local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

iterative weighted maximum likelihood denoising with probabilistic patch based weights image denoising is an important problem in image processing since noise may interfere with visual or automatic interpretation this paper presents a new approach for image denoising in the case of a known uncorrelated noise model the proposed filter is an extension of the nonlocal means nl means algorithm introduced by buades which performs a weighted average of the values of similar pixels pixel similarity is defined in nl means as the euclidean distance between patches rectangular windows centered on each two pixels in this paper a more general and statistically grounded similarity criterion is proposed which depends on the noise distribution model the denoising process is expressed as a weighted maximum likelihood estimation problem where the weights are derived in a data driven way these weights can be iteratively refined based on both the similarity between noisy patches and the similarity of patches extracted from the previous estimate we show that this iterative process noticeably improves the denoising performance especially in the case of low signal to noise ratio images such as synthetic aperture radar sar images numerical experiments illustrate that the technique can be successfully applied to the classical case of additive gaussian noise but also to cases such as multiplicative speckle noise the proposed denoising technique seems to improve on the state of the art performance in that latter case

regularity of the free boundary for the porous medium equation with r m 1 m 1 when u 0 then f 0 and both of the above equations become degenerate this degeneracy results in the interesting phenomenon of the finite speed of propagation if the initial data u is compactly supported in 1ztn the solution u t will remain compactly supported for all time t in this work we will show that under rather general assumptions on the initial data the free boundary

maximum likelihood estimation of rician distribution parameters the problem of parameter estimation from rician distributed data e g magnitude magnetic resonance images is addressed the properties of conventional estimation methods are discussed and compared to maximum likelihood ml estimation which is known to yield optimal results asymptotically in contrast to previously proposed methods ml estimation is demonstrated to be unbiased for high signal to noise ratio snr and to yield physical relevant results for low snr

a concave convex elliptic problem involving the fractional laplacian we study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the laplacian operator together with a concave convex term we completely characterize the range of parameters for which solutions of the problem exist and prove a multiplicity result we also prove an associated trace inequality and some liouville type results

one step local quasi likelihood estimation summary local quasi likelihood estimation is a useful extension of local least squares methods but its computational cost and algorithmic convergence problems make the procedure less appealing particularly when it is iteratively used in methods such as the back fitting algorithm cross validation and bootstrapping a one step local quasi likelihood estimator is introduced to overcome the computational drawbacks of the local quasi likelihood method we demonstrate that as long as the initial estimators are reasonably good the one step estimator has the same asymptotic behaviour as the local quasi likelihood method our simulation shows that the one step estimator performs at least as well as the local quasi likelihood method for a wide range of choices of bandwidths a data driven bandwidth selector is proposed for the one step estimator based on the pre asymptotic substitution method of fan and gijbels it is then demonstrated that the datadriven one step local quasi likelihood estimator performs as well as the maximum local quasilikelihood estimator by using the ideal optimal bandwidth

local polynomial kernel regression for generalized linear models and quasi likelihood functions abstract we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi likelihood contexts in the ordinary regression case local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity one noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators these properties are shown to carry over to generalized linear model and quasi likelihood settings we also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state of the art bandwidth selection methods

propagation separation approach for local likelihood estimation the paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models including e g the regression density poisson and binary response model the method extends the adaptive weights smoothing aws procedure introduced in polzehl and spokoiny 2000 in context of image denoising the main idea of the method is to describe a greatest possible local neighborhood of every design point x i in which the local parametric assumption is justified by the data the method is especially powerful for model functions having large homogeneous regions and sharp discontinuities the performance of the proposed procedure is illustrated by numerical examples for density estimation and classification we also establish some remarkable theoretical nonasymptotic results on properties of the new algorithm this includes the propagation property which particularly yields the root n consistency of the resulting estimate in the homogeneous case we also state an oracle result which implies rate optimality of the estimate under usual smoothness conditions and a separation result which explains the sensitivity of the method to structural changes

variable bandwidth kernel estimators of regression curves on montre qu en termes de l erreur en moyenne quadratique integree asymptotique les estimateurs du noyau avec un certain choix de largeur de bande local sont superieurs aux estimateurs ordinaires du noyau avec un choix de largeur de bande global si on utilise des largeurs de bandes optimales

local likelihood smoothing of sample extremes trends in sample extremes are of interest in many contexts an example being environmental statistics parametric models are often used to model trends in such data but they may not be suitable for exploratory data analysis this paper outlines a semiparametric approach to smoothing sample extremes based on local polynomial fitting of the generalized extreme value distribution and related models the uncertainty of fits is assessed by using resampling methods the methods are applied to data on extreme temperatures and on record times for the women s 3000 m race

a strong maximum principle for the laplace equation with mixed boundary condition abstract in this work we present a comparison result for two solutions of the laplace equation in a smooth bounded domain satisfying the same mixed boundary condition zero dirichlet data on part of the boundary and zero neumann data on the rest the result is in some sense a generalization of the hopf lemma to the case of mixed boundary conditions where the barrier function is not given explicitly but as the solution of the laplace equation with a constant right hand side and mixed boundary condition

