the stein stromberg covering theorem in metric spaces abstract in 8 naor and tao extended to the metric setting the o d log d bounds given by stein and stromberg for lebesgue measure in r d deriving these bounds first from a localization result and second from a random vitali lemma here we show that the stein stromberg original argument can also be adapted to the metric setting giving a third proof we also weaken the hypotheses and additionally we sharpen the estimates for lebesgue measure

jump and sharp cusp detection by wavelets summary a method is proposed to detect jumps and sharp cusps in a function which is observed with noise by checking if the wavelet transformation of the data has significantly large absolute values across fine scale levels asymptotic theory is established and practical implementation is discussed the method is tested on simulated examples and applied to stock market return data the analysis of change points which describe sudden localised changes has recently found increasing interest change points can be used to model practical problems arising in fields such as quality control economics medicine signal and image processing and physical sciences for example in electroencephalogram signals sharp cusps exhibit the accelerations and decelerations in the beating of the hearts many practical problems like this involve functions which have jumps and sharp cusps the recently developed theory of wavelets has drawn much attention from both math ematicians statisticians and engineers in the seminal work of donoho 1993 donoho johnstone 1994 1995a b and donoho johnstone et al 1995 orthonormal bases of compactly supported wavelets have been used to estimate functions the theory of wavelets permits decomposition of functions into localised oscillating components this is an ideal tool to study localised changes such as jumps and sharp cusps in one dimension as well as several dimensions unlike traditional smoothing methods based on a fixed spatial scale the wavelet method is a multiresolution approach and has local adaptivity in this

linear switching systems with slow growth of trajectories abstract we prove the existence of positive linear switching systems continuous time whose trajectories grow to infinity but slower than a given increasing function this implies that unlike the situation with linear ode the maximal growth of trajectories of linear systems may be arbitrarily slow for systems generated by a finite set of matrices this phenomenon is proved to be impossible in dimension 2 while in all bigger dimensions the sublinear growth may occur the corresponding examples are provided and several open problems are formulated

random martingales and localization of maximal inequalities abstract let x d be a metric measure space for r 0 consider the hardy littlewood maximal operator m r f x def sup r r 1 b x r b x r f d we show that if there is an n 1 such that one has the microdoubling condition b x 1 1 n r b x r for all x x and r 0 then the weak 1 1 norm of m r has the following localization property m r l 1 x l 1 x sup r 0 m r r n r l 1 x l 1 x an immediate consequence is that if x d is ahlfors david n regular then the weak 1 1 norm of m r is n log n generalizing a result of stein and stromberg 1983 47 we show that this bound is sharp by constructing a metric measure space x d that is ahlfors david n regular for which the weak 1 1 norm of m 0 is n log n the localization property of m r is proved by assigning to each f l 1 x a distribution over random martingales for which the associated random doob maximal inequality controls the weak 1 1 inequality for m r

remarks on the hardy littlewood maximal function we answer questions of a carbery m trinidad menarguez and f soria by proving firstly that for the centred hardy littlewood maximal function on the real line the best constant c for the weak type 1 1 inequality is strictly larger than 3 2 and secondly that c is strictly less than 2 known to be the best constant in the noncentred case

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

misspecified structural change threshold and markov switching models sudden perturbations of a large amplitude occur frequently in macroeconomic and financial time series a usual practice is to test linearity against a permanent structural change however changes can also be captured by nonlinear stationary models such that threshold and markov switching models in this paper we show that tests designed for a threshold alternative have also power against parameter instability originating from structural change or markov switching models on the other hand it is shown that tests for structural change have no power if the data are generated by a markov switching or threshold model therefore it appears that testing the null of parameter stability against a threshold alternative is a robust way to detect parameter instability in economic and financial time series a monte carlo analysis based on several models studied in the literature illustrates how the tests perform in practice

change point estimation using nonparametric regression we consider a regression model in which the mean function may have a discontinuity at an unknown point we propose an estimate of the location of the discontinuity based on one side nonparametric regression estimates of the mean function the change point estimate is shown to converge in probability at rate 0 n 1 and to have the same asymptotic distribution as maximum likelihood estimates considered by other authors under parametric regression models confidence regions for the location and size of the change are also discussed

