simultaneous nonparametric inference of time series we consider kernel estimation of marginal densities and regression functions of stationary processes it is shown that for a wide class of time series with proper centering and scaling the maximum deviations of kernel density and regression estimates are asymptotically gumbel our results substantially generalize earlier ones which were obtained under independence or beta mixing assumptions the asymptotic results can be applied to assess patterns of marginal densities or regression functions via the construction of simultaneous confidence bands for which one can perform goodness of fit tests as an application we construct simultaneous confidence bands for drift and volatility functions in a dynamic short term rate model for the u s treasury yield curve rates data

modified r matrix and separation of variables for the modified korteweg de vries mkdv hierarchy abstract in this article it is first given that a lax representation in terms of 2 2 matrices for the completeey integrable finite dimensional hamiltonian system cifhs h produced through the nonlinearization procedure for the mkdv hierarchy and then an associated non dynamical modified r matrix is constructed by making use of this r matrix and matrix trace equality a set of finite dimensional involutive functions fm m 0 1 f0 h which guarantees the integrability of hamiltonian systems h and the lax representations in terms of 2 2 matrices for the whole hamiltonian hierarchies fm m 0 1 are obtained moreover the involutive solutions of the mkdv hierarchy are given finally it is found that the hamiltonian jacobi equation for the hamiltonian system h can be separable under a group of new coordinates introduced by the 2 2 lax matrix

simultaneous confidence bands for linear regression and smoothing suppose we observe y i f x i varepsilon i i 1 ldots n we wish to find approximate 1 alpha simultaneous confidence regions for f x x in mathscr x our regions will be centered around linear estimates hat f x of nonparametric or nonparametric f x there is a large amount of previous work on this subject substantial restrictions have been usually placed on some or all of the dimensionality of x the class of functions f that can be considered the class of linear estimates hat f and the region mathscr x the method we present is an approximation to the tube formula dn can be used for multidimensional x and a wide class of linear estimates by considering the effect of bias we are able to relax assumptions on the class of functions f which are considered simultaneous and numerical computations are used to illustrate the performance

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

nonlinear system theory another look at dependence based on the nonlinear system theory we introduce previously undescribed dependence measures for stationary causal processes our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems the proposed dependence measures provide a natural framework for a limit theory for stationary processes in particular under conditions with quite simple forms we present limit theorems for partial sums empirical processes and kernel density estimates the conditions are mild and easily verifiable because they are directly related to the data generating mechanisms

quadrics on real riemannian spaces of constant curvature separation of variables and connection with gaudin magnet the integrable systems are considered which are connected with separation of variables in real riemannian spaces of constant curvature an isomorphism is given for these systems with hyperbolic gaudin magnet using this isomorphism the complete classification of separable coordinate systems is provided by means of the corresponding l operators for the gaudin magnet

confidence bands for regression functions abstract a method for obtaining conservative confidence bands for regression functions in low dimensions is given of particular importance is the applicability of the method to nonparametric regression computations examples and comparisons indicate the effectiveness of the method

confidence bands in nonparametric regression abstract new bias corrected confidence bands are proposed for nonparametric kernal regression these bands are constructed using only a kernel estimator of the regression curve and its data selected bandwidth they are shown to have asymptotically correct coverage properties and to behave well in a small sample study one consequence of the large sample developments is that bonferroni type bands for the regression curve at the design points also have conservative asymptotic coverage behavior with no bias correction

equivalence of two graphical calculi the author considers the integrable systems which are connected with separation of variables in the helmholtz operator on the real riemannian spaces of constant curvature an isomorphism is given for these systems with a quantum hyperbolic gaudin magnet using this isomorphism the complete classification of all separable coordinate systems on the manifolds considered is provided by means of the corresponding l operators for the gaudin magnet

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

quadrics on complex riemannian spaces of constant curvature separation of variables and the gaudin magnet integrable systems that are connected with orthogonal separation of variables in complex riemannian spaces of constant curvature are considered herein an isomorphism with the hyperbolic gaudin magnet previously pointed out by one of the authors extends to coordinates of this type the complete classification of these separable coordinate systems is provided by means of the corresponding l matrices for the gaudin magnet the limiting procedures or e calculus which relate various degenerate orthogonal coordinate systems play a crucial role in the classification of all such systems

relaxation of nonlinear oscillations in bcs superconductivity the diagonal case of the sl 2 richardson gaudin quantum pairing model is known to be solvable as an abel jacobi inversion problem the effect of random time dependent perturbations of the single particle spectrum on the exact solution is an open question of considerable physical relevance weak perturbations introduce a new slow time scale while preserving the nonlinear character of the dynamics in this paper such perturbations are considered it is shown that the long time asymptotics can be obtained by a deformation of the original integrable system equivalent to phase averaging over the fast time scale