locally adaptive bandwidth choice for kernel regression estimators abstract kernel estimators with a global bandwidth are commonly used to estimate regression functions on the other hand it is obvious that the choice of a local bandwidth can lead to better results because a larger class of kernel estimators is available evidently this may in turn affect variability the optimal bandwidths depend essentially on the regression function itself and on the residual variance and it is desirable to estimate them from the data in this article a local bandwidth estimator is studied a comparison with its global bandwidth equivalent is performed both in theory and in simulations as the main result it is shown that the possible gain in mean integrated squared error of the resulting regression estimator must be paid for by a larger variability of the estimator this may lead to worse results if the sample size is small an algorithm has been devised that puts special weight on stability aspects our simulation study shows that improvements over a global bandwidth estimator

regression models for identifying noise sources in magnetic resonance images stochastic noise susceptibility artifacts magnetic field and radiofrequency inhomogeneities and other noise components in magnetic resonance images mris can introduce serious bias into any measurements made with those images we formally introduce three regression models including a rician regression model and two associated normal models to characterize stochastic noise in various magnetic resonance imaging modalities including diffusion weighted imaging dwi and functional mri fmri estimation algorithms are introduced to maximize the likelihood function of the three regression models we also develop a diagnostic procedure for systematically exploring mr images to identify noise components other than simple stochastic noise and to detect discrepancies between the fitted regression models and mri data the diagnostic procedure includes goodness of fit statistics measures of influence and tools for graphical display the goodness of fit statistics can assess the key assumptions of the three r

power concavity on nonlinear parabolic flows our object in this paper is to show that the concavity of the power of a solution is preserved in the parabolic p laplace equation called power concavity and that the power is determined by the homogeneity of the parabolic operator in the parabolic p laplace equation for the density u the concavity of u p 2 p is considered which indicates why the log concavity has been considered in heat flow p 2 in addition the long time existence of the classical solution of the parabolic p laplacian equation can be obtained if the initial smooth data has concavity and a nondegenerate gradient along the initial boundary 2004 wiley periodicals inc

local partial likelihood estimation in proportional hazards regression fan gijbels and king ann statist 25 1997 1661 1690 considered the estimation of the risk function x in the proportional hazards model their proposed estimator is based on integrating the estimated derivative function obtained through a local version of the partial likelihood they proved the large sample properties of the derivative function but the large sample properties of the estimator for the risk function itself were not established in this paper we consider direct estimation of the relative risk function x 2 x 1 for any location normalization point x 1 the main novelty in our approach is that we select observations in shrinking neighborhoods of both x 1 and x 2 when constructing a local version of the partial likelihood whereas fan gijbels and king ann statist 25 1997 1661 1690 only concentrated on a single neighborhood resulting in the cancellation of the risk function in the local likelihood function the asymptotic properties of our estimator are rigorously established and the variance of the estimator is easily estimated the idea behind our approach is extended to estimate the differences between groups a simulation study is carried out

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

how to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems we present a model for nonlocal diffusion with neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary we study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero we prove that the solutions of this family of problems converge to a solution of the heat equation with neumann boundary conditions

rician distributed fmri asymptotic power analysis and cramer rao lower bounds traditional functional mri detection statistics for activation and hemodynamic response modeling assume gaussian data which is true only for high signal to noise ratio snr the correct distribution is rician in this correspondence we provide two new developments in the rician case first a derivation of the theoretical asymptotic power function as the number of samples goes to infinity second the derivation of a cramer rao lower bound this allows a correct assessment of the impact of various signal and noise levels on detection power for activation and or hemodynamic response parameter estimation accuracy based on our analysis we are able to extend existing definitions of snr by considering variation not only in baseline but also in drifts

a brezis nirenberg splitting approach for nonlocal fractional equations in this paper we consider problems modeled by the following nonlocal fractional equation su a x u f u in u 0in rn where s 0 1 is fixed is an open bounded subset of rn n 2s with lipschitz boundary s is the fractional laplace operator and is a real parameter under two different types of conditions on the functions a and f by using a famous critical point theorem in the presence of splitting established by brezis and nirenberg we obtain the existence of at least two nontrivial weak solutions for our problem

statistical estimation in generalized multiparameter likelihood models multiparameter likelihood models mlms with multiple covariates have a wide range of applications however they encounter the curse of dimensionality problem when the dimension of the covariates is large we develop a generalized multiparameter likelihood model that copes with multiple covariates and adapts to dynamic structural changes well it includes some popular models such as the partially linear and varying coefficient models as special cases we present a simple effective two step method to estimate both the parametric and the nonparametric components when the model is fixed the proposed estimator of the parametric component has the n 1 2convergence rate and the estimator of the nonparametric component enjoys an adaptivity property we suggest a data driven procedure for selecting the bandwidths and propose an initial estimator in profile likelihood estimation of the parametric part to ensure stability of the approach in general settings we further develop an automatic procedure to ide

