maximum likelihood estimate is described its performance is briefly discussed and some illustrative examples given

the geometry of mixture likelihoods a general theory in this paper certain fundamental properties of the maximum likelihood estimator of a mixing distribution are shown to be geometric properties of the likelihood set the existence support size likelihood equations and uniqueness of the estimator are revealed to be directly related to the properties of the convex hull of the likelihood set and the support hyperplanes of that hull it is shown using geometric techniques that the estimator exists under quite general conditions with a support size no larger than the number of distinct observations analysis of the convex dual of the likelihood set leads to a dual maximization problem a convergent algorithm is described the defining equations for the estimator are compared with the usual parametric likelihood equations for finite mixtures sufficient conditions for uniqueness are given part ii will deal with a special theory for exponential family mixtures

empirical geometry of multivariate data a deconvolution approach let y j j 1 n be independent observations in r m m 1 with common distribution q suppose that y j x j j j 1 n where x j j j 1 n are independent x j j 1 n have common distribution p and j j 1 n have common distribution so that q p the problem is to recover hidden geometric structure of the support of p based on the independent observations y j assuming that the distribution of the errors is known deconvolution statistical estimates of the fractal dimension and the hierarchical cluster tree of the support that converge with exponential rates are suggested moreover the exponential rates of convergence hold for adaptive versions of the estimates even in the case of normal noise j with unknown covariance in the case of the dimension estimation though the exponential rate holds only when the set of all possible values of the dimension is finite e g when the dimension is known to be integer if this set is infinite the optimal convergence rate of the estimator becomes very slow typically logarithmic even when there is no noise in the observations

on higher order corrections to the flow velocity covariance tensor second order log fluctuating conductivity variance 2 corrections to the head and velocity covariance functions are derived for a lognormal stationary hydraulic conductivity field the fourier transform method proposed by deng et al 1993 is used extensively to obtain numerical estimates of these functions for an exponential log fluctuating conductivity covariance it is shown that the velocity covariance is insensitive to second order corrections in the head field the velocity covariance on the other hand is highly sensitive to second order corrections in the velocity when the log fluctuating conductivity variance approaches unity a closed expression is derived for a second order correction to the velocity variance when there is no second order correction to the head field the longitudinal second order correction to the velocity variance is 0 4 2 different from the first order approximation in isotropic media 1 5 2 different in a highly stratified formation and no different when the ratio of vertical to horizontal integral scales approaches infinity the second order corrections to the horizontal and vertical transverse velocity variances are 2 2 different from the first order approximations for both isotropic and anisotropic systems

on nonoscillating integrals for computing inhomogeneous airy functions integral representations are considered of solutions of the inhomogeneous airy differential equation w z w pm1 pi the solutions of these equations are also known as scorer functions certain functional relations for these functions are used to confine the discussion to one function and to a certain sector in the complex plane by using steepest descent methods from asymptotics the standard integral representations of the scorer functions are modified in order to obtain non oscillating integrals for complex values of z in this way stable representations for numerical evaluations of the functions are obtained the methods are illustrated with numerical results

self consistent four point closure for transport in steady random flows ensemble averaging of advection dispersion equations describing transport of a passive scalar in incompressible random velocity fields requires a closure approximation commonly used approaches such as the direct interaction approximation and large eddy simulations as well as equivalent renormalization schemes employ so called two point or one loop closures these approaches have proven to be adequate for transport in zero mean unbiased time dependent random velocity fields with increasing accuracy for decreasing temporal coherence in the opposite limit of steady velocity fields with finite bias however these schemes fail to predict effective transport properties both quantitatively and qualitatively leading to an obvious inconsistency for transverse dispersion in two spatial dimensions for this case two point closures predict that macroscopic transverse dispersion increases as the square root of the disorder variance while it has been proven rigorously that there is no disorder induced contribution to macroscopic transverse dispersion for purely advective transport furthermore two point closures significantly underestimate the disorder induced contribution to longitudinal dispersion we derive a four point closure for stochastically averaged transport equations that goes beyond classical one loop schemes and demonstrate that it is exact for transverse dispersion and correctly predicts an increase of the longitudinal disorder induced dispersion coefficient with the square of the variance of the strong disorder the predicted values of asymptotic longitudinal dispersion coefficients are consistent with those obtained via monte carlo random walk simulations