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

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

confidence intervals for nonparametric curve estimates numerous nonparametric regression methods exist that yield consistent estimators of function curves often one is also interested in constructing confidence intervals for the unknown function when a function estimate is based on a single global smoothing parameter the resulting confidence intervals may hold their desired confidence level 1 on average but because bias in nonparametric estimation is not uniform they do not hold the desired level uniformly at all design points most research in this area has focused on mean squared error properties of the estimator for example mise itself a global measure in addition measures like mise are one step removed from the practical issue of coverage probability recent work that focuses on coverage probability has considered only coverage in an average sense ignoring the important issue of uniformity of coverage across the design space to deal with this problem a new estimator is developed which uses a local cross validation criterion lcv to determi

quantum euler manakov top on the 3 sphere s3 we construct the harmonic polynomials on the real s 3 sphere associated with the separation of the laplace operator in general ellipsoid coordinates they describe a basis of an irreducible unitary representation of the o 4 lie algebra coming from the quantum integrable system the euler manakov top on the 3 sphere s3 or equivalently a 2 site su 2 xyz gaudin magnet an isomorphism of this model with the 4 site su 1 1 xxx gaudin magnet is established the final two parameter spectral problem is efficiently solved

modelling nonlinear random vibrations using an amplitude dependent autoregressive time series model summary the behaviour of nonlinear deterministic vibrations has been studied by many authors and may typically include such features as jump phenomena and limit cycles nonlinear random vibrations in continuous time have also been studied and these may commonly give rise to the phenomenon of amplitude dependent frequency a discrete time series model is introduced which may be demonstrated to have properties similar to those of nonlinear random vibrations this model is of autoregressive form with amplitudedependent coefficients and may be estimated using an extension of a method for estimating linear time series models the model is fitted to the canadian lynx data and demonstrates that it may be possible to regard the periodic behaviour of this series as being generated by some underlying self exciting mechanism

quantum euler manakov top on the three sphere s3 the authors construct the harmonic polynomials on the real s3 sphere associated with the separation of the laplace operator in general ellipsoid coordinates they describe a basis of an irreducible unitary representation of the o 4 lie algebra coming from the quantum integrable system the euler manakov top on the 3 sphere s3 or equivalently a two site su 2 xyz gaudin magnet an isomorphism of this model with the four site su 1 1 xxx gaudin magnet is established the final two parameter spectral problem is efficiently solved

separation of variables on n dimensional riemannian manifolds i the n sphere sn and euclidean n space rn the following problem is solved what are all the different separable coordinate systems for the laplace beltrami eigenvalue equation on the n sphere sn and euclidean n space rn and how are they constructed this is achieved through a combination of differential geometric and group theoretic methods a graphical procedure for construction of these systems is developed that generalizes vilenkin s construction of polyspherical coordinates the significance of these results for exactly soluble dynamical systems on these manifolds is pointed out the results are also of importance for the analysis of the special functions appearing in the separable solutions of the laplace beltrami eigenvalue equation on these manifolds

linear r matrix algebra for classical separable systems we consider a hierarchy of the natural type hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates we give a lax representation in terms of 2 2 matrices for the whole hierarchy and construct the associated linear r matrix algebra with the r matrix dependent on the dynamical variables a yang baxter equation of dynamical type is proposed using the method of variable separation we provide the integration of the systems in classical mechanics constructing the separation equations and hence the explicit form of action variables the quantization problem is discussed with the help of the separation variables

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

variance function estimation in regression the effect of estimating the mean abstract the authors consider estimation of a variance function g in regression problems such estimation requires simultaneous estimation of the mean function f we obtain sharp results on the extent to which the smoothness of f influences best rates of convergence for estimating g for example in nonparametric regression with two derivatives on g classical rates of convergence are possible if and only if the unknown f satisfies a lipschitz condition of order 1 3 or more if a parametric model is known for g then g may be estimated n 1 2 consistently if and only if f is lipschitz of order 1 2 or more optimal rates of convergence are attained by kernel estimators

separation of variables for the type periodic toda lattice we prove separation of variables for the most general type periodic toda lattice with a lax matrix it is achieved by finding proper normalization for the corresponding baker akhiezer function separation of variables for all other periodic toda lattices associated with infinite series of root systems follows by taking appropriate limits