on some critical problems for the fractional laplacian operator abstract we study the effect of lower order perturbations in the existence of positive solutions to the following critical elliptic problem involving the fractional laplacian 2 u u q u n n u 0 in u 0 on where r n is a smooth bounded domain n 1 0 0 q n n 0 min n 2 for suitable conditions on depending on q we prove in the case q 1 there exist at least two solutions for every 0 and some 0 at least one if no solution if for q 1 we show existence of at least one solution for 0 1 and nonexistence for 1 when q 1 the existence is shown for every 0 also we prove that the solutions are bounded and regular

parabolic approach to nonlinear elliptic eigenvalue problems we consider the asymptotic profiles of the nonlinear parabolic flows ut um to show the geometric properties of the following elliptic nonlinear eigenvalue problems posed in a strictly convex domain for different values of p in the classical case p 1 we establish that log f is a strictly concave function when 0 p 1 the function is strictly concave while for there is at least a solution such that is strictly convex moreover eventual convexity properties for the parabolic flows are proved

bounds for the heat diffusion through windows of given area abstract we consider the lowest eigenvalue for the laplacian in a given lipschitz domain under mixed boundary conditions dirichlet in a subset of the boundary noninsulated window neumann otherwise this eigenvalue can be interpreted as heat leakage rate due to diffusion we give an explicit calculation for a model problem a rigorous lower bound that depends only on the area but not on the geometry of the window this bound confirms the observations from the model problem finally we show that no nontrivial upper bound is possible i e any small area for the window being prescribed its geometry can be made bad enough to cause heat leak rates arbitrarily close to the ones for no insulation anywhere the most important techniques are the aronszajn weinstein method of intermediate variational problems and the gaussian upper bounds for the heat kernel by e b davies

fast iterative image reconstruction for mri in the presence of field inhomogeneities in magnetic resonance imaging magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast fourier transform fft methods several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities but these methods assume that the field map that characterizes the off resonance frequencies is spatially smooth recently iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off resonance effects however straightforward implementations of such iterative methods suffer from inconveniently long computation times this paper describes a tool for accelerating iterative reconstruction of field corrected mr images a novel time segmented approximation to the mr signal equation we use a min max formulation to derive the temporal interpolator speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast fourier transform with normalized root mean squared approximation errors of 0 07 the proposed method provides fast accurate field corrected image reconstruction even when the field map is not smooth

wavelet based rician noise removal for magnetic resonance imaging it is well known that magnetic resonance magnitude image data obey a rician distribution unlike additive gaussian noise rician noise is signal dependent and separating signal from noise is a difficult task rician noise is especially problematic in low signal to noise ratio snr regimes where it not only causes random fluctuations but also introduces a signal dependent bias to the data that reduces image contrast this paper studies wavelet domain filtering methods for rician noise removal we present a novel wavelet domain filter that adapts to variations in both the signal and the noise

a nonlocal maximum likelihood estimation method for rician noise reduction in mr images postacquisition denoising of magnetic resonance mr images is of importance for clinical diagnosis and computerized analysis such as tissue classification and segmentation it has been shown that the noise in mr magnitude images follows a rician distribution which is signal dependent when signal to noise ratio snr is low it is particularly difficult to remove the random fluctuations and bias introduced by rician noise the objective of this paper is to estimate the noise free signal from mr magnitude images we model images as random fields and assume that pixels which have similar neighborhoods come from the same distribution we propose a nonlocal maximum likelihood nlml estimation method for rician noise reduction our method yields an optimal estimation result that is more accurate in recovering the true signal from rician noise than nl means algorithm in the sense of snr contrast and method error we demonstrate that nlml performs better than the conventional local maximum likelihood lml estimation method in preserving and defining sharp tissue boundaries in terms of a well defined sharpness metric while also having superior performance in method error

a majorize minimize framework for rician and non central chi mr images the statistics of many mr magnitude images are described by the non central chi ncc family of probability distributions which includes the rician distribution as a special case these distributions have complicated negative log likelihoods that are nontrivial to optimize this paper describes a novel majorize minimize framework for ncc data that allows penalized maximum likelihood estimates to be obtained by solving a series of much simpler regularized least squares surrogate problems the proposed framework is general and can be useful in a range of applications we illustrate the potential advantages of the framework with real and simulated data in two contexts 1 mr image denoising and 2 diffusion profile estimation in high angular resolution diffusion mri the proposed approach is shown to yield improved results compared to methods that model the noise statistics inaccurately and faster computation relative to commonly used nonlinear optimization techniques