log density estimation in linear inverse problems we estimate a probability density function p which is related by a linear operator k to a density function q in sequences of regular exponential families based on a random sample from q in this paper deconvolution and positron emission tomography are considered the logarithm of the density function is approximated by basis functions consisting of singular functions of k while direct maximum likelihood or minimum kullback leibler density estimation in exponential families selects the parameters to match the moments of the basis functions to the sample moments in the inverse problem the moment of each singular function is related to a corresponding moment of the direct problem by a factor given by a singular value a of k thus an appropriate analogue of the maximum likelihood estimate is obtained by matching moments with respect to p to 1 a times the empirical moments associated with the sample from q bounds on the kullback leibler distance between the true density and the estimators are obtained and rates of convergence are established for log density functions having a measure of smoothness the density estimator converges to the unknown density in the kullback leibler sense and in the l2 sense at a rate determined not only by the order of smoothness of the log density and the dimension of data but also by the decay rate of the singular values of the operator a minimax lower bound for deconvolution is provided under certain conditions numerical examples using simulated data are provided to illustrate the finite sample performance of the proposed method for deconvolution and positron emission tomography

a fast fourier transform stochastic analysis of the contaminant transport problem a three dimensional stochastic analysis of the contaminant transport problem is developed in the spirit of naff 1990 the new derivation is more general and simpler than previous analysis the fast fourier transformation is used extensively to obtain numerical estimates of the mean concentration and various spatial moments data from both the borden and cape cod experiments are used to test the methodology results are comparable to results obtained by other methods and to the experiments themselves

proportion of non zero normal means universal oracle equivalences and uniformly consistent estimators d0 we propose a general approach to construct the universal oracle equivalence of the proportion the construction is based on the underlying characteristic function the oracle equivalence reduces the problem of estimating the proportion to the problem of estimating the oracle which is relatively easier to handle in fact the oracle equivalence naturally yields a family of estimators for the proportion which are consistent under mild conditions uniformly across a wide class of parameters the approach compares favourably with recent works by meinshausen and rice and genovese and wasserman in particular the consistency is proved for an unprecedentedly broad class of situations the class is almost the largest that can be hoped for without further constraints on the model we also discuss various extensions of the approach report results on simulation experiments and make connections between the approach and several recent procedures in large scale multiple testing including the false discovery rate approach and the local false discovery rate approach

nonparametric estimation of a mixing density via the kernel method abstract editor s note this manuscript was first submitted in may 1996 and a first review with a request for revision was completed in september 1996 costas goutis lost his life in a scuba diving accident in july 1996 the associate editor jianqing fan graciously offered to carry out the required revisions this article presents a method for estimating the latent distribution of a mixture model the method is motivated by the standard kernel density estimation but instead of using an estimate based on the unobserved latent variables it takes the expectation with respect to their distribution conditional on the data the resulting estimator is continuous and hence appropriate when there is a strong belief in the continuity of the mixing distribution an asymptotic justification is presented and the associated computational problems are discussed the method is illustrated by an example of fission track analysis in which the density of the age of crystals is estimated

an exact nonlinear correction to transverse macrodispersivity for transport in heterogeneous formations the random two dimensional and steady in time eulerian velocity field is assumed to be given it pertains to flow through heterogeneous porous media of spatially variable permeability advective transport of a passive scalar is modeled by the lagrangian approach by seeking the statistical moments of the trajectory of a tagged particle the trajectories variances and the associated dispersion coefficients are evaluated by a perturbation expansion in the velocity variances and by corrsin s conjecture first order results obtained in the past show that for flow through porous formations of log permeability two point covariance of finite integral scale the transverse dispersion coefficient tends asymptotically to zero in contrast corrsin s conjecture leads to a finite dispersion coefficient quadratic in the velocity variance to investigate this effect an exact nonlinear correction to the transverse dispersion coefficient is derived for a normal velocity field it is shown that this term tends to zero asymptotically in contrast with the result based on corrsin s conjecture furthermore an illustrative computation shows the transport related nonlinear correction to be small at any time