generalized likelihood ratio statistics and wilks phenomenon likelihood ratio theory has had tremendous success in parametric inference due to the fundamental theory of wilks yet there is no general applicable approach for nonparametric inferences based on function estimation maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting even if they exist they are hard to find and can not be optimal as shown in this paper we introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics a new s wilks phenomenon is unveiled we demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow chi 2 distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including gaussian white noise models nonparametric regression models varying coefficient models and generalized varying coefficient models we further demonstrate that generalized likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by ingster they can even be adaptively optimal in the sense of spokoiny by using a simple choice of adaptive smoothing parameter our work indicates that the generalized likelihood ratio statistics are indeed general and powerful for nonparametric testing problems based on function estimation

simultaneous separation for the kowalevski and goryachev chaplygin gyrostats in the special case of zero square integral the kowalevski gyrostat and goryachev chaplygin gyrostat share a simple separation of variables originated from the 4 4 lax matrix

fully nonparametric estimation of scalar diffusion models we propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process we show how to identify the drift and diffusion function in situations where one or the other function is considered a nuisance parameter the asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens we prove almost sure consistency and weak convergence to mixtures of normal laws where the mixing variates depend on the chronological local time of the underlying diffusion process that is the random time spent by the process in the vicinity of a generic spatial point the estimation method and asymptotic results apply to both stationary and nonstationary recurrent processes

on the quantum inverse scattering method for the dst dimer abstract the quantum inverse scattering method is used to solve the spectral problem of the discrete self trapping dimer in the case of both a quadratic and a linear r matrix algebra representation the first case is solved by the algebraic bethe ansatz while in the case of the linear r matrix algebra we use the method of separation of variables in this last case it is shown that the wave functions of the quantum discrete self trapping dimer are related to the solutions of a heun s type equation and that the system is equivalent to the two site hyperbolic gaudin magnet separable in elliptic coordinates

bootstrap simultaneous error bars for nonparametric regression simultaneous error bars are constructed for nonparametric kernel estimates of regression functions the method is based on the bootstrap where resampling is done from a suitably estimated residual distribution the error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints applications to an economic problem are given and comparison to both pointwise and bonferroni type bars is presented through a simulation study

separation of variables for the ruijsenaars system we construct a separation of variables for the classical n particle ruijsenaars system the relativistic analog of the elliptic calogero moser system the separated coordinates appear as the poles of the properly normalised eigenvector baker akhiezer function of the corresponding lax matrix two different normalisations of the ba functions are analysed the canonicity of the separated variables is verified with the use of the r matrix technique the explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n 2 and n 3 taking the nonrelativistic limit we also construct a separation of variables for the elliptic calogero moser system

sieve bootstrap for smoothing in nonstationary time series we propose a sieve bootstrap procedure for time series with a deterministic trend the sieve for constructing the bootstrap is based on nonparametric trend estimation and autoregressive approximation for some noise process the bootstrap scheme itself does i i d resampling of estimated innovations from fitted autoregressive models we show the validity and indicate second order correctness of such sieve bootstrap approximations for the limiting distribution of nonparametric linear smoothers the resampling can then be used to construct nonparametric confidence intervals for the underlying trend in particular we show asymptotic validity for constructing confidence bands which are simultaneous within a neighborhood of size of the order of the smoothing bandwidth our resampling procedure yields satisfactory results in a simulation study for finite sample sizes we also apply it to the longest series of total ozone measurements from arosa switzerland and find a significant decreasing trend

confidence bands in nonparametric time series regression we consider nonparametric estimation of mean regression and conditional variance or volatility functions in nonlinear stochastic regression models simultaneous confidence bands are constructed and the coverage probabilities are shown to be asymptotically correct the imposed dependence structure allows applications in many linear and nonlinear auto regressive processes the results are applied to the s p 500 index data

an introduction to wavelets with applications to andrews plots a brief introduction to wavelets targeted to the statistician is given several types of wavelets are described and compared with the fourier functions it is shown how wavelets might extend the multivariate clustering technique of andrews plots

stationary problems for equation of the kdv type and dynamical r matrices we study a new quite general family of dynamical r matrices for an auxiliary loop algebra l su 2 related to restricted flows for equations of the kdv type this underlying r matrix structure allows us to reconstruct lax representations and to find variables of separation for a wide set of the integrable natural hamiltonian systems

strong invariance principles for dependent random variables we establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds applications to linear and some nonlinear processes are discussed strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions