krein adler transformations for shape invariant potentials and pseudo virtual states for 11 examples of one dimensional quantum mechanics with shape invariant potentials the darboux crum transformations in terms of multiple pseudo virtual state wavefunctions are shown to be equivalent to krein adler transformations deleting multiple eigenstates with shifted parameters these are based upon infinitely many polynomial wronskian identities of classical orthogonal polynomials i e the hermite laguerre and jacobi polynomials which constitute the main part of the eigenfunctions of various quantum mechanical systems with shape invariant potentials

on consistent nonparametric order determination and chaos summary we give a brief introduction to deterministic chaos and a link between chaotic deterministic models and stochastic time series models we argue that it is often natural to determine the embedding dimension in a noisy environment first in any systematic study of chaos setting the stochastic models within the framework of non linear autoregression we introduce the notion of a generalized partial autocorrelation and an order we approach the estimation of the embedding dimension via order determination of an unknown non linear autoregression by cross validation and give justification by proving its consistency under global boundedness as a by product we provide a theoretical justification of the final prediction error approach of auestad and tj0stheim some illustrations based on the henon map and several real data sets are given the bias of the residual sum of squares as essentially a noise variance estimator is quantified

nonparametric quantile estimations for dynamic smooth coefficient models in this article quantile regression methods are suggested for a class of smooth coefficient time series models we use both local polynomial and local constant fitting schemes to estimate the smooth coefficients in a quantile framework we establish the asymptotic properties of both the local polynomial and local constant estimators for mixing time series also a bandwidth selector based on the nonparametric version of the akaike information criterion is suggested together with a consistent estimate of the asymptotic covariance matrix furthermore the asymptotic behaviors of the estimators at boundaries are examined a comparison of the local polynomial quantile estimator with the local constant estimator is presented a simulation study is carried out to illustrate the performance of estimates an empirical application of the model to real data further demonstrates the potential of the proposed modeling procedures

multi indexed q racah polynomials as the second stage of the project multi indexed orthogonal polynomials we present in the framework of discrete quantum mechanics with real shifts in one dimension the multi indexed q racah polynomials they are obtained from the q racah polynomials by the multiple application of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state vectors in a similar way to the multi indexed laguerre and jacobi polynomials reported earlier the virtual state vectors are the solutions of the matrix schrodinger equation with negative eigenvalues except for one of the two boundary points

some properties of the variations of non additive set functions ii in this paper we continue the work of the preceding paper zhang fuzzy sets and systems to appear to introduce the concepts of the third kind of variations for non additive set functions the inclusion variations and discuss some properties of the variations in particular we investigate the inclusion variations of signed fuzzy measures which have closely to do with the uniqueness of jordan decomposition of non additive set functions we also show that these three kinds of variations inclusion variation disjoint variation and chain variation coincide when the considered set functions are additive finally we attempt to answer one of the open problems raised by pap null additive set functions kluwer academic publishers dordrecht 1995

testing for structural change in dynamic models the well known cusum test for structural change is investigated whe n there are lagged dependent variables among the regressors in a linear model the authors show that both a modified cusum test suggested b y j m dufour 1982 and the straightforward cusum test retain their asymptotic significance levels in dynamic models and find that the power depends crucially on the angle between the mean regressor and the structural shift copyright 1988 by the econometric society

triangular norms position paper ii general constructions and parameterized families this second part out of three of a series of position papers on triangular norms for part i see triangular norms position paper i basic analytical and algebraic properties fuzzy sets and systems in press deals with general construction methods based on additive and multiplicative generators and on ordinal sums also included are some constructions leading to non continuous t norms and a presentation of some distinguished families of t norms