on the asymptotic properties of a family of matrices in this paper we consider bounded families f of complex n n matrices after introducing the concept of asymptotic order we investigate how the norm of products of matrices behaves as the number of factors goes to infinity in the case of defective families f u sing the asymptotic order allows us to get a deeper knowledge of the asymptotic behaviour than just considering the so called generalized spectral radius with reference to the well known finiteness conjecture for finite families we also introduce the concepts of spectrum maximizing product and limit spectrum maximizing product showing that for finite families of 2 2matrices defectivity is equivalent to the existence of defective such limit products 2001 elsevier science inc all rights reserved

directional maximal operators and lacunarity in higher dimensions we introduce a notion of lacunarity in higher dimensions for which we can bound the associated directional maximal operators in l p bbb r n with p1 in particular we are able to treat the classes previously considered by nagel stein wainger sj ogren sj olin and carbery closely related to this we find a characterization of the sets of directions which give rise to bounded maximal operators the bounds enable lebesgue type differentiation of integrals in l rm loc p bbb r n replacing balls by tubes which point in these directions

discontinuous regression surfaces fitting we suggest a three stage procedure to recover discontinuous regression surfaces when noisy data are present in the first stage jump candidate points are detected using a jump detection criterion a local principal component line is then fitted through these points in a neighborhood of a design point this line provides a first order approximation to the true jump location curve in that neighborhood in the third stage observations on the same side of the line as the given point are combined using a weighted average procedure to fit the surface at that point if there are no jump candidate points in the neighborhood then all observations in that neighborhood are used in the surface fitting if however the center of the neighborhood is on a jump location curve only those observations on one side of the line are used thus blurring is automatically avoided around the jump locations this methodology requires o n k 2 computation where n is the sample size and k is the window width its assumptions on the model are flexible some numerical results are presented to evaluate the surface fit and to discuss the selection of the window widths

bandwidth selection for changepoint estimation in nonparametric regression nonparametric estimation of abrupt changes in a regression function involves choosing smoothing bandwidth parameters the performance of estimation procedures depends heavily on this choice so far little attention has been paid to the crucial issue of choosing appropriate bandwidth parameters in practice in this article we propose a bootstrap procedure for selecting the bandwidth parameters in a nonparametric two step estimation method this method results in a fully data driven procedure for estimating a finite but possibly unknown number of changepoints in a regression function we evaluate the performance of the data driven procedure via a simulation study which reveals that the fully automatic procedure performs quite well as an illustration we apply the procedure to some real data

pointwise theorems for amenable groups in this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along folner sequences that obey some restrictions these restrictions are mild enough so that such sequences exist for all amenable groups we also prove a generalization of the shannon mcmillan breiman theorem to all discrete amenable groups

linear estimators in change point problems observations xi are uncorrelated with means 6i i 1 n and variances 1 the linear estimators 0 tx for some n x n matrix t are widely used in smoothing problems where it is assumed that neighbouring parameter values are similar the smoothness assumption is violated in change point problems where neighbouring parameter values are equal except at some unspecified change points where there are jumps of unknown size from one parameter value to the next in the case of a single change point in one dimension for any linear estimator the expected sum of squared errors between estimates and parameters is of order v in for some choice of parameters compared to order 1 for the least squares estimate we show similar results for adaptive shift estimators in which the linear estimator uses a kernel estimated from the data finally for a change point problem in two dimensions the expected sum of squared errors is of order n3 4

linear smoothers and additive models we study linear smoothers and their use in building nonparametric regression models in the first part of this paper we examine certain aspects of linear smoothers for scatterplots examples of these are the running mean and running line kernel and cubic spline smoothers the eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother and several other topics such as the number of degrees of freedom of a smoother are discussed in the second part of the paper we describe how linear smoothers can be used to estimate the additive model a powerful nonparametric regression model using the backfitting algorithm we show that backfitting is the gauss seidel iterative method for solving a set of normal equations associated with the additive model we provide conditions for consistency and nondegeneracy and prove convergence for the backfitting and related algorithms for a class of smoothers that includes cubic spline smoothers

smooth optimum kernel estimators near endpoints summary kernel estimators for smooth curves like density spectral density or regression functions require modifications when estimating near endpoints of the support both for practical and asymptotic reasons the construction of such boundary kernels as solutions of a variational problem is addressed and representations in orthogonal polynomials are given including explicit solutions for the most important cases based on explicit formulae for certain functionals of the kernels it is shown that local bandwidth variation might be indicated near boundaries various bandwidth variation schemes are discussed and