low order approximations in deconvolution and regression with errors in variables summary we suggest two new methods which are applicable to both deconvolution and regression with errors in explanatory variables for nonparametric inference the two approaches involve kernel or orthogonal series methods they are based on defining a low order approximation to the problem at hand and proceed by constructing relatively accurate estimators of that quantity rather than attempting to estimate the true target functions consistently of course both techniques could be employed to construct consistent estimators but in many contexts of importance e g those where the errors are gaussian consistency is from a practical viewpoint an unattainable goal we rephrase the problem in a form where an explicit interpretable low order approximation is available the information that we require about the error distribution the error in variables distribution in the case of regression is only in the form of low order moments and so is readily obtainable by a rudimentary analysis of indirect measurements of errors e g through repeated measurements in particular we do not need to estimate a function such as a characteristic function which expresses detailed properties of the error distribution this feature of our methods coupled with the fact that all our estimators are explicitly defined in terms of readily computable averages means that the methods are particularly economical in computing time

on deconvolution of distribution functions the subject of this paper is the problem of nonparametric estimation of a continuous distribution function from observations with measurement errors we study minimax complexity of this problem when unknown distribution has a density belonging to the sobolev class and the error density is ordinary smooth we develop rate optimal estimators based on direct inversion of empirical characteristic function we also derive minimax affine estimators of the distribution function which are given by an explicit convex optimization problem adaptive versions of these estimators are proposed and some numerical results demonstrating good practical behavior of the developed procedures are presented

minimax and adaptive estimation of the wigner function in quantum homodyne tomography with noisy data we estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared quantum systems the state is represented through the wigner function a density on r 2 which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics the effect of the losses due to detection inefficiencies which are always present in a real experiment is the addition to the tomographic data of independent gaussian noise we construct a kernel estimator for the wigner function and prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions for the l2 risk we compute the upper bounds of a truncated kernel estimator over the same classes restricted to functions with sub gaussian asymptotic behaviour we construct adaptive estimators i e which do not depend on the smoothness parameters and prove that in some set ups they attain the minimax rates for the corresponding smoothness class

computation of inhomogeneous airy functions abstract the inhomogeneous airy functions gi x and hi x appear in various problems efficient chebyshev expansions are derived for these functions with coefficients given for up to twenty decimal places accuracy the use of generalised transforms significantly reduces the number of coefficients

exact constants for pointwise adaptive estimation under the riesz transform we consider nonparametric estimation of a multivariate function and its partial derivatives at a fixed point when the riesz transform of the function is observed in gaussian white noise we assume that the unknown function belongs to some sobolev class and construct an estimation procedure which achieves the best asymptotic minimax risk when the smoothness of the function is unknown

deconvolution with unknown error distribution we consider the problem of estimating a density fx using a sample y1 yn from fy fx fe where fe is an unknown density we assume that an additional sample e1 em from fe is observed estimators of fx and its derivatives are constructed by using nonparametric estimators of fy and fe and by applying a spectral cut off in the fourier domain we derive the rate of convergence of the estimators in case of a known and unknown error density fe where it is assumed that fx satisfies a polynomial logarithmic or general source condition it is shown that the proposed estimators are asymptotically optimal in a minimax sense in the models with known or unknown error density if the density fx belongs to a sobolev space h mathcal p and fe is ordinary smooth or supersmooth

