local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

variable window width kernel estimates of probability densities kernel density estimators which allow different amounts of smoothing at different locations are studied modifications of estimators proposed by breiman meisel and purcell 1977 and abramson 1982a which have variable window widths are seen to have very fast rates of convergence these rates have traditionally been obtained using a less natural higher order kernel which has the disadvantage of allowing an estimator which takes on negative values

a simple smooth backfitting method for additive models in this paper a new smooth backfitting estimate is proposed for additive regression models the estimate has the simple structure of nadaraya watson smooth backfitting but at the same time achieves the oracle property of local linear smooth backfitting each component is estimated with the same asymptotic accuracy as if the other components were known

local linear spatial regression a local linear kernel estimator of the regression function x g x e y i x i x x r d of a stationary d 1 dimensional spatial process y i x i i z n observed over a rectangular domain of the form in i i 1 i n z n 1 i k n k k 1 n n n 1 n n z n is proposed and investigated under mild regularity assumptions asymptotic normality of the estimators of g x and its derivatives is established appropriate choices of the bandwidths are proposed the spatial process is assumed to satisfy some very general mixing conditions generalizing classical time series strong mixing concepts the size of the rectangular domain in is allowed to tend to infinity at different rates depending on the direction in z n

local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

isospectral domains with mixed boundary conditions we construct a series of examples of planar isospectral domains with mixed dirichlet neumann boundary conditions this is a modification of a classical problem proposed by m kac

local linear regression estimation under long range dependence strong consistency and rates we consider the nonparametric estimation of multivariate regression functions and their derivatives for a regression model with long range dependent errors we adopt a local linear fitting approach and establish strong consistency and rates for the estimators of the regression function and its derivatives the rates of convergence depend on the amount of smoothing relative to the strength of the long range dependence lrd resulting in distinct rates for small and large bandwidths moreover the conditions determining this dichotomy are different for the estimates of the regression function than for the estimates of its derivatives

isospectral graphs with identical nodal counts according to a recent conjecture isospectral objects have different nodal count sequences gnutzmann et al 2005 j phys a math gen 38 8921 33 we study generalized laplacians on discrete graphs and use them to construct the first non trivial counterexamples to this conjecture in addition these examples demonstrate a surprising connection between isospectral discrete and quantum graphs

local polynomial regression analysis of clustered data this paper proposes a classical weighted least squares type of local polynomial smoothing for the analysis of clustered data with the key idea of using generalised inverses of correlation matrices the estimator has a simple closed form expression simplicity is achieved also for nonparametric generalised linear models with arbitrary link function via a transformation our approach can be characterised by local observations with local variances which yields intuitively correct results in the sense that correct incorrect specification of within cluster correlation has respective positive negative effects the approach is a natural extension of classical local polynomial smoothing consequently existing theory can be largely carried over and important issues such as bandwidth selection can be tackled in the classical fashion moreover the approach can handle various types of covariate such as cluster level subject level or partially cluster level numerical studies support the theoretical results the method is illustrated with a real example on luteinising hormone levels in cows

spline smoothing the equivalent variable kernel method on considere une approche spline pour la regression non parametrique et l estimation de courbe on montre que dans un certain sens le lissage spline correspond approximativement au lissage par une methode du noyau avec une largeur de bande dependant de la densite locale des points du plan de regression

variable bandwidth kernel estimators of regression curves on montre qu en termes de l erreur en moyenne quadratique integree asymptotique les estimateurs du noyau avec un certain choix de largeur de bande local sont superieurs aux estimateurs ordinaires du noyau avec un choix de largeur de bande global si on utilise des largeurs de bandes optimales

