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

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

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

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

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

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

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

a role for a brachistochrone in the broken ergodicity scenario abstract we show that the time evolution of ergodic decomposition is outlined by a variational principle for a rather universal class of complex systems with a quasicontinuous spectrum of intrinsic relaxation timescales in turn this variational approach enables us to single out a logarithmic scaling law for the barriers on the hierarchical level such a logarithmic dependence has already been shown to hold valid for certain complex systems moreover it represents a key factor to explain the experimentally ubiquitous debye kohlrausch relaxation law and plays a central role on different theoretical model descriptions within this field

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

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

quantum projectors and local operators in lattice integrable models in the framework of the quantum inverse scattering method we consider a problem of constructing local operators for one dimensional quantum integrable models especially for the lattice versions of the nonlinear schrodinger and sine gordon models we show that a certain class of local operators can be constructed from the matrix elements of the monodromy matrix in a simple way they are closely related to the quantum projectors and have nice commutation relations with half of the matrix elements of the elementary monodromy matrix the form factors of these operators can be calculated by using the standard algebraic bethe ansatz techniques

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

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

relaxation of metastable states in finite mean field kinetic ising systems the authors present simulational data and a theoretical analysis of the stochastic order parameter relaxation in finite mean field ising systems contrary to infinite mean field systems the finite ones exhibit qualitatively the same kind of relaxation as systems with finite interaction ranges they explore the equivalence of different versions of the kinetic ising model in the mean field case and study the lifetime of metastable states they focus on the vicinity of the mean field spinodal and present a phenomenological finite size scaling theory for the lifetime of the metastable state

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

determinant representation for dynamical correlation functions of the quantum nonlinear schrodinger equation painleve analysis of correlation functions of the impenetrable bose gas by m jimbo t miwa y mori and m sato 1 was based on the determinant representation of these correlation functions obtained by a lenard 2 the impenetrable bose gas is the free fermionic case of the quantum nonlinear schrodinger equation in this paper we generalize the lenard determinant representation for langle psi 0 0 psi dagger x t rangle to the non free fermionic case we also include time and temeprature dependence in forthcoming publications we shall perform the jmms analysis of this correlationl function this will give us a completely integrable equation and asymptotic for the quantum correlation function of interacting fermions

a simple lattice version of the nonlinear schrodinger equation and its deformation with an exact quantum solution a lattice version of the quantum nonlinear schrodinger nls equation is considered which has a significant simple form and fulfils most of the criteria desirable for such lattice variants of field models unlike most of the known lattice nls equations the present model belongs to a class which does not exhibit the usual symmetry properties however this lack of symmetry itself seems to be responsible for the remarkable simplification of the relevant objects in the theory such as the lax operator the hamiltonian and other commuting conserved quantities as well as their spectra the model allows exact quantum solution through the algebraic bethe ansatz and also a straightforward and natural generalization to the vector case thus giving a new exact lattice version of the vector nls model a deformation representing a new quantum integrable system involving tamm dancoff like q boson operators is constructed

hysteresis scaling for ising systems on fractal structures dynamical phase transitions in ising systems on sierpinski carpets and bond percolation lattices at percolation threshold are studied by means of standard monte carlo simulations we find that the area of hysteresis loop a can be scaled with respect to the sweep rate h of a linear driving field however the exponent in the scaling expression a hb is universal only for ising systems on sierpinski carpets we conclude that the hysteresis scaling is universal for the field driven first order phase transitions in ising systems on fractal structures based on scaling hypothesis we derive the expression of finite size effect on the hysteresis the exponent b is obtained by this method in some sierpinski carpets

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

aspects of the inverse problem for the toda chain we generalize babelon s approach to equations in dual variables so as to be able to treat new types of operators which we build out of the sub constituents of the model s monodromy matrix further we also apply sklyanin s recent monodromy matrix identities so as to obtain equations in dual variables for yet other operators the schemes discussed in this paper appear to be universal and thus in principle applicable to many models solvable through the quantum separation of variables

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

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

on form factors of the conjugated field in the nonlinear schrodinger model izergin korepin s lattice discretization of the nonlinear schrodinger model along with oota s inverse problem provides one with determinant representations for the form factors of the lattice discretized conjugated field operator we prove that these form factors converge in the zero lattice spacing limit to those of the conjugated field operator in the continuous model we also compute the large volume asymptotic behavior of such form factors in the continuous model these are in particular characterized by fredholm determinants of operators acting on closed contours we provide a way of defining these fredholm determinants in the case of generic parameters

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

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

langevin dynamic simulation of hysteresis in a field swept landau potential numerical simulations are done of langevin dynamics for a uniform orderparameter field swept landau model a 2 m2 b 4 m4 mh t to study hysteresis effects the field is swept at a constant rateh t h 0 ht the stochastic jump values of the field hj from an initially prepared metastable minimumm 0 are recorded on passage to a global minimum m the results are a the mean jump hj h increases hysteresis loop widens with h confirming a previous theoretical criterion based on rate competition between field sweep and inverse mean first passage time fpt b the broad jump distribution hj h is related to intrinsically large fpt fluctuations 2 2 2 o 1 and can be quantitatively understood possible experimental tests of the ideas are indicated