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

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

on adaptive estimation of linear functionals adaptive estimation of linear functionals over a collection of parameter spaces is considered a between class modulus of continuity a geometric quantity is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single para meter space a general construction of optimally adaptive estimators based on an ordered modulus of continuity is given the results are complemented by several illustrative examples

discrete transform approach to deconvolution problems summary if fourier series are used as the basis for inference in deconvolution problems the effects of the errors factorise out in a way that is easily exploited empirically this property is the consequence of elementary addition formulae for sine and cosine functions and is not readily available when one is using methods based on other orthogonal series or on continuous fourier transforms it allows relatively simple estimators to be constructed founded on the addition of finite series rather than on integration the performance of these methods can be particularly effective when edge effects are involved since cosine series estimators are quite resistant to boundary problems in this context we point to the advantages of trigonometric series methods for density deconvolution they have better mean squared error performance when edge effects are involved they are particularly easy to code and they admit a simple approach to empirical choice of smoothing parameter in which a version of thresholding familiar in wavelet based inference is used in place of conventional smoothing applications to other deconvolution problems are briefly discussed

data driven density estimation in the presence of additive noise with unknown distribution we study the following model of deconvolution y x varepsilon with i i d observations y 1 dots y n and varepsilon 1 dots varepsilon m the x j 1 leq j leq n are i i d with density f independent of the varepsilon j the aim of the paper is to estimate f without knowing the density f varepsilon of the varepsilon j we first define a projection estimator for which we provide bounds for the pointwise and the integrated l 2 risk we consider ordinary smooth and supersmooth noise varepsilon with regard to ordinary smooth and supersmooth densities f then we present an adaptive estimator of the density of f this estimator is obtained by penalization of the projection contrast which provides model selection lastly we present simulation experiments to illustrate the good performances of our estimator and study from the empirical point of view the importance of theoretical constraints

eulerian solutions of o σnv for the stochastic transport problem for conservative tracers coupled with o σ4ƒ solutions for the flow problem in an infinite domain an o vn eulerian stochastic theory cushman and hu 1997 for conservative transport is coupled with an o f4 eulerian solution to the flow problem the theory provides a self consistent recursive solution to the closure problem associated with eulerian methods the stochastic concentration is given to arbitrary order in v the variance of fluctuating velocity the balance law for mean concentration is not required in the analysis closed forms for mean concentration are given up to o v4 the mean concentration is transformed to wave vector and frequency domain via fast fourier transform to calculate numerically the mean concentration as well as its spatial moments the results show that the first order solution in v2 is equivalent to the nonlocal theory ofdeng et al 1993 second order corrections to flow and transport equations slightly decrease the second longitudinal moment but significantly increase the second transverse moment which is consistent with hsu et al s 1996 results the influence of second order corrections to skewness is not clear the second longitudinal moment obtained from the second order correction agrees with the monte carlo result but second order results for the second transverse moment and skewness significantly differ from those given in monte carlo simulations coupling the transport correction models with the velocity covariance generated through monte carlo simulation gives second transverse spatial moments that are very close to monte carlo simulations which suggests that the correction to flow is more important than the correction to transport the results also bring into question the accuracy of monte carlo simulations for flow

simulation of dispersion in heterogeneous porous formations statistics first order theories convergence of computations this paper discusses the results of numerical analysis of dispersion of passive solutes in two dimensional heterogeneous porous formations statistics of flow and transport variables the accuracy and the role of approximations implicit in existing first order theories and the convergence of computational results are investigated the results suggest that quite different rates of convergence with monte carlo runs hold for different spatial moments and that over 1000 realizations are required to stabilize second moments even for relatively mild heterogeneity sigma y 2 1 6 this has implications for the extent of the spatial domain for single realization numerical studies of the same type a comparison of the variance of plumes with the results of linear theories 0 05 sigma y 2 1 6 shows an unexpectedly broad validity field for the theoretical solution obtained from a suitable linearization of flow and transport reformulation of the same problem linearizing in tum the flow or the transport equations shows opposite deviations from the linear theory the interesting consequence is that the errors induced by linearizations in the flow or the transport equations have different signs and their effects on the moments of dispersing plumes are compensating thereby yielding consistent formulations unexpected features of the statistics of probability distributions of longitudinal and transverse velocities and travel times are also computed and discussed