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

finite sample variance of local polynomials analysis and solutions abstract fitting local polynomials in nonparametric regression has a number of advantages the attractive theoretical features are in a partial contradiction to variance properties for random design and to practical experience over a broad range of situations no upper bound can be given for the conditional variance the unconditional variance is infinite when using optimal weights with compact support properties are better for gaussian weights we analyze local polynomials for finite sample size both theoretically and numerically it turns out that difficulties arise in sparse regions in the realization of the design when the realization has locally a small variance and or a skew empirical distribution two small sample modifications of local polynomials are presented local increase of bandwidth in sparse regions of the design and local polynomial ridge regression both modifications combine a good finite sample behavior with the asymptotic advantages of local polynomials

local partial likelihood estimation in proportional hazards regression fan gijbels and king ann statist 25 1997 1661 1690 considered the estimation of the risk function x in the proportional hazards model their proposed estimator is based on integrating the estimated derivative function obtained through a local version of the partial likelihood they proved the large sample properties of the derivative function but the large sample properties of the estimator for the risk function itself were not established in this paper we consider direct estimation of the relative risk function x 2 x 1 for any location normalization point x 1 the main novelty in our approach is that we select observations in shrinking neighborhoods of both x 1 and x 2 when constructing a local version of the partial likelihood whereas fan gijbels and king ann statist 25 1997 1661 1690 only concentrated on a single neighborhood resulting in the cancellation of the risk function in the local likelihood function the asymptotic properties of our estimator are rigorously established and the variance of the estimator is easily estimated the idea behind our approach is extended to estimate the differences between groups a simulation study is carried out

quantum chaos on discrete graphs adapting a method developed for the study of quantum chaos on quantum metric graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76 spectral functions and trace formulae for discrete laplacians on graphs are derived this is achieved by expressing the spectral secular equation in terms of the periodic orbits of the graph and obtaining functions which belong to the class of functions proposed originally by ihara 1966 j mat soc japan 18 219 and expanded by subsequent authors stark and terras 1996 adv math 121 124 kotani and sunada 2000 j math sci univ tokyo 7 7 finally a model of classical dynamics on the discrete graph is proposed it is analogous to the corresponding classical dynamics derived for quantum graphs kottos and smilansky 1997 phys rev lett 79 4794 kottos and smilansky 1999 ann phys ny 274 76

comment on resolving isospectral drums by counting nodal domains in gnutzmann et al 2005 j phys a math gen 38 8921 33 the authors studied the 4 parameter family of isospectral flat 4 tori t a b c d discovered by conway and sloane with a particular method of counting nodal domains they were able to distinguish these tori numerically by computing the corresponding nodal sequences relative to a few explicit tuples a b c d in this note we confirm the expectation expressed in gnutzmann et al 2005 j phys a math gen 38 8921 33 by proving analytically that their nodal count distinguishes any 4 tuple of distinct positive real numbers

identifying the average treatment effect in ordered treatment models without unconfoundedness we show identification of the average treatment effect ate when treatment is specified by ordered choice in cross section or panel models treatment is determined by location of a latent variable containing a continuous instrument relative to two or more thresholds we place no functional form restrictions on latent errors and potential outcomes unconfoundedness of treatment does not hold and identification at infinity for the treated is not possible yet we still show nonparametric point identification and estimation of the ate we apply our model to reinvestigate the inverted u relationship between competition and innovation and find no inverted u in us data

second order approximation in the partially linear regression model we examine the second order properties of various quantities of interest in the partially linear regression model we obtain a stochastic expansion with remainder op n 2 4 where u 1 2 for the standardized semiparametric least squares estimator a standard error estimator and a studentized statistic we use the second order expansions to correct the standard error estimates for second order effects and to define a method of bandwidth choice a monte carlo experiment provides favorable evidence on our method of bandwidth choice

the isospectral fruits of representation theory quantum graphs and drums we present a method which enables one to construct isospectral objects such as quantum graphs and drums one aspect of the method is based on representation theory arguments which are shown and proved the complementary part concerns techniques of assembly which are both stated generally and demonstrated for this purpose quantum graphs are grist to the mill we develop the intuition that stands behind the construction as well as the practical skills of producing isospectral objects we discuss the theoretical implications which include sunada s theorem of isospectrality sunada 1985 ann math 121 169 arising as a particular case of this method a gallery of new isospectral examples is presented and some known examples are shown to result from our theory