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 dynamic regularized radial basis function network for nonlinear nonstationary time series prediction in this paper constructive approximation theorems are given which show that under certain conditions the standard nadaraya watson 1964 regression estimate nwre can be considered a specially regularized form of radial basis function networks rbfns from this and another related result we deduce that regularized rbfns are m s consistent like the nwre for the one step ahead prediction of markovian nonstationary nonlinear autoregressive time series generated by an i i d noise processes additionally choosing the regularization parameter to be asymptotically optimal gives regularized rbfns the advantage of asymptotically realizing minimum m s prediction error two update algorithms one with augmented networks infinite memory and the other with fixed size networks finite memory are then proposed to deal with nonstationarity induced by time varying regression functions for the latter algorithm tests on several phonetically balanced male and female speech samples show an average 2 2 db improvement in the predicted signal noise error ratio over corresponding adaptive linear predictors using the exponentially weighted rls algorithm further rls filtering of the predictions from an ensemble of three such rbfns combined with the usual autoregressive inputs increases the improvement to 4 2 db on average over the linear predictors

two step darboux transformations and exceptional laguerre polynomials it has been recently discovered that exceptional families of sturm liouville orthogonal polynomials exist that generalize in some sense the classical polynomials of hermite laguerre and jacobi in this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple step algebraic darboux transformations the construction is illustrated with an example of a 2 step darboux transformation of the classical laguerre polynomials which gives rise to a new orthogonal polynomial system indexed by two integer parameters for particular values of these parameters the classical laguerre and the type ii x l laguerre polynomials are recovered

construction of fuzzy σ algebras using triangular norms abstract generalizing the definitions given by the author fuzzy sets and systems4 1980 83 93 we introduce and study t fuzzy algebras t being any triangular norm the main result is that for a large class of triangular norms each t fuzzy algebra is generated i e consists of all functions x 0 1 being measurable with respect to some algebra on x

characterization of fuzzy measures constructed by means of triangular norms abstract following the ideas presented by the author e p klement j math anal appl 85 1982 543 565 finite t fuzzy measures are introduced t being a measurable triangular norm we show that a t fuzzy measure is always a fuzzy measure as considered earlier e p klement j math anal appl 25 1980 330 339 then we study the relation to the integral with respect to some classical measure finally for some special triangular norms t we give precise characterizations of the corresponding classes of t fuzzy measures

identification and wavelet estimation of weighted ate under discontinuous and kink incentive assignment mechanisms abstract this paper studies identification estimation and inference of a weighted average treatment effect w ate parameter in a class of switching regime models where the agent s selection of treatment is affected by either a discontinuous or kink incentive assignment mechanism and some unobservable characteristic for each assignment mechanism we i establish identification and propose a local wavelet estimator of the w ate ii establish asymptotic properties of the local wavelet estimator including optimal convergence rate and asymptotic normality and iii investigate the finite sample performance of the local wavelet estimators and compare them with local polynomial estimators via an extensive simulation study we also propose an identification robust wavelet estimator of the w ate

triangular norms position paper iii continuous t norms this third and last part of a series of position papers on triangular norms for parts i and ii see e p klement r mesiar e pap triangular norms position paper i basic analytical and algebraic properties fuzzy sets and systems in press e p klement r mesiar e pap triangular norms position paper ii general constructions and parameterized families submitted for publication presents the representation of continuous archimedean t norms by means of additive generators and the representation of continuous t norms by means of ordinal sums with archimedean summands both with full proofs finally some consequences of these representation theorems in the context of comparison and convergence of continuous t norms and of the determination of continuous t norms by their diagonal sections are mentioned

burmann expansion and test for additivity summary we propose a lagrange multiplier test for additivity based on the btirmann expansion of a conditional mean function the asymptotic null distribution of the test is shown to be x2 under some regularity conditions in contrast the lagrange multiplier test proposed by chen et al 1995 is based on the volterra expansion of the conditional mean function we discuss some desirable advantages of the bt rmann expansion over the volterra expansion for nonlinear time series modelling we also reported an empirical study which shows that in terms of empirical power the lagrange multiplier test motivated by the biirmann expansion outperforms the test of chen et al 1995 for the cases for which the lagrange multiplier test is designed for other cases for which none of the tests is specifically designed the empirical powers of the two tests are comparable finally we illustrated the use of the lagrange multiplier test with a blowfly experimental system