change point estimation under adaptive sampling we consider the problem of locating a jump discontinuity change point in a smooth parametric regression model with a bounded covariate it is assumed that one can sample the covariate at different values and measure the corresponding responses budget constraints dictate that a total of n such measurements can be obtained a multistage adaptive procedure is proposed where at each stage an estimate of the change point is obtained and new points are sampled from its appropriately chosen neighborhood it is shown that such procedures accelerate the rate of convergence of the least squares estimate of the change point further the asymptotic distribution of the estimate is derived using empirical processes techniques the latter result provides guidelines on how to choose the tuning parameters of the multistage procedure in practice the improved efficiency of the procedure is demonstrated using real and synthetic data this problem is primarily motivated by applications in engineering systems

consistencies and rates of convergence of jump penalized least squares estimators we study the asymptotics for jump penalized least squares regression aiming at approximating a regression function by piecewise constant functions besides conventional consistency and convergence rates of the estimates in l 2 0 1 our results cover other metrics like skorokhod metric on the space of cadlag functions and uniform metrics on c 0 1 we will show that these estimators are in an adaptive sense rate optimal over certain classes of approximation spaces special cases are the class of functions of bounded variation piecewise holder continuous functions of order 0 1 and the class of step functions with a finite but arbitrary number of jumps in the latter setting we will also deduce the rates known from change point analysis for detecting the jumps finally the issue of fully automatic selection of the smoothing parameter is addressed

inference of trends in time series summary we consider statistical inference of trends in mean non stationary models a test statistic is proposed for the existence of structural breaks in trends on the basis of a strong invariance principle of stationary processes we construct simultaneous confidence bands with asymptotically correct nominal coverage probabilities the results are applied to global warming temperature data and nile river flow data our confidence band of the trend of the global warming temperature series supports the claim that the trend is increasing over the last 150 years

a local polynomial jump detection algorithm in nonparametric regression we suggest a one dimensional jump detection algorithm based on local polynomial fitting for jumps in regression functions zero order jumps or jumps in derivatives first order or higherorder jumps if jumps exist in the mth order derivative of the underlying regression function then an m 1 order polynomial is fitted in a neighborhood of each design point we then characterize the jump information in the coefficients of the highest order terms of the fitted polynomials and suggest an algorithm for jump detection this method is introduced briefly for the general setup and then presented in detail for zero order and first order jumps several simulation examples are discussed we apply this method to the bombay india sea level pressure data

localization and dimension free estimates for maximal functions abstract in the recent paper a naor t tao random martingales and localization of maximal inequalities j funct anal 259 3 2010 731 779 naor and tao introduce a new class of measures with a so called micro doubling property and present via martingale theory and probability methods a localization theorem for the associated maximal functions as a consequence they obtain a weak type estimate in a general abstract setting for these maximal functions that is reminiscent of the n log n result of stein and stromberg in euclidean spaces the purpose of this work is twofold first we introduce a new localization principle that localizes not only in the time dilation parameter but also in space the proof uses standard covering lemmas and selection processes second we show that a uniform condition for micro doubling in the euclidean spaces provides dimension free estimates for their maximal functions in all l p with p 1 this is done introducing a new technique that allows to differentiate through dimensions

on the marginal instability of linear switched systems abstract stability properties for continuous time linear switched systems are at first determined by the largest lyapunov exponent associated with the system which is the analogue of the joint spectral radius for the discrete time case the purpose of this paper is to provide a characterization of marginally unstable systems i e systems for which the lyapunov exponent is equal to zero and there exists an unbounded trajectory and to analyze the asymptotic behavior of their trajectories our main contribution consists in pointing out a resonance phenomenon associated with marginal instability in the course of our study we derive an upper bound of the state at time t which is polynomial in t and whose degree is computed from the resonance structure of the system we also derive analogous results for discrete time linear switched systems

efficient algorithms for deciding the type of growth of products of integer matrices for a given finite set sigma of matrices with nonnegative integer entries we study the growth with t of max parallel to a 1 a t parallel to a i epsilon sigma we show how to determine in polynomial time whether this growth is bounded polynomial or exponential and we characterize all possible behaviors c 2007 elsevier inc all rights reserved

nonparametric inference on structural breaks this paper proposes estimators of location and size of structural breaks in a possibly dynamic nonparametric regression model the structural breaks can be located at given periods of time and or they can be explained by the values taken by some regressor as in threshold models no previous knowledge of the underlying regression function is required the paper also studies the case in which several regressors explain the breaks we derive the rate of convergence and provide central limit theorems for the estimators of the location s and size s a monte carlo experiment illustrates the performance of our estimators in small samples

asymptotic score statistic processes and tests for constant hazard against a change point alternative the problem of testing for a constant failure rate against alternatives with failure rates involving a single change point is considered the asymptotic significance level for tests based on maximal score statistics are shown to involve the solution to a first passage time problem for an ornstein uhlenbeck process an example illustrates the methodology

detecting gradual changes in locally stationary processes in a wide range of applications the stochastic properties of the observed time series change over time the changes often occur gradually rather than abruptly the properties are approximately constant for some time and then slowly start to change in such situations it is frequently of interest to locate the time point where the properties start to vary in contrast to the analysis of abrupt changes methods for detecting smooth or gradual change points are less developed and often require strong parametric assumptions in this paper we develop a fully nonparametric method to estimate a smooth change point in a locally stationary framework we set up a general procedure which allows to deal with a wide variety of stochastic properties including the mean auto covariances and higher order moments the theoretical part of the paper establishes the convergence rate of the new estimator in addition we examine its finite sample performance by means of a simulation study and illustrate the methodology by applications to temperature and financial return data

jump information criterion for statistical inference in estimating discontinuous curves nonparametric regression analysis when the regression function is discontinuous has many applications existing methods for estimating a discontinuous regression curve usually assume that the number of jumps in the regression curve is known beforehand which is unrealistic in some situations although there has been research on estimation of a discontinuous regression curve when the number of jumps is unknown the problem remains mostly open because such research often requires assumptions on other related quantities such as a known minimum jump size in this paper we propose a jump information criterion which consists of a term measuring the fidelity of the estimated regression curve to the observed data and a penalty related to the number of jumps and the jump sizes the number of jumps can then be determined by minimizing our criterion theoretical and numerical studies show that our method works well

a maximum principle for the stability analysis of positive bilinear control systems with applications to positive linear switched systems we consider a continuous time bilinear control system bcs with metzler matrices each entry in the transition matrix of such a system is nonnegative making the positive orthant an invariant set of the dynamics motivated by the stability analysis of positive linear switched systems plsss we define a control as optimal if for a fixed final time it maximizes the spectral radius of the transition matrix our main result is a first order necessary condition for optimality in the form of a maximum principle mp the proof of this mp combines the standard needle variation with a basic result from the perron frobenius theory of nonnegative matrices we describe several applications of this mp to the stability analysis of plsss under arbitrary switching

estimation of heteroscedasticity in regression analysis consider the regression model y i g t i varepsilon i 1 leq i leq n with nonrandom design variables t i and measurements y i for the unknown regression function g cdot we assume that the data are heteroscedastic i e e varepsilon 2 i sigma 2 i not equiv operatorname const and investigate how to estimate sigma 2 i if sigma 2 i sigma 2 t i with a smooth function sigma 2 cdot initial estimators tilde sigma 2 i can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent these estimates can be used to improve the estimation of the regression function g itself in parametric and nonparametric models further applications are suggested

jump detection in a regression curve and its derivative curve estimation from observed noisy data has broad applications in certain applications the underlying regression curve may have singularities including jumps and roofs valleys i e jumps in the first order derivative of the regression curve at some unknown positions representing structural changes of the related process detection of such singularities is important for understanding the structural changes in the literature a number of jump detection procedures have been proposed most of which are based on estimation of the one sided first order derivatives of the true regression curve in this paper motivated by certain related research in image processing we propose an alternative jump detection procedure besides the first order derivatives we suggest using helpful information about jumps in the second order derivatives as well theoretical justifications and numerical studies show that this jump detector works well in applications this procedure is then extended for detecting roofs v

the best constant for the centered hardy littlewood maximal inequality we find the exact value of the best possible constant c for the weak type 1 1 inequality for the one dimensional centered hardy littlewood maximal operator we prove that c is the largest root of the quadratic equation 12c 2 22c 5 0 thus obtaining c 1 5675208 this is the first time the best constant for one of the fundamental inequalities satisfied by a centered maximal operator is precisely evaluated