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

inverse eigenvalue problems on directed graphs the differential operators id and d2 p are constructed on certain finite directed weighted graphs two types of inverse spectral problems are considered first information about the graph weights and boundary conditions is extracted from the spectrum of d2 second the compactness of isospectral sets for d2 p is established by computation of the residues of the zeta function

on the bias of variable bandwidth curve estimators summary a major difficulty in understanding the properties of variable bandwidth methods breiman meisel purcell 1977 abramson 1982 is that extremely lengthy and complex algebra is needed to assess the influence of bias indeed the complexity is so great that it has forced investigators to use computer algebraic manipulation to determine formulae for bias in this note we completely eliminate these algebraic obstacles by presenting a simple easy to use formula which gives explicitly the bias of a variable bandwidth estimator to arbitrarily high order in very general problems

local multiplicative bias correction for asymmetric kernel density estimators abstract we consider semiparametric asymmetric kernel density estimators when the unknown density has support on 0 we provide a unifying framework which relies on a local multiplicative bias correction and contains asymmetric kernel versions of several semiparametric density estimators considered previously in the literature this framework allows us to use popular parametric models in a nonparametric fashion and yields estimators which are robust to misspecification we further develop a specification test to determine if a density belongs to a particular parametric family the proposed estimators outperform rival non and semiparametric estimators in finite samples and are easy to implement we provide applications to loss data from a large swiss health insurer and brazilian income data

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

equivalence between isospectrality and isolength spectrality for a certain class of planar billiard domains isospectrality of the planar domains which are obtained by successive unfolding of a fundamental building block is studied in relation to isolength spectrality of the corresponding domains although an explicit and exact trace formula such as poisson s summation formula or selberg s trace formula is not known to exist for such planar domains equivalence between isospectrality and isolength spectrality in a certain setting can be proved by employing the matrix representation of transplantation of eigenfunctions as an application of the equivalence transplantable pairs of domains which are all isospectral pair of planar domains and therefore counter examples of kac s question can one hear the shape of a drum are numerically enumerated and it is found that at least up to the domain composed of 13 building blocks transplantable pairs coincide with those constructed by the method due to sunada

asymmetric kernel regression kernel regression is one model that has been applied to explain or design radial basis neural networks practical application of the kernel regression method has shown that bias errors caused by the boundaries of the data can seriously effect the accuracy of this type of regression this paper investigates the correction of boundary error by substituting an asymmetric kernel function for the symmetric kernel function at data points close to the boundary the asymmetric kernel function allows a much closer approach to the boundary to be achieved without adversely effecting the noise filtering properties of the kernel regression

spectral problems with mixed dirichlet neumann boundary conditions isospectrality and beyond consider a bounded domain with the dirichlet condition on a part of the boundary and the neumann condition on its complement does the spectrum of the laplacian determine uniquely which condition is imposed on which part we present some results conjectures and problems related to this variation on the isospectral theme

reference curves based on non parametric quantile regression reference curves which take time into account such as those for age are often required in medicine but simple systematic and efficient statistical methods for constructing them are lacking classical methods are based on parametric fitting polynomial curves semi parametric methods are also widely used especially in europe here we propose a new methodology for the estimation of reference intervals for data sets based on non parametric estimation of conditional quantiles the derived methods should be applicable to all clinical or more generally biological variables that are measured on a continuous quantitative scale as an example we analyse a data set collected to establish reference curves for biophysical properties of the skin of healthy french women the results are compared to those obtained with royston s polynomial parametric method and the semi parametric lms approach

a bandwidth selector for local linear density estimators local linear density estimators achieve automatic boundary corrections and enjoy some typical optimal properties proper choice of the smoothing parameters is crucial for their performance a data based bandwidth selector is developed in the spirit of plug in rules consistency and asymptotic normality of the selected bandwidth are demonstrated the bandwidth is very efficient regardless of whether there are non smooth boundaries in the support of the density or not

a semiparametric approach to analysing dose response data in the analysis of a quantal dose response experiment with grouped data the most commonly used parametric procedure is logistic regression commonly referred to as logit analysis the adequacy of the fit by the logistic regression curve is tested using the chi square lack of fit test if the lack of fit test is not significant then the logistic model is assumed to be adequate and estimation of effective doses and confidence intervals on the effective doses can be made when the tolerance distribution of the dose response data is not known and cannot be assumed by the user one can use non parametric methods such as kernel regression or local linear regression to estimate the dose response curve effective doses and confidence intervals this research proposes another alternative based on semi parametric regression to analysing quantal dose response data called model robust quantal regression mrqr mrqr linearly combines the parametric and non parametric predictions with the use of a mixing parameter mrqr uses logistic regression as the parametric portion of the model and local linear regression as the non parametric portion of the model our research has shown that the mrqr procedure can improve the fit of the dose response curve by producing narrower confidence intervals for predictions while providing improved precision of estimates of the effective doses with respect to either logistic or local linear regression results

asymptotics for nonparametric and semiparametric fixed effects panel models abstract in this paper we investigate the problem of estimating nonparametric and semiparametric panel data models with fixed effects we focus on establishing the asymptotic results for estimators using smooth backfitting methods we consider two estimators for the smooth unknown function in nonparametric panel regressions one is a local linear estimator constructed similar as the one in mammen et al 2009 which was proposed for the additive nonparametric panel model the other is the local profile likelihood based estimator proposed by henderson et al 2008 hcl hereafter we build the link and compare the difference between these two estimators which are constructed under different sets of conditions we put both of these estimators in the smooth backfitting algorithm framework discussed in mammen et al 1999 following the recently developed theories on backfitting kernel estimates in mammen et al 2009 we establish the asymptotic normality of these estimators and hence verify the conjectures made by hcl and complement their paper further we consider a partially linear fixed effects panel data model with the nonparametric component estimated using the methods discussed in the first part of the paper we give the asymptotic result for the estimators of finite dimensional parameters which shows that the first step plug in estimators will not affect the asymptotic variance in the second step estimation

weighted local regression and kernel methods for nonparametric curve fitting abstract weighted local regression a popular technique for smoothing scatterplots is shown to be asymptotically equivalent to certain kernel smoothers since both methods are local weighted averages of the data it is proved that in the fixed design regression model given a weighted local regression procedure with any weight function there is a corresponding kernel method such that the quotients of weights distributed by both methods tend uniformly to 1 as the number of observations increases to infinity it is demonstrated by examples that in some instances the weights are nearly the same for both methods even for small samples the asymptotic equivalence allows the derivation of the leading terms of the mean squared error and of the local limit distribution for weighted local regression further a close correspondence is found between the orders of the polynomial to be locally fitted in weighted local regression and the order number of vanishing moments of the kernel employed in the kernel smoot

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

variable bandwidth and local linear regression smoothers in this paper we introduce an appealing nonparametric method for estimating the mean regression function the proposed method combines the ideas of local linear smoothers and variable bandwidth hence it also inherits the advantages of both approaches we give expressions for the conditional mse and mise of the estimator minimization of the mise leads to an explicit formula for an optimal choice of the variable bandwidth moreover the merits of considering a variable bandwidth are discussed in addition we show that the estimator does not have boundary effects and hence does not require modifications at the boundary the performance of a corresponding plug in estimator is investigated simulations illustrate the proposed estimation method

quantum graphs pt symmetry and reflection symmetry of the spectrum not necessarily self adjoint quantum graphs differential operators on metric graphs are considered assume in addition that the underlying metric graph possesses an automorphism symmetry p if the differential operator is p t symmetric then its spectrum has reflection symmetry with respect to the real line our goal is to understand whether the opposite statement holds namely whether the reflection symmetry of the spectrum of a quantum graph implies that the underlying metric graph possesses a non trivial automorphism and the differential operator is p t symmetric we give partial answer to this question by considering equilateral star graphs the corresponding laplace operator with robin vertex conditions possesses reflection symmetric spectrum if and only if the operator is p t symmetric with p being an automorphism of the metric graph