asymptotic properties of the eulerian truncation approximation analysis of the perfectly stratified transport problem it is well known that stochastic groundwater problems are difficult to solve without making approximations of one kind or another a popular approximation used in both subsurface flow and transport applications is the so called eulerian truncation this approximation which relies on a perturbation expansion of the governing equation neglects certain terms involving products of small fluctuations several recent publications e g dagan and neuman 1991 neuman and orr 1993 neuman 1993 question the validity of eulerian truncation arguing that it is inconsistent because neglected terms are of the same order as those retained in this paper we analyze the asymptotic properties of the eulerian truncation solution using a taylor series approach which clearly reveals the properties of different order approximations we show that the eulerian truncation does in fact generate an asymptotic ensemble mean expansion for the perfectly stratified example considered by dagan and neuman 1991 moreover we show that the eulerian truncation solution to this problem is identical to the well known cumulant expansion approximation which is also based on an asymptotic expansion

higher order corrections to the flow velocity covariance tensor revisited recently the authors presented an accurate in the variance of fluctuating log conductivity second order solution to the steady saturated flow problem the longitudinal second spatial moment was evaluated numerically on the basis of the perturbation solution and it was found that second order corrections to the head field had little effect on this moment here we use the same theory and evaluate the second transverse spatial moment we find in agreement with others that our theory suggests that the second order head corrections as well as the nonlinear velocity corrections are important

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

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

an investigation of the validity of first order stochastic dispersion theories in isotropie porous media monte carlo simulations are used to 1 investigate the accuracy of approximations that are implicit in first order stochastic dispersion theories and 2 identify the accuracy limits of first order dispersion theories in isotropic porous media the fickian theory of gelhar and axness 1983 as well as the fickian and non fickian theories of dagan 1984 and neuman and zhang 1990 are investigated all monte carlo simulations are in three dimensions confidence limits of ensemble averaged monte carlo results in isotropic porous media are established for 0 1 y 1 5 these results showed that first order theoretical estimates of the eulerian velocity covariance function are quite accurate for y 1 theoretical estimates of the non fickian longitudinal dispersivity do not deviate significantly from theory for at least y 1 5 theoretical estimation of the transverse dispersivity is limited to y 1 and the fickian longitudinal dispersivity is overestimated by the theory of gelhar and axness 1983 of all first order dispersion theories the theory of dagan 1984 is most robust in estimating the dispersivity tensor

robust and efficient recovery of a signal passed through a filter and then contaminated by non gaussian noise consider a channel where a continuous periodic input signal is passed through a linear filter and then is contaminated by an additive noise the problem is to recover this signal when we observe n repeated realizations of the output signal adaptive efficient procedures that are asymptotically minimax over all possible procedures are known for channels with gaussian noise and no filter the case of direct observation efficient procedures based on the smoothness of a recovered signal are known for the case of gaussian noise robust rate optimal procedures are known as well however there are no results on robust and efficient data driven procedures moreover the known results for the case of direct observation indicate that even a small deviation from gaussian noise may lead to a drastic change we show that for the considered case of indirect data and a particular class of so called supersmooth filters there exists a procedure of recovery of an input signal that possesses the desired properties namely it is adaptive to the smoothness of the input signal robust to the distribution of the noise globally and pointwise efficient that is its minimax global and pointwise risks converge with the best constant and rate over all possible estimators as n spl rarr spl infin and universal in the sense that for a wide class of linear not necessarily bounded operators the efficient estimator is a plug in one furthermore we explain how to employ the obtained asymptotic results for the practically important case of small n large noise

adaptive demixing in poisson mixture models let x 1 x 2 x n be an i i d sample from the poisson mixture distribution p x 1 x 0 s x e s f s ds rates of convergence in mean integrated squared error mise of orthogonal series estimators for the mixing density f supported on a b are studied for the holder class of densities whose r th derivative is lipschitz a the mise converges at the rate log n log log n 2 r for sobolev classes of densities whose r th derivative is square integrable the mise converges at the rate log n log log n 2r the estimator is adaptive over both these classes for the sobolev class a lower bound on the minimax rate of convergence is log n log log n 2r and so the orthogonal polynomial estimator is rate optimal

