a nonoverlapping domain decomposition method for orthogonal spline collocation problems a nonoverlapping domain decomposition approach is used on uniform and matching grids to first define and then to compute the orthogonal spline collocation solution of the dirichlet boundary value problem for poisson s equation on an l shaped region we prove existence and uniqueness of the collocation solution and derive optimal order hs norm error bounds for s 0 1 2 the collocation solution on two interfaces is computed using the preconditioned conjugate gradient method and the collocation solution on three squares is computed by a matrix decomposition method that uses fast fourier transforms the total cost of the algorithm is o n2 log n where the number of unknowns in the collocation solution is o n2

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

a fast domain decomposition poisson solver on a rectangle for hermite bicubic orthogonal spline collocation a fast domain decomposition solver is presented for the piecewise hermite bicubic orthogonal spline collocation solution of poisson s equation on a rectangle the rectangle is divided into parallel strips and the collocation solution is first obtained on the interfaces by solving a collection of independent tridiagonal linear systems a recently developed fast fourier transform solver for piecewise hermite bicubic orthogonal spline collocation is then used to compute the collocation solution on each strip on an n times n uniform partition the proposed domain decomposition solver requires o n 2 log log n arithmetic operations assuming that the strips have the same width and that their number is proportional to n log n the solver is also highly parallel in nature

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

discontinuous versus smooth regression given measurements x i y i i 1 n we discuss methods to assess whether an underlying regression function is smooth continuous or differentiable or whether it has discontinuities the variance of the measurements is assumed to be unknown and is estimated simultaneously by regressing squared differences of the data formed with various span sizes on the span size itself we obtain an asymptotic linear model with dependent errors the parameters of this asymptotic linear model include the sum of the squared jump sizes as well as the variance of the measurements both parameters can be consistently estimated with mean squared error rates of convergence of n 2 3 for the sum of squared jump sizes and n 1 for the error variance we derive the asymptotic constants of the mean squared error mse and discuss the dependence of mse on the maximum span size l the test for the existence of jumps is formulated for the null hypothesis that the sum of squared jump sizes is 0 the asymptotic distribution of the test statistic is obtained essentially via a central limit theorem for u statistics we motivate and illustrate the methods with data surrounded by a scientific controversy concerning the question whether the growth of children occurs smoothly or rather in jumps

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

hierarchical ts fuzzy system and its universal approximation an efficient tool to deal with the rule explosion problem is the hierarchical system by which a fuzzy system can be decomposed into a number of hierarchically connected low dimensional systems in this paper a generalized hierarehical tagaki sugeno ts system is built it is shown that the input output i o relationship of this generalized hierarehical system can be represented as one of a standard ts fuzzy system and the system approximation capability is analyzed by taking piecewise linear functions as a bridge by constructive method it is proven that the hierarchical fuzzy systems hfs s can be universal approximators for the given approximation accuracy an estimation formula about the number of the rules needed in the hfs is established finally some simulation examples confirm that the hfs s with smaller size rule base can approximate the given functions with high accuracy the results obtained here provide us with the theoretical basis for various applications of hfs s

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

orthogonal spline collocation methods for partial differential equations abstract this paper provides an overview of the formulation analysis and implementation of orthogonal spline collocation osc also known as spline collocation at gauss points for the numerical solution of partial differential equations in two space variables advances in the osc theory for elliptic boundary value problems are discussed and direct and iterative methods for the solution of the osc equations examined the use of osc methods in the solution of initial boundary value problems for parabolic hyperbolic and schrodinger type systems is described with emphasis on alternating direction implicit methods the osc solution of parabolic and hyperbolic partial integro differential equations is also mentioned finally recent applications of a second spline collocation method modified spline collocation are outlined

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

detections of changes in return by a wavelet smoother with conditional heteroscedastic volatility in this paper we propose two estimators an integral estimator and a discretized estimator for the wavelet coefficient of regression functions in nonparametric regression models with heteroscedastic variance these estimators can be used to test the jumps of the regression function the model allows for lagged dependent variables and other mixing regressors the asymptotic distributions of the statistics are established and the asymptotic critical values are analytically obtained from the asymptotic distribution we also use the test to determine consistent estimators for the locations of change points the jump sizes and locations of change points can be consistently estimated using wavelet coefficients and the convergency rates of these estimators are derived we perform some monte carlo simulations to check the powers and sizes of the test statistics finally we give practical examples in finance and economics to detect changes in stock returns and short term interest rates using the empirical wavelet method

some approximation properties of adaptive fuzzy systems with variable universe of discourse in the last 20years while most research on fuzzy approximation theory has focused on nonadaptive fuzzy systems little work has been done on adaptive fuzzy systems this paper introduces an algorithm for adaptive fuzzy systems with variable universe of discourse vud by means of contraction expansion factors universe of discourse can be modified online and fuzzy rules can reproduce automatically to adapt to the modified universe of discourse thus dependence on the size of initial rule base is greatly reduced using stone weierstrass theorem vud adaptive fuzzy systems are proved to be universal approximators with two order approximation accuracy in addition the convergence properties of approximation error are discussed and a sufficient condition is presented to partition universe of discourse and to calculate the size of rule base an example is also given to illustrate the approximation power of vud adaptive fuzzy systems

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

approximation capabilities of hierarchical fuzzy systems derived from practical application in location analysis and pricing and based on the approach of hierarchical structure analysis of continuous functions this paper investigates the approximation capabilities of hierarchical fuzzy systems by first introducing the concept of the natural hierarchical structure it is proved that continuous functions with natural hierarchical structure can be naturally and effectively approximated by hierarchical fuzzy systems to overcome the curse of dimensionality in both the number of rules and parameters then based on kolmogorov s theorem it is shown that any continuous function can be represented as a superposition of functions with the natural hierarchical structure and can then be approximated by hierarchical fuzzy systems to achieve the universal approximation property further the conditions under which the hierarchical fuzzy approximation is superior to the standard fuzzy approximation in overcoming the curse of dimensionality are analyzed

wavelet analysis of change points in a non parametric regression with heteroscedastic variance abstract in this paper we develop wavelet methods for detecting and estimating jumps and cusps in the mean function of a non parametric regression model an important characteristic of the model considered here is that it allows for conditional heteroscedastic variance a feature frequently encountered with economic and financial data wavelet analysis of change points in this model has been considered in a limited way in a recent study by chen et al 2008 with a focus on jumps only one problem with the aforementioned paper is that the test statistic developed there has an extreme value null limit distribution the results of other studies have shown that the rate of convergence to the extreme value distribution is usually very slow and critical values derived from this distribution tend to be much larger than the true ones here we develop a new test and show that the test statistic has a convenient null limit n 0 1 distribution this feature gives the proposed approach an appealing advantage over the existing approach another attractive feature of our results is that the asymptotic theory developed here holds for both jumps and cusps implementation of the proposed method for multiple jumps and cusps is also examined the results from a simulation study show that the new test has excellent power and the estimators developed also yield very accurate estimates of the positions of the discontinuities

fuzzy systems as universal approximators an additive fuzzy system can uniformly approximate any real continuous function on a compact domain to any degree of accuracy an additive fuzzy system approximates the function by covering its graph with fuzzy patches in the input output state space and averaging patches that overlap the fuzzy system computes a conditional expectation e y x if we view the fuzzy sets as random sets each fuzzy rule defines a fuzzy patch and connects commonsense knowledge with state space geometry neural or statistical clustering systems can approximate the unknown fuzzy patches from training data these adaptive fuzzy systems approximate a function at two levels at the local level the neural system approximates and tunes the fuzzy rules at the global level the rules or patches approximate the function

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

sufficient conditions on general fuzzy systems as function approximators abstract in a constructive way we have found sufficient conditions under which general fuzzy systems can uniformly approximate any real continuous function on a compact domain to any degree of accuracy more importantly we have revealed the underlying mechanism of such function approximation the fuzzy systems can uniformly approximate any polynomials which according to the weierstrass approximation theorem can uniformly approximate any continuous function on a compact domain these findings lead to the following practical results 1 explicit fuzzy rules of fuzzy systems can now be easily obtained according to functions to be approximated and 2 formulas are derived which can calculate the number of input fuzzy sets output fuzzy sets and fuzzy rules needed in order to satisfy any given approximation accuracy the number increases as required approximation error decreases and as approximation error approaches zero the number approaches these formulas also reveal that the number is minimal when the maximum number of intersection between adjacent input fuzzy sets is one practical implications of these results will be discussed especially in the context of fuzzy control and fuzzy modeling two examples are provided to demonstrate how to design fuzzy systems to approximate given functions with required approximation accuracy

approximation theory of fuzzy systems based upon genuine many valued implications siso cases it is proved that the single input and single output siso fuzzy systems based upon genuine many valued implications are universal approximators it is shown theoretically that fuzzy control systems based upon genuine many valued implications are equivalent to those based upon t norm implications the general approach to construct fuzzy systems is given it is also shown that defuzzifier based upon center of areas is not appropriate to the fuzzy systems based upon genuine many valued implications

variable universe stable adaptive fuzzy control of a nonlinear system abstract a kind of stable adaptive fuzzy control of a nonlinear system is implemented based on the variable universe method proposed first in 1 first of all the basic structure of variable universe adaptive fuzzy controllers is briefly introduced then the contraction expansion factor which is a key tool of the variable universe method is defined by means of the integral regulation idea and then a kind of adaptive fuzzy controller is designed by using such a contraction expansion factor the simulation on the first order nonlinear system is done and as a result its simulation effect is quite good in comparison with the corresponding result in 2 3 second it is proved that the variable universe adaptive fuzzy control is asymptotically stable by use of liapunov theory the simulation on a second order nonlinear system shows that its simulation effect is also quite good in comparison with the corresponding result in 2 besides a useful tool called symbolic factor is proposed which may be of universal significance it can greatly reduce the setting time and enhance the robustness of the system

sugeno type controllers are universal controllers abstract we show that given any continuous real valued function on a compact set there is a sugeno type controller that will approximate this function to any degree of accuracy an application of this result to fuzzy control is presented which shows that this type of sugeno controller is a universal controller

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

hierarchical fuzzy systems for function approximation on discrete input spaces with application this paper investigates the capabilities of hierarchical fuzzy systems to approximate functions on discrete input spaces first it is shown that any function on a discrete space has an arbitrary separable hierarchical structure and can be naturally approximated by hierarchical fuzzy systems as a by product of this result a discrete version of kolmogorov s theorem is obtained second it is proven that any function on a discrete space can be approximated to any degree of accuracy by hierarchical fuzzy systems with any desired separable hierarchical structure that is functions on discrete spaces can be approximated more simply and flexibly than those on continuous spaces third a hierarchical fuzzy system identification method is proposed in which human knowledge and numerical data are combined for system construction and identification finally the proposed method is applied to the market condition performance modeling problem in site selection decision support and shows the better performance in both accuracy and interpretability than the regression and neural network approaches in additions the reason and mechanism why hierarchical fuzzy systems outperform regression and neural networks in this type of application are analyzed

stable fuzzy adaptive control for a class of nonlinear systems abstract in this paper two adaptive fuzzy control schemes for a class of nonlinear systems are proposed each scheme employs a fuzzy control with an adaptive law and a compensation control it is proved that the closed loop system is asymptotically stable based on lyapunov synthesis approach while the identification of the controlled plant is unnecessary finally these schemes are applied to control a chaotic system the simulation results show that our schemes are superior to the scheme of 12

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

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 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

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

nonparametric estimation of structural change points in volatility models for time series we propose a hybrid estimation procedure that combines the least squares and nonparametric methods to estimate change points of volatility in time series models its main advantage is that it does not require any specific form of marginal or transitional densities of the process we also establish the asymptotic properties of the estimators when the regression and conditional volatility functions are not known the proposed tests for change points of volatility are shown to be consistent and more powerful than the nonparametric ones in the literature finally we provide simulations and empirical results using the hong kong stock market index hsi series

nonparametric simultaneous testing for structural breaks in this paper we consider a regression model with errors that are martingale differences this modeling includes the regression of both independent and time series data the aim is to study the appearance of structural breaks in both the mean and the variance functions assuming that such breaks may occur simultaneously in both the functions we develop nonparametric testing procedures that simultaneously test for structural breaks in the conditional mean and the conditional variance the asymptotic distribution of an adaptive test statistic is established as well as its asymptotic consistency and efficiency simulations illustrate the performance of the adaptive testing procedure an application to the analysis of financial time series also demonstrates the usefulness of the proposed adaptive test in practice

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 half half plot the half half hh plot is a new graphical method to investigate qualitatively the shape of a regression curve the empirical hh plot counts observations in the lower and upper quarter of a strip that moves horizontally over the scatterplot the plot displays jumps clearly and reveals further features of the regression curve we prove a functional central limit theorem for the empirical hh plot with rate of convergence in a simulation study the good performance of the plot is demonstrated the method is also applied to two case studies the proofs and one more case study are deferred to a supplement which is available online

universal fuzzy controllers a collection b of fuzzy controllers is a universal fuzzy controller if and only if given any controllable process p it can also be controlled by some fuzzy controller in b this paper identifies a new type of universal fuzzy controller

markov chain monte carlo in conditionally gaussian state space models summary a bayesian analysis is given for a state space model with errors that are finite mixtures of normals and with coefficients that can assume a finite number of different values a sequence of indicator variables determines which components the errors belong to and the values of the coefficients the computation is carried out using markov chain monte carlo with the indicator variables generated without conditioning on the states previous approaches use the gibbs sampler to generate the indicator variables conditional on the states in many problems however there is a strong dependence between the indicator variables and the states causing the gibbs sampler to converge unacceptably slowly or even not to converge at all the new sampler is implemented in o n operations where n is the sample size permitting an exact bayesian analysis of problems that previously had no computationally tractable solution we show empirically that the new sampler can be much more efficient than previous approaches and illustrate its applicability to robust nonparametric regression with discontinuities and to a time series change point problem