a unifying approach to nonparametric regression estimation abstract there exists a variety of nonparametric regression estimators with cubic smoothing splines k nearest neighbor k nn estimators and various types of kernel estimators among the most popular a class of kernel estimators with local bandwidth depending on the density of the design points is introduced where the degree of design adaptation may be expressed by a single parameter in 0 1 the adaptation to the design is such that the bandwidth is made larger to a degree depending on where the design is thin special values of this parameter correspond approximately to the ordinary fixed width kernel estimator the smoothing spline and the k nn estimator hence this method offers a synthesis of some classical methods the same method allows estimation of derivatives by using appropriate kernels the influence of this degree of design adaptation on the integrated mean squared error is investigated there is no uniformly optimal solution the optimal solution depends in a complex way on de

locally weighted polynomial regression parameter choice and application to forecasts of the great salt lake 1 relationships between hydrologic variables are often nonlinear usually the functional form of such a relationship is not known a priori a multivariate nonparametric regression methodology is provided here for approximating the underlying regression function using locally weighted polynomials locally weighted polynomials consider the approximation of the target function through a taylor series expansion of the function in the neighborhood of the point of estimate cross validatory procedures for the selection of the size of the neighborhood over which this approximation should take place and for the order of the local polynomial to use are provided and shown for some simple situations the utility of this nonparametric regression approach is demonstrated through an application to nonparametric short term forecasts of the biweekly great salt lake volume blind forecasts up to 1 year in the future using the 1847 2004 time series of the great salt lake are presented

identification and estimation of nonseparable single index models in panel data with correlated random effects the identification in a nonseparable single index models with correlated random effects is considered in panel data with a fixed number of time periods the identification assumption is based on the correlated random effects structure under this assumption the parameters of interest are identified up to a multiplicative constant and could be estimated by an average difference of derivatives estimator based on the local polynomial smoothing we suggest to linearly combine the estimators obtained for different orders of differences and derive the variance minimizing weighting scheme the asymptotic distribution of the proposed estimators is derived both for stationary and non stationary explanatory variables along with a test of the stationarity finally monte carlo experiments reveal finite sample properties of the proposed estimator

a method of estimating the average derivative abstract we derive a simple semi parametric estimator of the direct average derivative e d m x where m x is the regression function and s the support of the density of x is compact we partition s into disjoint bins and the local slope d m x within these bins is estimated by using ordinary least squares our average derivative estimate a is then obtained by taking the weighted average of these least squares slopes we show that this estimator is asymptotically normally distributed we also propose a consistent estimator of the variance of a using monte carlo simulation experiments based on a censored regression model with tobit model as a special case we produce small sample results comparing our estimator with the hardle stoker 1989 investigating smooth multiple regression by the method of average derivatives journal of american statistical association 84 408 986 995 method we conclude that a performs better that the hardle stoker estimator for bounded and discontinuous covariates

resolving isospectral drums by counting nodal domains several types of systems have been put forward during the past few decades to show that there exist isospectral systems which are metrically different one important class consists of laplace beltrami operators for pairs of flat tori in r n with n 4 we propose that the spectral ambiguity can be resolved by comparing the nodal sequences the numbers of nodal domains of eigenfunctions arranged by increasing eigenvalues in the case of isospectral flat tori in four dimensions where a four parameter family of isospectral pairs is known we provide heuristic arguments supported by numerical simulations to support the conjecture that the isospectrality is resolved by the nodal count thus one can count the shape of a drum if it is designed as a flat torus in four dimensions less