local maximum likelihood estimation and inference local maximum likelihood estimation is a nonparametric counterpart of the widely used parametric maximum likelihood technique it extends the scope of the parametric maximum likelihood method to a much wider class of parametric spaces associated with this nonparametric estimation scheme is the issue of bandwidth selection and bias and variance assessment this paper provides a unified approach to selecting a bandwidth and constructing confidence intervals in local maximum likelihood estimation the approach is then applied to least squares nonparametric regression and to nonparametric logistic regression our experiences in these two settings show that the general idea outlined here is powerful and encouraging

the brezis nirenberg result for the fractional laplacian the aim of this paper is to deal with the non local fractional counterpart of the laplace equation involving critical non linearities studied in the famous paper of brezis and nirenberg 1983 namely our model is the equation su u u 2 2u in u 0 in rn where s is the fractional laplace operator s 0 1 is an open bounded set of rn n 2s with lipschitz boundary 0 is a real parameter and 2 2n n 2s is a fractional critical sobolev exponent in this paper we first study the problem in a general framework indeed we consider the equation lku u u 2 2u f x u 0 in u 0 in rn where lk is a general non local integrodifferential operator of order s and f is a lower order perturbation of the critical power u 2 2u in this setting we prove an existence result through variational techniques then as a concrete example we derive a brezis nirenberg type result for our model equation that is we show that if 1 s is the first eigenvalue of the non local operator s with homogeneous dirichlet boundary datum then for any 0 1 s there exists a non trivial solution of the above model equation provided n 4s in this sense the present work may be seen as the extension of the classical brezis nirenberg result to the case of non local fractional operators

large time geometrical properties of solutions of the barenblatt equation of elasto plastic filtration abstract we study the geometric behavior for large times of the solutions of the following equation u t u t u 0 1 posed in the whole space r n for n 1 when the initial data are nonnegative continuous and compactly supported we prove that after a finite time log u becomes a concave function in the space variable and converges to all orders of differentiability to a certain parabolic shape so called barenblatt type profile which was described in kamin et al 1991 20 extensions to more general fully nonlinear equations are considered

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

multiscale local change point detection with applications to value at risk this paper offers a new approach to modeling and forecasting of non stationary time series with applications to volatility modeling for financial data the approach is based on the assumption of local homogeneity for every time point there exists a historical interval of homogerzeity in which the volatility parameter can be well approximated by a constant the proposed procedure recovers this interval from the data using the local change point lcp analysis afterward the estimate of the volatility can be simply obtained by local averaging the approach carefully addresses the question of choosing the tuning parameters of the procedure using the so called propagation condition the main result claims a new oracle inequality in terms of the modeling bias which measures the quality of the local constant approximation this result yields the optimal rate of estimation for smooth and piecewise constant volatility functions then the new procedure is applied to some data sets and a comparison with a standard garch model is also provided finally we discuss applications of the new method to the value at risk problem the numerical results demonstrate a very reasonable performance of the new method

nonlocal regularization of inverse problems a unified variational framework we introduce a unifying energy minimization framework for nonlocal regularization of inverse problems in contrast to the weighted sum of square differences between image pixels used by current schemes the proposed functional is an unweighted sum of inter patch distances we use robust distance metrics that promote the averaging of similar patches while discouraging the averaging of dissimilar patches we show that the first iteration of a majorize minimize algorithm to minimize the proposed cost function is similar to current nonlocal methods the reformulation thus provides a theoretical justification for the heuristic approach of iterating nonlocal schemes which re estimate the weights from the current image estimate thanks to the reformulation we now understand that the widely reported alias amplification associated with iterative nonlocal methods are caused by the convergence to local minimum of the nonconvex penalty we introduce an efficient continuation strategy to overcome this problem the similarity of the proposed criterion to widely used nonquadratic penalties e g total variation and lp semi norms opens the door to the adaptation of fast algorithms developed in the context of compressive sensing we introduce several novel algorithms to solve the proposed nonlocal optimization problem thanks to the unifying framework these fast algorithms are readily applicable for a large class of distance metrics

preadjusted non parametric estimation of a conditional distribution function summary the paper deals with non parametric estimation of a conditional distribution function we suggest a method of preadjusting the original observations non parametrically through location and scale to reduce the bias of the estimator we derive the asymptotic properties of the estimator proposed a simulation study investigating the finite sample performances of the estimators discussed is provided and reveals the gain that can be achieved it is also shown how the idea of the preadjusting opens the path to improved estimators in other settings such as conditional quantile and density estimation and conditional survival function estimation in the case of censored data