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

minimax estimation of sharp change points we define the sharp change point problem as an extension of earlier problems in change point analysis related to nonparametric regression as particular cases these include estimation of jump points in smooth curves more generally we give a systematic treatment of the correct rate of convergence for estimating the position of a cusp of an arbitrary order we propose a test function for the local regularity of a signal that characterizes such a point as a global maximum in the sample implementation of our method from observations of the signal at discrete time positions i n i 1 n we use a wavelet transformation to approximate the position of the change point in the no noise case we study the noise effect in the worst case scenario over a wide class of functions having a unique irregularity of order a and propose a sequence of estimators which converge at the rate n 1 1 2 as n tends to infinity finally we analyze the likelihood ratio of the problem and show that this is actually the minimax rate of convergence examples of thresholding empirical wavelet coefficients to estimate the position of sharp change points are also presented

discussion on the meeting on chaos robert m may university of oxford until relatively recently attempts to understand how the world works have been based often implicitly on the assumption that simple laws expressed as simple equations will result in simple predictable behaviour conversely complex dynamics were thought typically to result from complicated concatenations of causes roulette wheels may ultimately obey deterministic laws but their behaviour can only effectively be described in stochastic terms the recognition that many simple dynamical systems of which the quadratic map may be the simplest example can exhibit behaviour which not merely looks random but which is so sensitive to initial conditions as to be effectively unpredictable is revolutionizing this view the flip side of the chaos coin is the even more recent recognition that apparently random time series may be produced by very simple low dimensional deterministic systems operating in chaotic regimes thus researchers with backgrounds in theoretical physics or classical applied mathematics are converging on territory previously occupied solely by a subset of statisticians to ask whether we can distinguish the apparent randomness of deterministic chaos from the real randomness of the roulette wheel both kinds of noisy system are unpredictable in the long term but chaotic systems may differ from truly stochastic systems by admitting short term predictability this question has both fundamental intellectual importance and obvious potential applications in many fields not least economics the set of papers brought together for this meeting give in my opinion a superb overview of the current state of the art the authors have avoided becoming bogged down in semantics as casdagli observes it is philosophically possible to argue that ultimately everything is deterministic or oppositely that ultimately noise is all pervasive rather they have taken the commonsense view that whether we classify a system as exhibiting deterministic chaos described by some underlying non linear model or noise described in stochastic terms depends on the details broadly simple low dimensional chaotic dynamical systems may be recognized as deterministic and short term predictable whereas complicated high dimensional systems are best recognized in purely stochastic terms with only probabilistic prediction at any timescale in so far as short term prediction is central to distinctions between low and high dimensional chaotic dynamical systems an exercise that for physicists and applied mathematicians began with the desire to distinguish chaos from noise begins to take pragmatic shape as new ideas about forecasting on the basis of past time series data inter alia papers at this meeting offer important insights into how best to determine the dimension variously defined of a putative attractor given a relatively low dimensional chaotic system plus noise in particular richard smith has clarified the distinction between problems posed by observational noise and system noise broomhead has offered onion skin layers of complications introduced by filtering the data and cheng and tong have explored new approaches to non linear autoregressive methods rather than to attempt any overview and synthesis of the present papers i wish to highlight what i see as some of the likely next steps some touched on at the meeting some not

exactly solvable potentials with finitely many discrete eigenvalues of arbitrary choice we address the problem of possible deformations of exactly solvable potentials having finitely many discrete eigenvalues of arbitrary choice as kay and moses showed in 1956 reflectionless potentials in one dimensional quantum mechanics are exactly solvable with an additional time dependence these potentials are identified as the soliton solutions of the korteweg de vries kdv hierarchy an n soliton potential has the time t and 2n positive parameters k1 kn and cj j 1 n corresponding to n discrete eigenvalues kj2 the eigenfunctions are elementary functions expressed by the ratio of determinants the darboux crum krein adler transformations or the abraham moses transformations based on eigenfunction deletions produce lower soliton number potentials with modified parameters cj we explore various identities satisfied by the eigenfunctions of the soliton potentials which reflect the uniqueness theorem of gel fand levitan marchenko equations for separable degenerate kernels

singularity detection and processing with wavelets the mathematical characterization of singularities with lipschitz exponents is reviewed theorems that estimate local lipschitz exponents of functions from the evolution across scales of their wavelet transform are reviewed it is then proven that the local maxima of the wavelet transform modulus detect the locations of irregular structures and provide numerical procedures to compute their lipschitz exponents the wavelet transform of singularities with fast oscillations has a particular behavior that is studied separately the local frequency of such oscillations is measured from the wavelet transform modulus maxima it has been shown numerically that one and two dimensional signals can be reconstructed with a good approximation from the local maxima of their wavelet transform modulus as an application an algorithm is developed that removes white noises from signals by analyzing the evolution of the wavelet transform maxima across scales in two dimensions the wavelet transform maxima indicate the location of edges in images