nodal domains on isospectral quantum graphs the resolution of isospectrality we present and discuss isospectral quantum graphs which are not isometric these graphs are the analogues of the isospectral domains in which were introduced recently in gordon et al 1992 bull am math soc 27 134 8 chapman 1995 am math mon 102 124 buser et al 1994 int math res not 9 391 400 okada and shudo 2001 j phys a math gen 34 5911 22 jakobson et al 2006 j comput appl math 194 141 55 and levitin et al 2006 j phys a math gen 39 2073 82 all based on sunada s construction of isospectral domains sunada t 1985 ann math 121 196 86 after presenting some of the properties of these graphs we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity

locally parametric nonparametric density estimation this paper develops a nonparametric density estimator with parametric overtones suppose f x is some family of densities indexed by a vector of parameters we define a local kernel smoothed likelihood function which for each x can be used to estimate the best local parametric approximant to the true density this leads to a new density estimator of the form f x o x thus inserting the best local parameter estimate for each new value of x when the bandwidth used is large this amounts to ordinary full likelihood parametric density estimation while for moderate and small bandwidths the method is essentially nonparametric using only local properties of data and the model alternative ways more general than via the local likelihood are also described the methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class properties of this new semiparametric estimator are investigated our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias the new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework the methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

asymptotically isospectral quantum graphs and generalised trigonometric polynomials abstract the theory of almost periodic functions is used to investigate spectral properties of schrodinger operators on metric graphs also known as quantum graphs in particular we prove that two schrodinger operators may have asymptotically close spectra if and only if the corresponding reference laplacians are isospectral our result implies that a schrodinger operator is isospectral to the standard laplacian on a may be different metric graph only if the potential is identically equal to zero

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

schrodinger operators on graphs and geometry iii general vertex conditions and counterexamples schrodinger operators on metric graphs with general vertex conditions are studied explicit spectral asymptotics is derived in terms of the spectrum of reference laplacians a geometric version of

a stable adaptive control design for discrete plants with unmodelled dynamics this paper presents a simple discrete model reference adaptive control design which maintains stability around the equilibrium of the tracking error in the presence of unmodelled dynamics only a weak a priori knowledge namely upper bounds of the absolute value of the unmodelled parameters and orders of the unmodelled dynamics is required for the implementation the adaptive controller has an additional parameter to protect the scheme against instability

on bandwidth variation in kernel estimates a square root law we consider kernel estimation of a smooth density f at a point but depart from the usual approach in admitting an adaptive dependence of the sharpness of the kernels on the underlying density proportionally varying the bandwidths like f 1 2 at the contributing readings lowers the bias to a vanishing fraction of the usual value and makes for performance seen in well known estimators that force moment conditions on the kernel and so sacrifice positivity of the curve estimate issues of equivariance and variance stabilitization are treated

diagnostic analysis and computational strategies for estimating discrete time duration models a monte carlo study abstract this paper uses monte carlo analysis to study important and contentious issues in estimating single spell discrete time duration models we find simulated annealing dominates gradient methods for recovering true models we recommend a partially flexible step function for duration dependence combined with likelihood ratio tests for determining support points of unobserved heterogeneity we find that ignoring time changing features of explanatory variables introduces substantial biases in model coefficient and average partial effect estimates these biases do not diminish as sample size increases