an expanded theoretical treatment of iteration dependent majorize minimize algorithms the majorize minimize mm optimization technique has received considerable attention in signal and image processing applications as well as in statistics literature at each iteration of an mm algorithm one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate the next iterate is obtained by minimizing this tangent majorant function resulting in a sequence of iterates that reduces the cost function monotonically a well known special case of mm methods are expectation maximization algorithms in this paper we expand on previous analyses of mm due to fessler and hero that allowed the tangent majorants to be constructed in iteration dependent ways also this paper overcomes an error in one of those earlier analyses there are three main aspects in which our analysis builds upon previous work first our treatment relaxes many assumptions related to the structure of the cost function feasible set and tangent majorants for example the cost function can be nonconvex and the feasible set for the problem can be any convex set second we propose convergence conditions based on upper curvature bounds that can be easier to verify than more standard continuity conditions furthermore these conditions allow for considerable design freedom in the iteration dependent behavior of the algorithm finally we give an original characterization of the local region of convergence of mm algorithms based on connected e g convex tangent majorants for such algorithms cost function minimizers will locally attract the iterates over larger neighborhoods than typically is guaranteed with other methods this expanded treatment widens the scope of the mm algorithm designs that can be considered for signal and image processing applications allows us to verify the convergent behavior of previously published algorithms and gives a fuller understanding overall of how these algorithms behave

a cure for noisy magnetic resonance images chi square unbiased risk estimation in this paper we derive an unbiased expression for the expected mean squared error associated with continuously differentiable estimators of the noncentrality parameter of a chi square random variable we then consider the task of denoising squared magnitude magnetic resonance mr image data which are well modeled as independent noncentral chi square random variables on two degrees of freedom we consider two broad classes of linearly parameterized shrinkage estimators that can be optimized using our risk estimate one in the general context of undecimated filterbank transforms and the other in the specific case of the unnormalized haar wavelet transform the resultant algorithms are computationally tractable and improve upon most state of the art methods for both simulated and actual mr image data

nonparametric density estimation with a parametric start the traditional kernel density estimator of an unknown density is by construction completely nonparametric in the sense that it has no preferences and will work reasonably well for all shapes the present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities for example the normal while not losing much in precision when the true density is far from the parametric class the idea is to multiply an initial parametric density estimate with a kernel type estimate of the necessary correction factor this works well in cases where the correction factor function is less rough than the original density itself extensive comparisons with the kernel estimator are carried out including exact analysis for the class of all normal mixtures the new method with a normal start wins quite often even in many cases where the true density is far from normal procedures for choosing the smoothing parameter of the estimator are also discussed the new estimator should be particularly useful in higher dimensions where the usual nonparametric methods have problems the idea is also spelled out for nonparametric regression

estimation in generalised varying coefficient models with unspecified link functions abstract in this paper we study the generalised varying coefficient models where the link function is unspecified and the response variable can be either continuous or discrete as the link function is unspecified the models under investigation become unidentifiable in this paper we derive an identification condition for the generalised varying coefficient models which is much weaker and more reasonable than that given by kuruwita et al 2011 whose model can be seen as a special case of our modelling framework under the identification condition we introduce a nonparametric iterative procedure to estimate the functional coefficient with its direction and norm as well as the unspecified link function and then establish the asymptotic properties of the resulting nonparametric estimators furthermore a weighted least squares based algorithm is provided to implement the iterative estimation procedure the simulation studies and empirical application show that our estimation methodology works quite well in both small and median sample cases

the kernel estimate of a regression function in likelihood based models abstract smoothing splines have a penalized likelihood motivation good and gaskins 1971 allowing direct application to nonparametric regression in likelihood based models the notion of a weighted likelihood for the nonparametric kernel estimation of a regression function is proposed generalizing the local likelihood theory of tibshirani and hastie 1987 let the data be of the form xi yi i 1 n where xi 0 1 d are lattice points and the yi are independent random variables from a family of distributions with parameter i g xi with g having continuous partial derivatives of order k 2 the goal is to arrive at a nonparametric estimate o of o g xo for a fixed point xo 0 1 d we consider the estimator o that maximizes the weighted likelihood function w n i 1 w xo xi b log f yi with f the density of yi w a symmetric kernel with compact support and b the bandwidth that controls the degree of smoothing sufficient conditions for consistency and asympt

convexity and all time c regularity of the interface in flame propagation we consider the following one phase free boundary problem find u such that u 0 and with qt n 0 t under the condition that o is convex and log uo is concave we show that the convexity of t and the concavity of log u t are preserved under the flow for 0 t t as long as t and u on t are smooth as a consequence we show the existence of a smooth up to the interface solution on 0 t tc with tc denoting the extinction time of t we also provide a new proof of a short time existence with c2 initial data on the general domain 2002 john wiley sons inc

inference for continuous semimartingales observed at high frequency the econometric literature of high frequency data often relies on moment estimators which are derived from assuming local constancy of volatility and related quantities we here study this local constancy approximation as a general approach to estimation in such data we show that the technique yields asymptotic properties consistency normality that are correct subject to an ex post adjustment involving asymptotic likelihood ratios these adjustments are derived and documented several examples of estimation are provided powers of volatility leverage effect and integrated betas the first order approximations based on local constancy can be over the period of one observation or over blocks of successive observations it has the advantage of gaining in transparency in defining and analyzing estimators the theory relies heavily on the interplay between stable convergence and measure change and on asymptotic expansions for martingales