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

nonparametric tests of linearity for time series summary we introduce tests of linearity for time series based on nonparametric estimates of the conditional mean and the conditional variance the tests are compared to a number of parametric tests and to nonparametric tests based on the bispectrum asymptotic expressions give bad approximations and the null distribution under linearity is constructed using resampling of the best linear approximation the new tests perform well on the examples tested

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

non monotonic fuzzy measures and the choquet integral abstract this paper discusses non monotonic fuzzy measures which are set functions without monotonicity and the choquet integral with respect to non monotonic fuzzy measures a concrete example shows that the choquet integral is meaningful in the case of non monotonic fuzzy measures as well as in the case of ordinary fuzzy measures every comonotonically additive positively homogeneous functional of bounded variation can be represented as a choquet integral with respect to a non monotonic fuzzy measure of bounded variation the space of such functionals is a real banach space isometrically isomorphic to the space of non monotonic fuzzy measures of bounded variation

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

extensions of solvable potentials with finitely many discrete eigenstates we address the problem of rational extensions of six examples of shape invariant potentials having finitely many discrete eigenstates the overshoot eigenfunctions are proposed as candidates unique in this group for the virtual state wavefunctions which are an essential ingredient for multi indexed and iso spectral extensions of these potentials they have exactly the same form as the eigenfunctions but their degrees are much higher than nmax so that their energies are lower than the groundstate energy

factorization method and new potentials with the oscillator spectrum a one parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator this is a new derivation of a class of potentials previously obtained by abraham and moses with the help of the gelfand levitan formalism

nonparametric identification of nonlinear time series selecting significant lags abstract in this article we suggest a nonparametric procedure for selecting significant lags in the model description of a general nonlinear stationary time series the procedure can be applied to both the conditional mean and the conditional variance and is valid for heteroscedastic series the procedure is illustrated by simulations and sunspot data lynx data and blowfly data are analyzed it is indicated that projectors can be used in conjunction with the procedure for selecting significant lags to check the adequacy of an additive time series model

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

local global neural networks a new approach for nonlinear time series modeling we propose the local global neural networks model within the context of time series models this formulation encompasses some already existing nonlinear models and also admits the mixture of experts approach we emphasize the linear expert case and extensively discuss the theoretical aspects of the model stationarity conditions existence consistency and asymptotic normality of the parameter estimates and model identifiability the proposed model consists of a mixture of stationary and nonstationary linear models and is able to describe intermittent dynamics the system spends a large fraction of time in a bounded region but sporadically develops an instability that grows exponentially for some time and then suddenly collapses intermittency is a commonly observed behavior in ecology and epidemiology fluid dynamics and other natural systems a model building strategy is also considered and the parameters are estimated by concentrated maximum likelihood the procedure is illustrated with two real

triangular norms position paper i basic analytical and algebraic properties we present the basic analytical and algebraic properties of triangular norms we discuss continuity as well as the important classes of archimedean strict and nilpotent t norms triangular conorms and de morgan triples are also mentioned finally a brief historical survey on triangular norms is given

nonlinear additive models for environmental time series with applications to ground level ozone data analysis abstract environmental time series usually vary systematically in response to meteorological conditions and thus often are not stationary in this article a class of additive models are introduced for environmental time series in which both mean levels and variances of the series are nonlinear functions of relevant meteorological variables backfitting algorithms in nonlinear regression are adopted to estimate the unknown functions in the model and the maximum likelihood method is used to estimate the parameters in the noise component asymptotic properties of the parameter estimates including consistency and limiting distribution are derived under mild conditions the model is applied to daily maxima of ground level ozone concentrations in the chicago area for possible long term trend assessment compared to alternative models the proposed models gave more accurate estimations for the 95th and 99th percentiles of the ozone distribution

nonparametric transfer function models in this paper a class of nonparametric transfer function models is proposed to model nonlinear relationships between input and output time series the transfer function is smooth with unknown functional forms and the noise is assumed to be a stationary autoregressive moving average arma process the nonparametric transfer function is estimated jointly with the arma parameters by modelling the correlation in the noise the transfer function can be estimated more efficiently the parsimonious arma structure improves the estimation efficiency in finite samples the asymptotic properties of the estimators are investigated the finite sample properties are illustrated through simulations and one empirical example

