the hermite transform applications it is demonstrated how the hermite transform can be used for image coding and analysis hierarchical coding structures based on increasingly specified basic patterns i e general 2 d patterns general 1 d patterns and specific 1 d patterns such as edges and corners are presented in the image coding application the relation with existing pyramid coders is described a new coding scheme based on local one dimensional image approximations is introduced in the image analysis application the relation between the hermite transform and existing line edge detection schemes is described it is shown that by concentrating on more specific patterns the coding efficiency can be increased since fewer coefficients have to be coded meanwhile sufficient descriptive power can be maintained for approximating the most interesting features in natural images

testing additivity in generalized nonparametric regression models with estimated parameters we develop several kernel based consistent tests of an hypothesis of additivity in nonparametric regression we allow for discrete covariates and parameters estimated from a semiparametric gmm criterion function the additivity hypothesis is of interest because it delivers interpretability and reasonably fast convergence rates for nonparametric estimators the asymptotic distribution of the parameter estimators are found we also derive the asymptotic distribution of the additivity test statistics under a sequence of local alternatives we give a ranking of the different tests based on local asymptotic power the practical performance is investigated through simulations based on the data set used in linton and hardle 1996

empirical likelihood ratio confidence regions than are those based on student s t statistic an effective method is presented for computing empirical profile likelihoods for the mean of a vector random variable the method is a reduction by convex duality to an unconstrained minimization of a convex function on a low dimensional domain algorithms exist for finding the unique global minimum at a superlinear rate of convergence a byproduct is a noncombinatorial algorithm for determining whether a given point lies within the convex hull of a finite set of points the multivariate empirical likelihood regions are justified for functions of several means such as variances correlations and regression parameters and for statistics with linear estimating equations an algorithm is given for computing profile empirical likelihoods for these statistics

sieve empirical likelihood ratio tests for nonparametric functions generalized likelihood ratio statistics have been proposed in fan zhang and zhang ann statist 29 2001 153 193 as a generally applicable method for testing norparametic hypotheses about nonparametric functions the likelihood ratio statistics are constructed based on the assumption that the distributions of stochastic errors are in a certain parametric family we extend their work to the case where the error distribution is completely unspecified via newly proposed sieve empirical likelihood ratio selr tests the approach is also applied to test conditional estimating equations on the distributions of stochastic errors it is shown that the proposed selr statistics follow asymptotically resealed chi 2 distributions with the scale constants and the degrees of freedom being independent of the nuisance parameters this demonstrates that the wilks phenomenon observed in fan zhang and zhang ann statist 29 2001 153 193 continues to hold under more relaxed models and a larger class of techniques the asymptotic power of the proposed test is also derived which achieves the optimal rate for nonparametric hypothesis testing the proposed approach has two advantages over the generalized likelihood ratio method it requires one only to specify some conditional estimating equations rather than the entire distribution of the stochastic error and the procedure adapts automatically to the unknown error distribution including heteroscedasticity a simulation study is conducted to evaluate our proposed procedure empirically

hermite neural network correlation and application a method for obtaining the correlation of two hermite neural networks is developed the method is based on the fact that a hermite function is unchanged by the fourier transform which allows an expression for the correlation to be obtained directly from the network weights without the need for the fourier transform comparative results with other neural network correlation methods are presented on simulated radar signals

extending the scope of empirical likelihood this article extends the scope of empirical likelihood methodology ill three directions to allow for plug in estimates of nuisance parameters in estimating equations slower than root n rates of convergence and settings in which there are a relatively large number of estimating equations compared to the sample size calibrating empirical likelihood confidence regions with plug in is sometimes intractable due to the complexity of the asymptotics so we introduce a bootstrap approximation that call be used in such situations we provide a range of examples from survival analysis and nonparametric statistics to illustrate the main results

on the second order properties of empirical likelihood with moment restrictions this paper considers the second order properties of empirical likelihood el for a parameter defined by moment restrictions which is the inferential framework of the generalized method of moments it is shown that the el defined for this general framework still admits the delicate second order property of bartlett correction this represents a substantial extension of all the established cases of bartlett correction for the el an empirical bartlett correction is proposed which is shown to work effectively in improving the coverage accuracy of confidence regions for the parameter

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

a semiparametric maximum likelihood estimator this paper presents a procedure for analyzing a model in which the parameter vector has two parts a finite dimensional component 8 and a nonparametric component a the procedure does not require parametric modeling of but assumes that the true density of the data satisfies an index restriction the idea is to construct a parametric model passing through the true model and to estimate by setting the score for the parametric model to zero the score is estimated nonparametrically and the estimator is shown to be n consistent and asymptotically normal the estimator is then shown to attain the semiparametric efficiency bound characterized in begun et al 1983 for multivariate nonlinear regression simultaneous equations partially specified regression index regression censored regression switching regression and disequilibrium models in which the error densities are unknown

smooth optimum kernel estimators near endpoints summary kernel estimators for smooth curves like density spectral density or regression functions require modifications when estimating near endpoints of the support both for practical and asymptotic reasons the construction of such boundary kernels as solutions of a variational problem is addressed and representations in orthogonal polynomials are given including explicit solutions for the most important cases based on explicit formulae for certain functionals of the kernels it is shown that local bandwidth variation might be indicated near boundaries various bandwidth variation schemes are discussed and

consistent model specification tests for time series econometric models in this paper we consider general hypothesis testing problems for nonparametric and semiparametric time series econometric models we apply the general methodology to construct a consistent test for omitted variables and a consistent test for a partially linear model the proposed tests are shown to have asymptotic normal distributions under their respective null hypotheses we also discuss the problems of testing portfolio conditional mean variance efficiency and testing a semiparametric single index model monte carlo simulations are conducted to examine the finite sample performances of the nonparametric omitted variable test and the test for a partially linear specification

comparing nonparametric versus parametric regression fits in general there will be visible differences between a parametric and a nonparametric curve estimate it is therefore quite natural to compare these in order to decide whether the parametric model could be justified an asymptotic quantification is the distribution of the integrated squared difference between these curves we show that the standard way of bootstrapping this statistic fails we use and analyse a different form of bootstrapping for this task we call this method the wild bootstrap and apply it to fitting engel curves in expenditure data analysis

consistent model specification tests omitted variables and semiparametric functional forms in this paper we develop several consistent tests in the context of a nonparametric regression model these include tests for the significance of a subset of regressors and tests for the specification of the semiparametric functional form of the regression function where the latter covers tests for a partially linear and a single index specification against a general nonparametric alternative one common feature to the construction of all these tests is the use of the central limit theorem for degenerate u statistics of order higher than two as a result they share the same advantages over most of the corresponding existing tests in the literature a they do not depend on any ad hoc modifications such as sample splitting random weighting etc b under the alternative hypotheses the test statistics in this paper diverge to positive infinity at a faster rate than those based on ad hoc modifications

test of significance based on wavelet thresholding and neyman s truncation abstract traditional nonparametric tests such as the kolmogorov smirnov test and the cramer von mises test are based on the empirical distribution functions although these tests possess root n consistency they effectively use only information contained in the low frequencies this leads to low power in detecting fine features such as sharp and short aberrants as well as global features such as high frequency alternations the drawback can be repaired via smoothing based test statistics in this article we propose two such kind of test statistics based on the wavelet thresholding and the neyman truncation we provide extensive evidence to demonstrate that the proposed tests have higher power in detecting sharp peaks and high frequency alternations while maintaining the same capability in detecting smooth alternative densities as the traditional tests similar conclusions can be made for two sample nonparametric tests of distribution functions in that case the traditional linear rank tests such as th

a test for model specification of diffusion processes we propose a test for model specification of a parametric diffusion process based on a kernel estimation of the transitional density of the process the empirical likelihood is used to formulate a statistic for each kernel smoothing bandwidth which is effectively a studentized l2 distance between the kernel transitional density estimator and the parametric transitional density implied by the parametric process to reduce the sensitivity of the test on smoothing bandwidth choice the final test statistic is constructed by combining the empirical likelihood statistics over a set of smoothing bandwidths to better capture the finite sample distribution of the test statistic and data dependence the critical value of the test is obtained by a parametric bootstrap procedure properties of the test are evaluated asymptotically and numerically by simulation and by a real data example