local likelihood and local partial likelihood in hazard regression in survival analysis the relationship between a survival time and a covariate is conveniently modeled with the proportional hazards regression model this model usually assumes that the covariate has a log linear effect on the hazard function in this paper we consider the proportional hazards regression model with a nonparametric risk effect we discuss estimation of the risk function and its derivatives in two cases when the baseline hazard function is parametrized and when it is not parametrized in the case of a parametric baseline hazard function inference is based on a local version of the likelihood function while in the case of a nonparametric baseline hazard we use a local version of the partial likelihood this results in maximum local likelihood estimators and maximum local partial likelihood estimators respectively we establish the asymptotic normality of the estimators it turns out that both methods have the same asymptotic bias and variance in a common situation even though the local likelihood method uses information about the baseline hazard function

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

a free boundary problem for the heat equation arising in flame propagation we introduce a new free boundary problem for the heat equation of interest in combustion theory it is obtained in the description of laminar flames as an asymptotic limit for high activation energy the problem asks for the determination of a domain in space time q c rn x 0 t and a function u x t 0 defined in q such that ut au in q u takes certain initial conditions u x 0 uo x for x e qo oqfn t 0 and two conditions are satisfied on the free boundary r a2 n t 0 u o and u1 1 where u1 denotes the derivative of u along the spatial exterior normal to r we approximate this problem by means of a certain regularization on the boundary and prove the existence of a weak solution under suitable assumptions on the initial data

local quasi likelihood with a parametric guide generalized linear models and the quasi likelihood method extend the ordinary regression models to accommodate more general conditional distributions of the response nonparametric methods need no explicit parametric specification and the resulting model is completely determined by the data themselves however nonparametric estimation schemes generally have a slower convergence rate such as the local polynomial smoothing estimation of nonparametric generalized linear models studied in fan heckman and wand j amer statist assoc 90 1995 141 150 in this work we propose a unified family of parametrically guided nonparametric estimation schemes this combines the merits of both parametric and nonparametric approaches and enables us to incorporate prior knowledge asymptotic results and numerical simulations demonstrate the improvement of our new estimation schemes over the original nonparametric counterpart

linear transforms for fourier data on the sphere application to high angular resolution diffusion mri of the brain this paper presents a novel family of linear transforms that can be applied to data collected from the surface of a 2 sphere in three dimensional fourier space this family of transforms generalizes the previously proposed funk radon transform frt which was originally developed for estimating the orientations of white matter fibers in the central nervous system from diffusion magnetic resonance imaging data the new family of transforms is characterized theoretically and efficient numerical implementations of the transforms are presented for the case when the measured data is represented in a basis of spherical harmonics after these general discussions attention is focused on a particular new transform from this family that we name the funk radon and cosine transform fract based on theoretical arguments it is expected that fract based analysis should yield significantly better orientation information e g improved accuracy and higher angular resolution than frt based analysis while maintaining the strong characterizability and computational efficiency of the frt simulations are used to confirm these theoretical characteristics and the practical significance of the proposed approach is illustrated with real diffusion weighted mri brain data these experiments demonstrate that in addition to having strong theoretical characteristics the proposed approach can outperform existing state of the art orientation estimation methods with respect to measures such as angular resolution and robustness to noise and modeling errors

the brezis nirenberg problem for the fractional laplacian with mixed dirichlet neumann boundary conditions abstract in this work we study the existence of solutions to the critical brezis nirenberg problem when one deals with the spectral fractional laplace operator and mixed dirichlet neumann boundary conditions i e s u u u 2 s 1 u 0 in u 0 on d u 0 on n where r n is a regular bounded domain 1 2 s 1 2 s is the critical fractional sobolev exponent 0 r is the outwards normal to d n are smooth n 1 dimensional submanifolds of such that d n d n and d n is a smooth n 2 dimensional submanifold of

relative efficiencies of kernel and local likelihood density estimators summary local likelihood methods enjoy advantageous properties such as good performance in the presence of edge effects that are rarely found in other approaches to nonparametric density estimation however as we argue in this paper standard kernel methods can have distinct advantages when edge effects are not present we show that whereas the integrated variances of the two methods are virtually identical the integrated squared bias of a conventional kernel estimator is less than that of a local log linear estimator by as much as a factor of 4 moreover the greatest bias improvements offered by kernel methods occur when they are needed most i e when the effect of bias is particularly high similar comparisons can also be made when high degree local log polynomial fits are assessed against high order kernel methods for example although as is well known high degree local polynomial fits offer potentially infinite efficiency gains relative to their kernel competitors the converse is also true indeed the asymptotic value of the integrated squared bias of a local log quadratic estimator can exceed any given constant multiple of that for the competing kernel method in all cases the densities that suffer problems in the context of local log likelihood methods can be chosen to be symmetric either unimodal or bimodal either infinitely or compactly supported and to have arbitrarily many derivatives as functions on the real line they are not pathological however our results reveal quantitative differences between global performances of local log polynomial estimators applied to unimodal or multimodal distributions