a consistent characteristic function based test for conditional independence abstract y is conditionally independent of z given x if pr f y x z f y x 1 for all y on its support where f denotes the conditional density of y given x z or x this paper proposes a nonparametric test of conditional independence based on the notion that two conditional distributions are equal if and only if the corresponding conditional characteristic functions are equal we extend the test of su and white 2005 a hellinger metric nonparametric test for conditional independence discussion paper department of economics ucsd in two directions 1 our test is less sensitive to the choice of bandwidth sequences 2 our test has power against deviations on the full support of the density of x y z we establish asymptotic normality for our test statistic under weak data dependence conditions simulation results suggest that the test is well behaved in finite samples applications to stock market data indicate that our test can reveal some interesting nonlinear dependence that a traditional linear granger causality test fails to detect

application of a least absolute shrinkage and selection operator to aeroelastic flight test data identification of non linear systems involves estimating unknown parameters and model regressor selection selection of a subset of candidate terms that best describes the observed output model selection is an important step in black box modelling of any observed process this procedure is concerned with selecting a subset of parameters to give a parsimonious description of the system which may afford greater insight into the functionality of the system or a simpler controller design in this study a least absolute shrinkage and selection operator lasso technique is investigated for computing efficient model descriptions of non linear aeroelastic systems the lasso minimises the residual sum of squares by the addition of an l1 penalty term on the parameter vector of the traditional l2 minimisation problem its use for model selection is a natural extension of this constrained minimisation approach to pseudolinear regression problems which produces some model parameters that are exactly zero and the

estimating continuous time stochastic volatility models of the short term interest rate abstract we obtain consistent parameter estimates of continuous time stochastic volatility diffusions for the u s risk free short term interest rate sampled weekly over 1954 1995 using the efficient method of moments procedure of gallant and tauchen the preferred model displays mean reversion and incorporates level effects and stochastic volatility in the diffusion function extensive diagnostics indicate that the cox ingersoll ross model with an added stochastic volatility factor provides a good characterization of the short rate process further they suggest that recently proposed garch models fail to approximate the discrete time short rate dynamics while level egarch models perform reasonably well

multi indexed wilson and askey wilson polynomials as the third stage of the project multi indexed orthogonal polynomials we present in the framework of discrete quantum mechanics with pure imaginary shifts in one dimension the multi indexed wilson and askey wilson polynomials they are obtained from the original wilson and askey wilson polynomials by multiple applications of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state solutions of types i and ii in a similar way to the multi indexed laguerre jacobi and q racah polynomials reported earlier

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

residual variance and residual pattern in nonlinear regression summary a nonparametric estimator of residual variance in nonlinear regression is proposed it is based on local linear fitting asymptotically the estimator has a small bias but a larger variance compared with the parametric estimator in linear regression finite sample properties are investigated in a simulation study including a comparison with other nonparametric estimators the method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis a further application is checking the fit of parametric models this is illustrated for longitudinal growth data

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

additivity tests for nonlinear autoregression summary additivity is commonly used in the statistical literature to simplify data analysis especially in analysis of variance and in multivariate smoothing in this paper we propose three procedures for testing additivity in nonlinear time series analysis the first procedure combines some smoothing techniques with analysis of variance the second is a lagrange multiplier test using nonparametric estimation and the third is a permutation test which uses smoothing techniques to obtain the test statistic and its reference distribution we investigate properties of the proposed tests and use simulation to check their performance in finite samples applications of the tests to nonlinear time series analysis are discussed and illustrated by real examples

memory universal prediction of stationary random processes we consider the problem of one step ahead prediction of a real valued stationary strongly mixing random process xi sub i spl infin sup spl infin the best mean square predictor of x sub 0 is its conditional mean given the entire infinite past x sub i sub i spl infin sup 1 given a sequence of observations x sub 1 x sub 2 x sub n we propose estimators for the conditional mean based on sequences of parametric models of increasing memory and of increasing dimension for example neural networks and legendre polynomials the proposed estimators select both the model memory and the model dimension in a data driven fashion by minimizing certain complexity regularized least squares criteria when the underlying predictor function has a finite memory we establish that the proposed estimators are memory universal the proposed estimators which do not know the true memory deliver the same statistical performance rates of integrated mean squared error as that delivered by estimators that know the true memory furthermore when the underlying predictor function does not have a finite memory we establish that the estimator based on legendre polynomials is consistent

additive fuzzy measures and integrals iii abstract a point of view concerning fuzzy measures is explained to this end a new concept of disjointness for fuzzy is introduced and studied also a concept of an additive class of fuzzy sets is defined to be a class of fuzzy sets closed under some additive operations the fuzzy measures are defined to be sum preserving real functions over such additive classes some basic properties of the fuzzy measures are derived in contrast with other homonymous concepts studied in literature our fuzzy measures lead to an additive fuzzy integral see the part ii of the paper

identification of nonlinear time series first order characterization and order determination summary we study the possibility of identifying nonlinear time series using nonparametric estimates of the conditional mean and conditional variance it is shown that most nonlinear models satisfy the assumptions needed to apply nonparametric asymptotic theory sampling variations of the conditional quantities are studied by simulation and explained by asymptotic arguments for a number of first order nonlinear autoregressive processes the conditional mean and variance can be used for identification purposes but one must be aware of bias and misspecification effects we also propose a criterion for determining the order of a general nonlinear model the criterion is justified in parts by heuristics but encouraging results are obtained from a limited set of simulation experiments several

nonparametric pricing of interest rate derivative securities we propose a nonparametric estimation procedure for continuous time stochastic models because prices of derivative securities depend crucially on the form of the instantaneous volatility of the underlying process we leave the volatility function unrestricted and estimate it nonparametrically only discrete data are used but the estimation procedure still does not rely on replacing the continuous time model by some discrete approximation instead the drift and volatility functions are forced to match the densities of the process we estimate the stochastic differential equation followed by the short term interest rate and compute nonparametric prices for bonds and bond options

on the equivalence of the integral and the differential exact solution generation methods for the one dimensional schrodinger equation it is demonstrated that the integral exact solution generation methods for the one dimensional schrodinger equation based on the gelfand levitan formalism are in some cases equivalent to the differential ones based on the n order darboux transformation some new exact solvable potentials are generated from the effective coulomb potential and the harmonic oscillator potential a new form of n soliton potential i e reflectionless potential with n discrete energy levels disposed in a desirable manner based on an explicit expression for an n order wronski determinant constructed from hyperbolic functions and its orthonormal discrete spectrum eigenfunctions are given

some properties of the variations of non additive set functions on t tribes the inclusion variation and chain variation on t tribes play an important role in t measures and fuzzy games theory where they were used for additive set functions on t tribe in this paper classical variation theory is extended to non additive set functions on ttribes after defining the inclusion variation disjoint and chain variation on t tribes we investigate some properties of these three variations in detail such as the null null additivity exhaustivity order continuity continuity from the left and so on an alternative proof of the jordan decomposition theorem is given thus some results pertaining to variations in classical measure theory are generalized in some case as for t measures on t tribe its different formal variations can be extended to ts measures on ts tribes

blind identification of manufacturing variation patterns by combining source separation criteria blind source separation recently has been investigated for blindly identifying variation patterns in multivariate manufacturing data to aid in tracking down and eliminating root causes of manufacturing variation many different criteria can be used in blind separation algorithms the performance and applicability of which depend on conditions that generally are not known a priori we present a method for automatically combining the different criteria so as to directly minimize the mean squared estimation error the resulting algorithm is more effective and more robust than counterparts that use other means of combining the criteria

nonparametric multistep ahead prediction in time series analysis we consider the problem of multistep ahead prediction in time series analysis by using nonparametric smoothing techniques forecasting is always one of the main objectives in time series analysis research has shown that non linear time series models have certain advantages in multistep ahead forecasting traditionally nonparametric k step ahead least squares prediction for non linear autoregressive ar d models is done by estimating e x t k mid x t cdots x t d 1 via nonparametric smoothing of x t k on x t cdots x t d 1 directly we propose a multistage nonparametric predictor we show that the new predictor has smaller asymptotic mean squared error than the direct smoother though the convergence rate is the same hence the predictor proposed is more efficient some simulation results advice for practical bandwidth selection and a real data example are provided copyright 2004 royal statistical society