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

tracking a smooth fault line in a response surface we suggest a sequential or tracking algorithm for estimating a smooth fault line in a response surface the method starts with an approximation to a point on the line and from there the line is tracked as it meanders through the plane the technique differs from recent approaches in that it does not require a large part of the plane to be searched for evidence of a fault line this offers potential computational savings and produces a method that is invariant under rotations of coordinate axes except insofar as a rotation might affect the estimated starting point and the relative orientation of the grid on which calculations are done that feature is important if design points are not located on a regular grid we investigate properties of the method under very general conditions on the design allowing poisson cluster processes jiggled grid processes and deterministic regular lattices uniform rates of convergence are derived in all these settings for the case of noisy data and shown to be within logarithmic factors of optimal pointwise convergence rates in the no noise setting

nonlinear context adaptation in the calibration of fuzzy sets abstract in this note we elaborate on the concept and use of context adaptation the underlying idea hinges upon a nonlinear transformation of an actual reference unit universe of discourse into a subset of reals say a b that is implied by actually available data current context assuming a collection of fuzzy sets a a1 a2 an defined over 0 1 the adaptation gives rise to a new frame of cognition a a1 a2 an expressed over a b owing inherent nonlinearity of the developed mapping different elements fuzzy sets of a can be stretched or expanded according to the given experimental data proposed is a neural network as a relevant optimization tool

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

a comparative study on sufficient conditions for takagi sugeno fuzzy systems as universal approximators universal approximation is the basis of theoretical research and practical applications of fuzzy systems studies on the universal approximation capability of fuzzy systems have achieved great progress in recent years in this paper linear takagi sugeno ts fuzzy systems that use linear functions of input variables as rule consequent and their special case named simplified fuzzy systems that use fuzzy singletons as rule consequent are investigated on condition that overlapped fuzzy sets are employed new sufficient conditions for simplified fuzzy systems and linear ts fuzzy systems as universal approximators are given respectively then a comparative study on existing sufficient conditions is carried out with numeric examples

approximation theory of fuzzy systems spl minus siso case in this paper the approximation properties of mimo fuzzy systems generated by the product inference are discussed we first give an analysis of fuzzy basic functions fbf s and present several properties of fbf s based on these properties of fbf s we obtain several basic approximation properties of fuzzy systems 1 basic approximation property which reveals the basic approximation mechanism of fuzzy systems 2 uniform approximation bounds which give the uniform approximation bounds between the desired control or decision functions and fuzzy systems 3 uniform convergent property which shows that fuzzy systems with defined approximation accuracy can always be obtained by dividing the input space into finer fuzzy regions and 4 universal approximation property which shows that fuzzy systems are universal approximators and extends some previous results on this aspect the similarity between fuzzy systems and mathematical approximation is discussed and an idea to improve approximation accuracy is suggested based on uniform approximation bounds

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

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

threshold estimation based on a p value framework in dose response and regression settings we use p values to identify the threshold level at which a regression function leaves its baseline value a problem motivated by applications in toxicological and pharmacological dose response studies and environmental statistics we study the problem in two sampling settings one where multiple responses can be obtained at a number of different covariate levels and the other the standard regression setting involving limited number of response values at each covariate our procedure involves testing the hypothesis that the regression function is at its baseline at each covariate value and then computing the potentially approximate p value of the test an estimate of the threshold is obtained by fitting a piecewise constant function with a single jump discontinuity known as a stump to these observed p values as they behave in markedly different ways on the two sides of the threshold the estimate is shown to be consistent and its finite sample properties are studied through simulations our approach is computationally simple and extends to the estimation of the baseline value of the regression function heteroscedastic errors and to time series it is illustrated on some real data applications

multilevel preconditioners for non self adjoint or indefinite orthogonal spline collocation problems efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the solution of an orthogonal spline collocation osc discretization of a dirichlet boundary value problem with a non self adjoint or an indefinite operator the osc solution is sought in the hermite space of piecewise bicubic polynomials it is proved that the proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the normal osc equation the preconditioners are used with the preconditioned conjugate gradient method and numerical results are presented that demonstrate their efficiency

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

enhanced adaptive fuzzy control with optimal approximation error convergence in this paper an enhanced adaptive fuzzy control afc strategy with guaranteed convergence of an optimal fuzzy approximation error fae is presented for a class of uncertain nonlinear systems in the general brunovsky form based on the fuzzy logic system fls with variable universes of discourse relaxed sufficient conditions that guarantee the optimal fae being convergent are given and the upper bound of the optimal fae is obtained the control singularity problem resulting from the unknown affine term is resolved by a novel fuzzy approximation equation and the parameter adaptive law of the fls is derived by the lyapunov synthesis by means of the optimal fae bound result it is proved that the closed loop system achieves partially asymptotic stability under a certain selection of control parameters the proposed approach retains all advantages of a previous similar approach under relaxed constraint conditions thus it provides a more flexible solution to the afc with optimal fae convergence simulation studies have demonstrated high precision tracking performance with smooth control input of the proposed approach

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