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

boundary crossing probabilities and statistical applications abstract this paper surveys recent results involving boundary crossing probabilities and related statistical applications the first part is concerned with problems of sequential analysis especially repeated significance tests and their application to sequential clinical trials involving survival data the second part develops the probability theory motivated by the problems of part 1 a method for computing first passage distributions of brownian motion to linear boundaries is introduced and then modified to handle problems in discrete time and those involving nonlinear boundaries the third part is concerned with fixed sample statistical problems especially change point problems which involve boundary crossing probabilities examples are given of problems for which methods of part 2 appear adequate and of problems which require new methods

estimation of a function with discontinuities via local polynomial fit with an adaptive window choice we propose a method of adaptive estimation of a regression function which is near optimal in the classical sense of the mean integrated error at the same time the estimator is shown to be very sensitive to discontinuities or change points of the underlying function f or its derivatives for instance in the case of a jump of a regression function beyond the intervals of length in order n 1 log n around change points the quality of estimation is essentially the same as if locations of jumps were known the method is fully adaptive and no assumptions are imposed on the design number and size of jumps the results are formulated in a nonasymptotic way and can therefore be applied for an arbitrary sample size

the weak type 1 1 bounds for the maximal function associated to cubes grow to infinity with the dimension let md be the centered hardy littlewood maximal function associated to cubes in r d with lebesgue measure and let cd denote the lowest constant appearing in the weak type 1 1 inequality satised by md we show that cd 1 as d 1 thus answering for the case of cubes a longstanding

asymptotics for change point models under varying degrees of mis specification change point models are widely used by statisticians to model drastic changes in the pattern of observed data least squares maximum likelihood based estimation of change points leads to curious asymptotic phenomena when the change point model is correctly specified such estimates generally converge at a fast rate n and are asymptotically described by minimizers of a jump process under complete mis specification by a smooth curve i e when a change point model is fitted to data described by a smooth curve the rate of convergence slows down to n1 3 and the limit distribution changes to that of the minimizer of a continuous gaussian process in this paper we provide a bridge between these two extreme scenarios by studying the limit behavior of change point estimates under varying degrees of model mis specification by smooth curves which can be viewed as local alternatives we find that the limiting regime depends on how quickly the alternatives approach a change point model we unravel a family of intermediate limits that can transition at least qualitatively to the limits in the two extreme scenarios the theoretical results are illustrated via a set of carefully designed simulations we also demonstrate how inference for the change point parameter can be performed in absence of knowledge of the underlying scenario by resorting to subsampling techniques that involve estimation of the convergence rate

edge preserving smoothers for image processing abstract classical smoothers have limited usefulness in image processing because sharp edges tend to be blurred there is a literature on edge preserving smoothers but these all require moderately large smooth stretches here we discuss an approach to this problem called sigma filtering and propose an improvement based on running m estimation both computational and theoretical aspects are developed for image processing the methods have a niche between standard filtering approaches and bayes markov random field analysis

adaptive weights smoothing with applications to image restoration we propose a new method of nonparametric estimation which is based on locally constant smoothing with an adaptive choice of weights for every pair of data points some theoretical properties of the procedure are investigated then we demonstrate the performance of the method on some simulated univariate and bivariate examples and compare it with other nonparametric methods finally we discuss applications of this procedure to magnetic resonance and satellite imaging

optimal change point estimation from indirect observations we study nonparametric change point estimation from indirect noisy observations focusing on the white noise convolution model we consider two classes of functions that are smooth apart from the change point we establish lower bounds on the minimax risk in estimating the change point and develop rate optimal estimation procedures the results demonstrate that the best achievable rates of convergence are determined both by smoothness of the function away from the change point and by the degree of ill posedness of the convolution operator optimality is obtained by introducing a new technique that involves as a key element detection of zero crossings of an estimate of the properly smoothed second derivative of the underlying function

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

a note on marginal stability of switched systems in this note we present criteria for marginal stability and marginal instability of switched systems for switched nonlinear systems we prove that uniform stability is equivalent to the existence of a common weak lyapunov function cwlf that is generally not continuous for switched linear systems we present a unified treatment for marginal stability and marginal instability for both continuous time and discrete time switched systems in particular we prove that any marginally stable system admits a norm as a cwlf by exploiting the largest invariant set contained in a polyhedron several insightful algebraic characteristics are revealed for marginal stability and marginal instability

on the centered hardy littlewood maximal operator we will study the centered hardy littlewood maximal operator acting on positive linear combinations of dirac deltas we will use this to obtain improvements in both the lower and upper bounds or the best constant c in the l 1 weak l1 inequality for this operator in fact we will show that 11 61 12 1 5675208 c 5 3 1 66