a flexible coefficient smooth transition time series model we consider a flexible smooth transition autoregressive star model with multiple regimes and multiple transition variables this formulation can be interpreted as a time varying linear model where the coefficients are the outputs of a single hidden layer feedforward neural network this proposal has the major advantage of nesting several nonlinear models such as the self exciting threshold autoregressive setar the autoregressive neural network ar nn and the logistic star models furthermore if the neural network is interpreted as a nonparametric universal approximation to any borel measurable function our formulation is directly comparable to the functional coefficient autoregressive far and the single index coefficient regression models a model building procedure is developed based on statistical inference arguments a monte carlo experiment showed that the procedure works in small samples and its performance improves as it should in medium size samples several real examples are also addressed

solvable rational extensions of the morse and kepler coulomb potentials we show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential this is made by using darboux backlund transformations based on unphysical regular riccati schrodinger functions which are obtained from specific symmetries associated with the considered family of potentials

generalizes darboux transformations classification of inverse scattering methods for the radial schrodinger equation a wide class of darboux transformations of a sturm liouville equation providing a unified treatment of exactly solvable models in quantum scattering theory is considered systematically a classification of darboux transformations is given and the relations between darboux transformations and standard inverse scattering procedures for the radial schrodinger equation including the matrix method of newton sabatier are studied in particular the definitions properties and matrix generalizations of darboux transformations associated with marchenko type integral equations are studied in detail

non atomic fuzzy measures and games in this paper liapounoff s theorem for fuzzy measures is proved and it is applied to extend a theorem of aumann and shapley 2 concerning the existence of the value for non atomic games precisely it is proved that the range of any non atomic n vector fuzzy measure in the sense of 6 is convex and compact and it is shown that this result allows to extend theorem b from 2 p 23 from games with unambiguous coalitions to games with fuzzy coalitions called fuzzy games defined on s algebras of fuzzy sets which are not necessarily s algebras of ideal sets

local polynomial estimators of the volatility function in nonparametric autoregression abstract in this paper we consider a class of dynamic models in which both the conditional mean and the conditional variance volatility are unknown functions of the past we first derive probabilistic conditions under which nonparametric estimation of these functions is possible we then construct an estimator based on local polynomial fitting we examine the rates of convergence of these estimators and give a result on their asymptotic normality the local polynomial fitting of the volatility function is applied to different foreign exchange rate series we find an asymmetric u shaped smiling face form of the volatility function

a new family of shape invariantly deformed darboux poschl teller potentials with continuous l we present a new family of shape invariant potentials which could be called a continuous version of the potentials corresponding to the exceptional x j1 jacobi polynomials constructed recently by the present authors in a certain limit it reduces to a continuous family of shape invariant potentials related to the exceptional x l1 laguerre polynomials the latter was known as one example of the conditionally exactly solvable potentials on a half line

specification estimation and evaluation of smooth transition autoregressive models abstract this article considers the application of two families of nonlinear autoregressive models the logistic lstar and exponential estar autoregressive models this includes the specification of the model based on simple statistical tests linearity testing against smooth transition autoregression determining the delay parameter and choosing between lstar and estar models are discussed estimation by nonlinear least squares is considered as well as evaluating the properties of the estimated model the proposed techniques are illustrated by examples using both simulated and real time series

nonlinear additive arx models abstract we consider in this article a class of nonlinear additive autoregressive models with exogenous variables for nonlinear time series analysis and propose two modeling procedures for building such models the procedures proposed use two backfitting techniques the ace and bruto algorithms to identify the nonlinear functions involved and use the methods of best subset regression and variable selection in regression analysis to determine the final model simulated and real examples are used to illustrate the analysis

nonparametric identification of nonlinear time series projections abstract we study the possibility of identifying general linear and nonlinear time series models using nonparametric methods the kernel estimators of the conditional mean and variance are used as a basis and the properties of these quantities as model indicators are briefly discussed some drawbacks are pointed out and motivated by these we introduce projections as tools of identification the projections are especially useful for additive modeling expressions for the asymptotic bias and variance are obtained the projection of the conditional variance is suggested as a tool for identifying heteroscedastic time series the results are illustrated by simulations for both the estimators of the projections and the estimators of the conditional mean and variance