closed form expressions for time frequency operations involving hermite functions the product convolution correlation wigner distribution function wdf and ambiguity function af of two hermite functions of arbitrary order n and m are derived and expressed as a bounded weighted sum of n m hermite functions it was already known that these mathematical operations performed on gaussians hermite functions of the zeroth order lead to a result which can be expressed as a gaussian function again we generalize this reciprocity to hermite functions of arbitrary order the product convolution correlation wdf and af operations performed on two hermite functions of arbitrary order lead to remarkably similar closed form expressions where the difference between the operations is primarily determined by distinct phase changes of the weights of the hermite functions in the result the closed form expressions are generalized to the class of square integrable functions a key insight from the closed form expressions is applied to the design of orthogonal time frequency localized communication signals which are characterized by an af with rotational symmetry in addition to this application the theoretical expressions may prove useful for signal analysis in fields ranging from communications radar and image processing to quantum mechanics

the hermite transform theory the author introduces a scheme for the local processing of visual information called the hermite transform the problem is addressed from the point of view of image coding and therefore the scheme is presented as an analysis resynthesis system the objectives of the present work however are not restricted to coding the analysis part is designed so that it can also serve applications in the area of computer vision indeed derivatives of gaussians which have found widespread application in feature detection over the past few years play a central role in the hermite analysis it is also argued that the proposed processing scheme is in close agreement with current insight into the image processing that is carried out by the human visual system in particular it is demonstrated that the hermite transform is in better agreement with human visual modeling than gabor expansions

empirical likelihood methods with weakly dependent processes this paper studies the method of empirical likelihood in models with weakly dependent processes in such cases if the likelihood function is formulated as if the data process were independent obviously empirical likelihood fails we propose to use empirical likelihood of blocks of observations to solve this problem in a nonparametric manner this method of blockwise empirical likelihood preserves the dependence of data and the resulting likelihood ratios can be used to construct asymptotically valid confidence intervals we consider general estimating equations for which an efficient estimator is derived by maximizing blockwise empirical likelihood we also introduce blocks of blocks empirical likelihood to conduct inference for parameters of the infinite dimensional joint distribution of data the smooth function model is used for such cases we show that blockwise empirical likelihood of the smooth function model with weakly dependent processes is bartlett correctable a wide variety of problems such as time series regressions and spectral densities can be treated using our methodology

multivariate boundary kernels and a continuous least squares principle summary whereas there are many references on univariate boundary kernels the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail the use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean squared error or sup norm deviation for functions with compact support we develop a class of boundary kernels which work for any support regardless of the complexity of its boundary our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated these boundary kernels provide a natural continuation of non negative kernels used in the interior onto the boundary they are obtained as solutions of the same kernel generating variational problem which also produces the kernel function used in the interior as its solution we discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares along the way we establish a continuous least squares principle and a continuous analogue of the gauss markov theorem

testing conditional moment restrictions let x z be a pair of observable random vectors we construct a new smoothed empirical likelihood based test for the hypothesis e g z x 0 w p 1 where g is a vector of known functions and an unknown finite dimensional parameter we show that the test statistic is asymptotically normal under the null hypothesis and derive its asymptotic distribution under a sequence of local alternatives furthermore the test is shown to possess an optimality property in large samples simulation evidence suggests that it also behaves well in small samples

closed form correlation functions of generalized hermite wavelets a closed form expression is given for the correlation functions of generalized hermite wavelets constructed from an also generalized definition of hermite polynomials due to their gaussianity these wavelets can be used as a tool in the analysis or design of systems involving nonsinusoidal wavelets as well as to model impulsive waveforms found in real world applications and signal processing problems as such the formula is potentially applicable to various areas of science