matching and semi parametric iv estimation a distance based measure of migration and the wages of young men abstract our paper estimates the effect of us internal migration on wage growth for young men between their first and second job our analysis of migration extends previous research by i exploiting the distance based measures of migration in the national longitudinal surveys of youth 1979 nlsy79 ii allowing the effect of migration to differ by schooling level and iii using propensity score matching to estimate the average treatment effect on the treated atet for movers and iv using local average treatment effect late estimators with covariates to estimate the average treatment effect ate and atet for compliers we believe the conditional independence assumption cia is reasonable for our matching estimators since the nlsy79 provides a relatively rich array of variables on which to match our matching methods are based on local linear local cubic and local linear ridge regressions local linear and local ridge regression matching produce relatively similar point estimates and standard errors while local cubic regression matching badly over fits the data and provides very noisy estimates we use the bootstrap to calculate standard errors since the validity of the bootstrap has not been investigated for the matching estimators we use and has been shown to be invalid for nearest neighbor matching estimators we conduct a monte carlo study on the appropriateness of using the bootstrap to calculate standard errors for local linear regression matching the data generating processes in our monte carlo study are relatively rich and calibrated to match our empirical models or to test the sensitivity of our results to the choice of parameter values the estimated standard errors from the bootstrap are very close to those from the monte carlo experiments which lends support to our using the bootstrap to calculate standard errors in our setting from the matching estimators we find a significant positive effect of migration on the wage growth of college graduates and a marginally significant negative effect for high school dropouts we do not find any significant effects for other educational groups or for the overall sample our results are generally robust to changes in the model specification and changes in our distance based measure of migration we find that better data matters if we use a measure of migration based on moving across county lines we overstate the number of moves while if we use a measure based on moving across state lines we understate the number of moves further using either the county or state measures leads to much less precise estimates we also consider semi parametric late estimators with covariates frolich 2007 using two sets of instrumental variables we precisely estimate the proportion of compliers in our data but because we have a small number of compliers we cannot obtain precise late estimates

a kernel based parametric method for conditional density estimation a conditional density function which describes the relationship between response and explanatory variables plays an important role in many analysis problems in this paper we propose a new kernel based parametric method to estimate conditional density an exponential function is employed to approximate the unknown density and its parameters are computed from the given explanatory variable via a nonlinear mapping using kernel principal component analysis kpca we develop a new kernel function which is a variant to polynomial kernels to be used in kpca the proposed method is compared with the nadaraya watson estimator through numerical simulation and practical data experimental results show that the proposed method outperforms the nadaraya watson estimator in terms of revised mean integrated squared error rmise therefore the proposed method is an effective method for estimating the conditional densities

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

multivariate functional coefficient regression models for nonlinear vector time series data summary vector time series data are widely found in practice in this paper we propose a multivariate functional coefficient regression model with heteroscedasticity for modeling such data the local linear smoother is employed to estimate the unknown coefficient matrices asymptotic normality 10 of the proposed estimators is established and bandwidth selection is considered to deal with cointegration widely observed in financial markets we propose an error corrected multivariate functional coefficient model simulations show that the proposed estimation procedures capture nonlinear structures of coefficients well analysis of united states interest rates illustrates the proposed methodology 15

nonparametric mixture of regression models motivated by an analysis of u s house price index hpi data we propose nonparametric finite mixture of regression models we study the identifiability issue of the proposed models and develop an estimation procedure by employing kernel regression we further systematically study the sampling properties of the proposed estimators and establish their asymptotic normality a modified em algorithm is proposed to carry out the estimation procedure we show that our algorithm preserves the ascent property of the em algorithm in an asymptotic sense monte carlo simulations are conducted to examine the finite sample performance of the proposed estimation procedure an empirical analysis of the u s hpi data is illustrated for the proposed methodology

multivariate regression estimation of continuous time processes from sampled data local polynomial fitting approach let y x y t x t spl infin t spl infin be real valued continuous time jointly stationary processes and let t sub j be a renewal point processes on 0 spl infin with a finite mean rate independent of y x we consider the estimation of regression function r x sub 0 x sub 1 x sub m 1 spl tau sub 1 spl tau sub m of spl psi y spl tau sub m given x 0 x sub 0 x spl tau sub 1 x sub 1 x spl tau sub m 1 sub x 1 for arbitrary lags 0 spl tau sub 1 spl tau sub m on the basis of the discrete time observations y t sub j x t sub j t sub j sub j 1 sup n we estimate the regression function and all its partial derivatives up to a total order p spl ges 1 using high order local polynomial fitting we establish the weak consistency of such estimates along with rates of convergence we also establish the joint asymptotic normality of the estimates for the regression function and all its partial derivatives up to a total order p spl ges 1 and provide explicit expressions for the bias and covariance matrix of the asymptotically normal distribution