change curve estimation via wavelets abstract the recently developed theory of wavelets has a remarkable ability to zoom in on very short lived frequency phenomena such as transients in signals and singularities in functions and hence provides an ideal tool to study localized changes this article proposes a wavelet method for estimating jump and sharp cusp curves of a function in the plane the method involves first computing wavelet transformation of data and then estimating jump and sharp cusp curves by wavelet transformation across fine scales asymptotic theory is established and simulations are carried out to lend some credence to the asymptotic theory the wavelet estimate is nearly optimal and can be computed by fast algorithms the method is applied to a real image

triangular norm based measures and their markov kernel representation abstract we approach the problem whether left continuous triangular norm based valuations called t measures or t probability measures defined on triangular normbased tribes of the unit cube can be disintegrated by markov kernels we prove that each t measure based on a fundamental triangular norm these triangular norms t together with their corresponding triangular conorms s satisfy the functional equation t x y s x y x y can be uniquely represented as a sum of a disintegrable t measure and a hard core which is either identically zero or which is monotonically irreducible i e cannot be disintegrated

fuzzy measurability and integrability developpement d une nouvelle theorie relative au probleme de la mesure des objets flous continuite des fonctions sigma additives sur une classe sigma additive d ensembles flous extension de la notion d integrale floue methode de calcul des integrales floues problemes ouverts

functional coefficient regression models for nonlinear time series abstract the local linear regression technique is applied to estimation of functional coefficient regression models for time series data the models include threshold autoregressive models and functional coefficient autoregressive models as special cases but with the added advantages such as depicting finer structure of the underlying dynamics and better postsample forecasting performance also proposed are a new bootstrap test for the goodness of fit of models and a bandwidth selector based on newly defined cross validatory estimation for the expected forecasting errors the proposed methodology is data analytic and of sufficient flexibility to analyze complex and multivariate nonlinear structures without suffering from the curse of dimensionality the asymptotic properties of the proposed estimators are investigated under the mixing condition both simulated and real data examples are used for illustration

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

an extended class of orthogonal polynomials defined by a sturm liouville problem abstract we present two infinite sequences of polynomial eigenfunctions of a sturm liouville problem as opposed to the classical orthogonal polynomial systems these sequences start with a polynomial of degree one we denote these polynomials as x 1 jacobi and x 1 laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval 1 1 or the half line 0 respectively and they are a basis of the corresponding l 2 hilbert spaces moreover we prove a converse statement similar to bochner s theorem for the classical orthogonal polynomial systems if a self adjoint second order operator has a complete set of polynomial eigenfunctions p i i 1 then it must be either the x 1 jacobi or the x 1 laguerre sturm liouville problem a rodrigues type formula can be derived for both of the x 1 polynomial sequences

non polynomial extensions of solvable potentials a la abraham moses abraham moses transformations besides darboux transformations are well known procedures to generate extensions of solvable potentials in one dimensional quantum mechanics here we present the explicit forms of infinitely many seed solutions for adding eigenstates at arbitrary real energy through the abraham moses transformations for typical solvable potentials e g the radial oscillator the darboux poschl teller and some others these seed solutions are simple generalisations of the virtual state wavefunctions which are obtained from the eigenfunctions by discrete symmetries of the potentials the virtual state wavefunctions have been an essential ingredient for constructing multi indexed laguerre and jacobi polynomials through multiple darboux crum transformations in contrast to the darboux transformations the virtual state wavefunctions generate non polynomial extensions of solvable potentials through the abraham moses transformations

tapered block bootstrap summary we introduce and study tapered block bootstrap methodology that yields an improvement over the well known block bootstrap for time series of kiinsch 1989 the asymptotic validity and the favourable bias properties of the tapered block bootstrap are shown the important practical issues of optimally choosing the window shape and the block size are addressed in detail while some finite sample simulations are presented validating the good performance of the tapered block bootstrap

exceptional orthogonal polynomials exactly solvable potentials and supersymmetry we construct two new exactly solvable potentials giving rise to bound state solutions to the schrodinger equation which can be written in terms of the recently introduced laguerre or jacobi type x1 exceptional orthogonal polynomials these potentials extending either the radial oscillator or the scarf i potential by the addition of some rational terms turn out to be translationally shape invariant as their standard counterparts and isospectral to them