global optimization of statistical functions with simulated annealing abstract many statistical methods rely on numerical optimization to estimate a model s parameters unfortunately conventional algorithms sometimes fail even when they do converge there is no assurance that they have found the global rather than a local optimum we test a new optimization algorithm simulated annealing on four econometric problems and compare it to three common conventional algorithms not only can simulated annealing find the global optimum it is also less likely to fail on difficult functions because it is a very robust algorithm the promise of simulated annealing is demonstrated on the four econometric problems

hazard models with varying coefficients for multivariate failure time data statistical estimation and inference for marginal hazard models with varying coefficients for multivariate failure time data are important subjects in survival analysis a local pseudo partial likelihood procedure is proposed for estimating the unknown coefficient functions a weighted average estimator is also proposed in an attempt to improve the efficiency of the estimator the consistency and asymptotic normality of the proposed estimators are established and standard error formulas for the estimated coefficients are derived and empirically tested to reduce the computational burden of the maximum local pseudo partial likelihood estimator a simple and useful one step estimator is proposed statistical properties of the one step estimator are established and simulation studies are conducted to compare the performance of the one step estimator to that of the maximum local pseudo partial likelihood estimator the results show that the one step estimator can save computational cost without compromising performance both asymptotically and empirically and that an optimal weighted average estimator is more efficient than the maximum local pseudo partial likelihood estimator a data set from the busselton population health surveys is analyzed to illustrate our proposed methodology

local likelihood based on kernel censoring by drawing an analogy with likelihood for censored data a local likelihood function is proposed which gives more weight to observations near a region of interest in the sample space resulting methods can be used for assessing local departures from a parametric model and for semiparametric density estimation some theory and three examples is given

locally parametric nonparametric density estimation this paper develops a nonparametric density estimator with parametric overtones suppose f x is some family of densities indexed by a vector of parameters we define a local kernel smoothed likelihood function which for each x can be used to estimate the best local parametric approximant to the true density this leads to a new density estimator of the form f x o x thus inserting the best local parameter estimate for each new value of x when the bandwidth used is large this amounts to ordinary full likelihood parametric density estimation while for moderate and small bandwidths the method is essentially nonparametric using only local properties of data and the model alternative ways more general than via the local likelihood are also described the methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class properties of this new semiparametric estimator are investigated our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias the new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework the methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression

variance reduction in multiparameter likelihood models local likelihood modeling is a unified and effective approach to establishing the dependence of a response variable which can be of various types on independent variables therefore these models have become popular in a wide range of applications there is an increasing interest in employing multiparameter local likelihood models to investigate trends of sample extremes in environmental statistics when sample maxima are modeled by a generalized extreme value distribution the sample size is small in general and local likelihood estimation exhibits a large variation in this article variance reduction techniques are employed to improve the efficiency of the inference a simulation study and an application to annual maximum temperatures show that our methods are very effective in finite samples

spatial aggregation of local likelihood estimates with applications to classification this paper presents a new method for spatially adaptive local constant likelihood estimation which applies to a broad class of nonparametric models including the gaussian poisson and binary response models the main idea of the method is given a sequence of local likelihood estimates weak estimates to construct a new aggregated estimate whose pointwise risk is of order of the smallest risk among all weak estimates we also propose a new approach toward selecting the parameters of the procedure by providing the prescribed behavior of the resulting estimate in the simple parametric situation we establish a number of important theoretical results concerning the optimality of the aggregated estimate in particular our oracle result claims that its risk is up to some logarithmic multiplier equal to the smallest risk for the given family of estimates the performance of the procedure is illustrated by application to the classification problem a numerical study demonstrates its reasonable performance in simulated and real life examples

local likelihood density estimation local likelihood was introduced by tibshirani and hastie as a method of smoothing by local polynomials in non gaussian regression models in this paper an extension of these methods to density estimation is discussed and comparison with other methods of density estimation presented the local likelihood method has particularly strong advantages over kernel methods when estimating tails of densities and in multivariate settings suppose constraints are incorporated in a simple manner asymptotic properties of the estimate are discussed a method for computing the estimate is outlined c code to implement the estimation procedure described in this paper together with s interfaces for graphical display of results are available at http cm bell labs com stat project locfit index html

kernel estimators for univariate binary regression we present a rather thorough investigation of the use of kernel based nonparametric estimators of the binary regression function in the case of a single covariate we consider various versions of nadaraya watson and local linear estimators some involving a single bandwidth and others involving two bandwidths the locally linear logistic estimator proves to be a good single bandwidth estimator although the basic nadaraya watson estimator also fares quite well two bandwidth methods show great potential when bandwidths are selected with knowledge of the target function but much of their potential vanishes when data based bandwidths are used likelihood cross validation and plug in approaches are the data based bandwidth selection methods tested both prove quite useful with a preference for the latter adaptive two bandwidth methods retain particularly good performance only in certain special situations and separate estimation of the two bandwidths as for optimal density estimation is never recommended

positive solutions of nonlinear problems involving the square root of the laplacian abstract we consider nonlinear elliptic problems involving a nonlocal operator the square root of the laplacian in a bounded domain with zero dirichlet boundary conditions for positive solutions to problems with power nonlinearities we establish existence and regularity results as well as a priori estimates of gidas spruck type in addition among other results we prove a symmetry theorem of gidas ni nirenberg type

a unified maximum likelihood framework for simultaneous motion and t_ 1 estimation in quantitative mr t_ 1 mapping in quantitative mr t1 mapping the spin lattice relaxation time t1 of tissues is estimated from a series of t1 weighted images as the t1 estimation is a voxel wise estimation procedure correct spatial alignment of the t1 weighted images is crucial conventionally the t1 weighted images are first registered based on a general purpose registration metric after which the t1 map is estimated however as demonstrated in this paper such a two step approach leads to a bias in the final t1 map in our work instead of considering motion correction as a preprocessing step we recover the motion free t1 map using a unified estimation approach in particular we propose a unified framework where the motion parameters and the t1 map are simultaneously estimated with a maximum likelihood ml estimator with our framework the relaxation model the motion model as well as the data statistics are jointly incorporated to provide substantially more accurate motion and t1 parameter estimates experiments with realistic monte carlo simulations show that the proposed unified ml framework outperforms the conventional two step approach as well as state of the art model based approaches in terms of both motion and t1 map accuracy and mean square error furthermore the proposed method was additionally validated in a controlled experiment with real t1 weighted data and with two in vivo human brain t1 weighted data sets showing its applicability in real life scenarios

boosting method for local learning in statistical pattern recognition we propose a local boosting method in classification problems borrowing from an idea of the local likelihood method our proposal local boosting includes a simple device for localization for computational feasibility we proved the bayes risk consistency of the local boosting in the framework of probably approximately correct learning inspection of the proof provides a useful viewpoint for comparing ordinary boosting and local boosting with respect to the estimation error and the approximation error both boosting methods have the bayes risk consistency if their approximation errors decrease to zero compared to ordinary boosting local boosting may perform better by controlling the trade off between the estimation error and the approximation error ordinary boosting with complicated base classifiers or other strong classification methods including kernel machines may have classification performance comparable to local boosting with simple base classifiers for example decision stumps local boosting however has an advantage with respect to interpretability local boosting with simple base classifiers offers a simple way to specify which features are informative and how their values contribute to a classification rule even though locally several numerical studies on real data sets confirm these advantages of local boosting

pet ct image registration in the chest using free form deformations we have implemented and validated an algorithm for three dimensional positron emission tomography transmission to computed tomography registration in the chest using mutual information as a similarity criterion inherent differences in the two imaging protocols produce significant nonrigid motion between the two acquisitions a rigid body deformation combined with localized cubic b splines is used to capture this motion the deformation is defined on a regular grid and is parameterized by potentially several thousand coefficients together with a spline based continuous representation of images and parzen histogram estimates our deformation model allows closed form expressions for the criterion and its gradient a limited memory quasi newton optimization algorithm is used in a hierarchical multiresolution framework to automatically align the images to characterize the performance of the method 27 scans from patients involved in routine lung cancer staging were used in a validation study the registrations were assessed visually by two expert observers in specific anatomic locations using a split window validation technique the visually reported errors are in the 0 to 6 mm range and the average computation time is 100 min on a moderate performance workstation

nonnegative matrix factorization for rapid recovery of constituent spectra in magnetic resonance chemical shift imaging of the brain we present an algorithm for blindly recovering constituent source spectra from magnetic resonance mr chemical shift imaging csi of the human brain the algorithm which we call constrained nonnegative matrix factorization cnmf does not enforce independence or sparsity instead only requiring the source and mixing matrices to be nonnegative it is based on the nonnegative matrix factorization nmf algorithm extending it to include a constraint on the positivity of the amplitudes of the recovered spectra this constraint enables recovery of physically meaningful spectra even in the presence of noise that causes a significant number of the observation amplitudes to be negative we demonstrate and characterize the algorithm s performance using sup 31 p volumetric brain data comparing the results with two different blind source separation methods bayesian spectral decomposition bsd and nonnegative sparse coding nnsc we then incorporate the cnmf algorithm into a hierarchical decomposition framework showing that it can be used to recover tissue specific spectra given a processing hierarchy that proceeds coarse to fine we demonstrate the hierarchical procedure on sup 1 h brain data and conclude that the computational efficiency of the algorithm makes it well suited for use in diagnostic work up

semilinear elliptic problems with mixed dirichlet neumann boundary conditions this work deals with the analysis of problems u uq urin u 0in b u 0on where b u means mixed boundary conditions we prove results about existence multiplicity and a priori estimates