wavelet deconvolution this paper studies the issue of optimal deconvolution density estimation using wavelets the approach taken here can be considered as orthogonal series estimation in the more general context of the density estimation we explore the asymptotic properties of estimators based on thresholding of estimated wavelet coefficients minimax rates of convergence under the integrated square loss are studied over besov classes b sub spl sigma pq of functions for both ordinary smooth and supersmooth convolution kernels the minimax rates of convergence depend on the smoothness of functions to be deconvolved and the decay rate of the characteristic function of convolution kernels it is shown that no linear deconvolution estimators can achieve the optimal rates of convergence in the besov spaces with p 2 when the convolution kernel is ordinary smooth and super smooth if the convolution kernel is ordinary smooth then linear estimators can be improved by using thresholding wavelet deconvolution estimators which are asymptotically minimax within logarithmic terms adaptive minimax properties of thresholding wavelet deconvolution estimators are also discussed

density estimation for the case of supersmooth measurement error abstract the problem is to estimate the probability density of a random variable contaminated by an independent measurement error i explore one of the worst case scenario when the characteristic function of this measurement error decreases exponentially and thus optimal estimators converge only with logarithmic rate the particular example of such measurement error is any random variable contaminated by normal cauchy or another stable random variable for this setting and circular data i suggest an asymptotically efficient data driven estimator that is adaptive to both smoothness of estimated density and distribution of measurement error moreover this estimator is universal in sense that its derivatives and integral are sharp estimators of the corresponding derivatives and the cumulative distribution function and these estimators are sharp both globally and pointwise for the case of small sample sizes i suggest a modified estimator that mimics an optimal linear pseudoestimator i explore this est

a monte carlo assessment of eulerian flow and transport perturbation models monte carlo studies of flow and transport in two dimensional synthetic conductivity fields are employed to evaluate first order flow and eulerian transport theories hydraulic conductivity is assumed to obey fractional brownian motion fbm statistics with infinite integral scale or to have an exponential covariance structure with finite integral scale the flow problem is solved via a block centered finite difference scheme and a random walk approach is employed to solve the transport equation for a conservative tracer the model is tested for mass conservation and convergence of computed statistics and found to yield accurate results it is then used to address several issues in the context of flow and transport the validity of the first order relation between the fluctuating velocity covariance and the fluctuating log conductivity is examined the simulations show that for exponential covariance this approximation is justified in the mean flow direction for log conductivity variance f2 of the order of unity however as f2 increases the relation for the transverse velocity component deviates from the fully nonlinear monte carlo results eulerian transport models neglect triplet correlation functions that appear in the nonlocal macroscopic flux the relative importance of the triplet correlation term for conservative chemicals is examined this term appears to be small relative to the convolution flux term in mildly heterogeneous media as f2 increases or the integral scale grows the triplet correlation becomes significant in purely convective transport the triplet correlation term is significant if the heterogeneity is evolving the exact nonlocal macroscale flux for the purely convective case significantly differs from that of the convective dispersive transport this is in agreement with recent theoretical analysis and numerical studies and it suggests that neglecting local scale dispersion may lead to large errors localization errors in the flux term are evaluated using monte carlo simulations the nonlocal in time model significantly differs from the fully nonlocal model for small variance and integral scale there is a slight difference between the fully localized flux and the fully nonlocal convolution flux this is also in agreement with recent theories that suggest that moments through the second for the two models are nearly identical for conservative tracers the fully localized model does not perform well in the purely convective cases

monte carlo studies of flow and transport in fractal conductivity fields comparison with stochastic perturbation theory a monte carlo simulation of flow and transport is employed to study tracer migration in porous media with evolving scales of heterogeneity fractal media transport is studied with both conservative and reactive chemicals in media that possess physical as well as chemical heterogeneity linear kinetic equations are assumed to relate the sorbed phase and the aqueous phase concentrations the fluctuating log conductivity possesses the power law spectrum of a fractional brownian motion fbm chemical heterogeneity is represented as spatially varying reaction rates that also are assumed to obey fbm statistics and may be correlated to the conductivity field the model is based on a finite difference approximation to the flow problem and a random walk particle tracking approach for solving the solute transport equation the model is used to make comparisons with the nonlocal transport equations recently developed by deng et al 1993 and hu et al 1995 1997 the results presented herein support these nonlocal models for a wide range of heterogeneous systems however the infinite integral scale associated with the fractal conductivity has a significant effect on the prediction of the nonlocal theories this suggests that integral scale should play a role in stochastic eulerian perturbation theories the importance of the local scale dispersion depends to a great extent on the magnitude of the local dispersivities the effect of neglecting local dispersion decreases with the decrease in local dispersivity