nonparametric analysis of changes in hazard rates for censored survival data an alternative to change point models summary as a nonparametric estimator for the point of the most rapid change of a hazard rate we propose the location of an extremum of a nonparametric estimate of the derivative or equivalently of a zero of a nonparametric estimate of the second derivative using the kernel method for the nonparametric estimation of derivatives of the hazard rate the asymptotic local limiting distribution and uniform consistency are applied to prove consistency and to find the limiting distribution of these estimators under random censoring and to construct confidence intervals both for the derivatives of the hazard rate and for the point of most rapid change an application to leukaemia data illustrates this concept and we discuss its relations to change point modelling the monte carlo method is used to assess the reliability of finite sample analyses and in particular of the given example

on the functional estimation of jump diffusion models abstract we provide a general asymptotic theory for the fully functional estimates of the infinitesimal moments of continuous time models with discontinuous sample paths of the jump diffusion type minimal requirements are placed on the dynamic properties of the underlying jump diffusion process i e stationarity is not required our theoretical framework justifies consistent in a statistical sense nonparametric extraction of the parameters and functions that drive the dynamic evolution of the process of interest i e the potentially nonaffine and level dependent intensity of the jump arrival being an example from the estimated infinitesimal conditional moments as suggested in johannes 2003 the statistical and economic role of jumps in continuous time interest rate models journal of finance forthcoming

jump surface estimation edge detection and image restoration surface estimation is important in many applications when conventional smoothing procedures e g running averages local polynomial kernel smoothing procedures smoothing spline procedures are used for estimating jump surfaces from noisy data jumps are blurred at the same time when noise is removed in recent years new smoothing methodologies have been proposed in the statistical literature for detecting jumps in surfaces and for estimating jump surfaces with jumps preserved we provide a review of these methodologies because a monochrome image can be considered a jump surface of the image intensity function with jumps at the outlines of objects edge detection and image restoration problems in image processing are closely related to the jump surface estimation problem in statistics we also review major methodologies on edge detection and image restoration and discuss connections and differences among these methods and related methods in the statistical literature

change points in nonparametric regression analysis an otherwise smooth regression model are proposed the assumptions needed are much weaker than those made in parametric models the proposed estimators apply as well to the detection of discontinuities in derivatives and therefore to the detection of change points of slope and of higher order curvature the proposed estimators are based on a comparison of left and right one sided kernel smoothers weak convergence of a stochastic process in local differences to a gaussian process is established for properly scaled versions of estimators of the location of a change point the continuous mapping theorem can then be invoked to obtain asymptotic distributions and corresponding rates of convergence for change point estimators these rates are typically faster than n 1 2 rates of global lp convergence of curve estimates with appropriate kernel modifications adapting to estimated change points are derived as a consequence it is shown that these rates of convergence are the same as if the location of the change point was known the methods are illustrated by means of the well known data on the annual flow volume of the nile river between 1871 and 1970 1 introduction nonparametric regression methods are usually applied in order to obtain a smooth fit of a regression curve without having to specify a parametric class of regression functions sometimes a generally smooth curve might contain an isolated discontinuity or change point in the curve or in a possibly higher order derivative and in many cases interest focuses on the occurrence of such change points in parametric approaches to the regression change point problem simple linear regressions before and after a possible change point are assumed and then the possibility of a discontinuity in the form of a jump or of a jump in the first derivative or equivalently a slope change is incorporated into the model see for instance hinkley 1969 and brown durbin and evans 1975 the analysis of change points which describe sudden localized changes typically occurring in economics medicine and the physical sciences has recently found increasing interest general smoothness assumptions allowing for a large class of regression functions to be considered seem to be more appropriate in a variety of applied problems than parametric modelling an

the asymptotic distributions of kernel estimators of the mode in a decreasing sequence of intervals centered on the true mode the normalized kernel estimate of the density converges weakly to a nonstationary gaussian random process the expected value of this process is a parabola through the origin the covariance function of this process depends on the smoothness of the kernel when the kernel is mean square differentiable the location of the maximum of this process has a normal distribution when the kernel is discontinuous the location of the maximum has a distribution related to a solution of the heat equation

smoothing with split linear fits we introduce a family of smoothing algorithms that can produce discontinuous output unlike most commonly used smoothers that tend to blur discontinuities in the data this smoother can be used for smoothing with edge detection we cite examples of other approaches to two dimensional smoothing with edge detection in image processing and apply our one dimensional smoother to sea surface temperature data where the discontinuities arise from changes in ocean currents