random walk simulation of transport in heterogeneous porous media local mass conservation problem and implementation methods the random walk method for simulating solute transport in porous media is typically based on the assumption that the velocity and velocity dependent dispersion tensor vary smoothly in space however in cases where sharp interfaces separate materials with contrasting hydraulic properties these quantities may be discontinuous normally velocities are interpolated to arbitrary particle locations when finite difference or finite element methods are used to solve the flow equation the use of interpolation schemes that preserve discontinuities in velocity at material contacts can result in a random walk model that does not locally conserve mass unless a correction is applied at these contacts test simulations of random walk particle tracking with and without special treatment of material contacts demonstrate the problem techniques for resolving the problem including interpolation schemes and a reflection principle are reviewed and tested results from simulations of transport in porous media with discontinuities in the dispersion tensor show which methods satisfy continuity simulations of transport in two dimensional heterogeneous porous media demonstrate the potentially significant effect of using a nonconservative model to compute spatial moments and breakthrough of a solute plume

bayesian model selection in finite mixtures by marginal density decompositions we consider the problem of estimating the number of components d and the unknown mixing distribution in a finite mixture model in which d is bounded by some fixed finite number n our approach relies on the use of a prior over the space of mixing distributions with at most n components by decomposing the resulting marginal density under this prior we discover a weighted bayes factor method for consistently estimating d that can be implemented by an iid generalized weighted chinese restaurant gwcr monte carlo algorithm we also discuss a gibbs sampling method the blocked gibbs sampler for estimating d and also the mixing distribution we show that our resulting posterior is consistent and achieves the frequentist optimal op n 1 4 rate of estimation we compare the performance of the new gwcr model selection procedure with that of the akaike information criterion and the bayes information criterion implemented through an em algorithm applications of our methods to five real datasets and simulation

on estimating mixing densities in discrete exponential family models this paper concerns estimating a mixing density function g and its derivatives based on iid observations from f x f x g d where f x is a known exponential family of density functions with respect to the counting measure on the set of nonnegative integers fourier methods are used to derive kernel estimators upper bounds for their rate of convergence and lower bounds for the optimal rate of convergence if f x i 0 e x 1 x for some positive numbers 0 and e then our estimators achieve the optimal rate of convergence log n m for estimating the mth derivative of g under a lipschitz condition of order a m the optimal rate of convergence is almost achieved when x f x 0 e x 1 estimation of the mixing distribution function is also considered

stochastic uncertainty analysis for solute transport in randomly heterogeneous media using a karhunen loeve based moment equation approach 1 a new approach has been developed for solving solute transport problems in randomly heterogeneous media using the karhunen loeve based moment equation klme technique proposed by zhang and lu 2004 the klme approach combines the karhunen loeve decomposition of the underlying random conductivity field and the perturbative and polynomial expansions of dependent variables including the hydraulic head flow velocity dispersion coefficient and solute concentration the equations obtained in this approach are sequential and their structure is formulated in the same form as the original governing equations such that any existing simulator such as modular three dimensional multispecies transport model for simulation of advection dispersion and chemical reactions of contaminants in groundwater systems mt3dms can be directly applied as the solver through a series of two dimensional examples the validity of the klme approach is evaluated against the classical monte carlo simulations results indicate that under the flow and transport conditions examined in this work the klme approach provides an accurate representation of the mean concentration for the concentration variance the accuracy of the klme approach is good when the conductivity variance is 0 5 as the conductivity variance increases up to 1 0 the mismatch on the concentration variance becomes large although the mean concentration can still be accurately reproduced by the klme approach our results also indicate that when the conductivity variance is relatively large neglecting the effects of the cross terms between velocity fluctuations and local dispersivities as done in some previous studies can produce noticeable errors and a rigorous treatment of the dispersion terms becomes more appropriate