on local smoothing of nonparametric curve estimators abstract we develop new local versions of familiar smoothing methods such as cross validation and smoothed cross validation in the contexts of density estimation and regression these new methods are locally adaptive in the sense that they capture smooth local fluctuations in the curve by using smoothly varying bandwidths that change as the character of the curve changes moreover the new methods are accurate easy to apply and computationally expedient

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

power of the scan statistic in detecting a changed segment in a bernoulli sequence summary fu curnow 1990 derive recursive equations to find the level of significance and power of a likelihood ratio test for a changed segment of specified length based on the scan statistic the maximum number of successes within the specified length their method is computationally feasible for segment lengths of 20 or less we present and evaluate highly accurate approximations as well as bounds for the power function of this test that are computationally feasible even for very large segment lengths we also evaluate power when the duration of the increased length used in the test statistic does not correspond to the actual length scientists dealing with data sequences modelled by independent bernoulli trials frequently seek criteria to detect changes in the underlying process certain criteria have been developed to be sensitive to situations in which the probability of success on an individual trial changes at a point in the sequence fu curnow 1990 consider a version of a two change point problem with known distance m between the change points and with the trial probability of success at the two ends being equal they give recurrence relations for the probability of type i error and the power computational difficulties arise on even large computers when the number of trials is large in the present paper we derive highly accurate approximations that allow computation of power for all segment lengths consider a sequence of n independent 0 1 random variables x1 x2 xn let 1 denote success with pr x 1 i t we wish to test the null hypothesis of a constant probability of success ho i t i t 1 n against the alternative that some contiguous sequence of m events starting at an unknown trial z has a higher probability of success fu curnow 1990 note that the generalized likelihood ratio test for this alternative rejects ho for large values of the scan statistic

smoothed cross validation for bandwidth selection of a kernel density estimator a generalization of the widely studied least squares cross validation method is considered the essential idea is to do a particular type of presmoothing of the data this is seen to be essentially the same as using the smoothed bootstrap estimate of the mean integrated squared error analysis reveals that a rather large amount of presmoothing yields excellent asymptotic performance the rate of convergence to the optimum is known to be best possible under a wide range of smoothness conditions the method is more appealing than other selectors with this property because its motivation is not heavily dependent on precise asymptotic analysis and because its form is simple and intuitive theory is also given for choice of the amount of presmoothing and this is used to derive a data based method for this choice

locally adaptive bandwidth choice for kernel regression estimators abstract kernel estimators with a global bandwidth are commonly used to estimate regression functions on the other hand it is obvious that the choice of a local bandwidth can lead to better results because a larger class of kernel estimators is available evidently this may in turn affect variability the optimal bandwidths depend essentially on the regression function itself and on the residual variance and it is desirable to estimate them from the data in this article a local bandwidth estimator is studied a comparison with its global bandwidth equivalent is performed both in theory and in simulations as the main result it is shown that the possible gain in mean integrated squared error of the resulting regression estimator must be paid for by a larger variability of the estimator this may lead to worse results if the sample size is small an algorithm has been devised that puts special weight on stability aspects our simulation study shows that improvements over a global bandwidth estimator

local bandwidth selection for kernel estimation of population densities with line transect sampling summary seber 1986 biometrics42 267 292 suggested an approach to biological population density estimation using kernel estimates of the probability density of detection distances in line transect sampling chen 1996a applied statistics45 135 150 and others have employed cross validation to choose a global bandwidth for the kernel estimator or have suggested adaptive kernel estimation chen 1996b biometrics52 1283 1294 because estimation of the density is required at only a single point we investigate a local bandwidth selection procedure that is a modification of the method of schucany 1995 journal of the american statistical association90 535 540 for nonparametric regression we report on simulation results comparing the proposed method and a local normal scale rule with cross validation and adaptive estimation the local bandwidths and normal scale rule produce estimates with mean squares that are half the size of the others in most cases consistency results are also provided

indirect cross validation for density estimation a new method of bandwidth selection for kernel density estimators is proposed the method termed indirect cross validation icv makes use of so called selection kernels least squares cross validation lscv is used to select the bandwidth of a selection kernel estimator and this bandwidth is appropriately rescaled for use in a gaussian kernel estimator the proposed selection kernels are linear combinations of two gaussian kernels and need not be unimodal or positive a theory is developed showing that the relative error of icv bandwidths can converge to 0 at a rate of n 1 4 which is substantially better than the n 1 10 rate of lscv interestingly the selection kernels that are best for purposes of bandwidth selection are very poor if used to actually estimate the density function this property appears to be part of the larger and well documented paradox to the effect that the harder the estimation problem the better cross validation performs the icv method uniformly outperforms lscv in a simul

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

spinorial infinite equations fitting metric affine gravity two different approaches are used to construct infinite component spinor equations based on the multiplicity free irreducible representations of sl 4 r these manifield equations are sl 2 c invariant they exist in special relativity and can directly be coupled to gravitation in the metric affine theory i e in einstein s general relativity with nonpropagating torsion and nonmetricity in the first approach the maximal compact subgroup s 4 of sl 4 r is physical a vector operator x is constructed directly in the infinite dimensional reducible representation ddisc 1 2 0 ddisc 0 1 2 in the second approach sl 2 c and a vector operator are embedded directly in sl 4 r via the dirac representation a manifield equation is then constructed in a manner analogous to the majorana equation by taking an infinite dimensional irreducible multiplicity free representation of sl 4 r spinorial in j1 in the j1 j2 reduction over s 4 both manifields can fit the observed ma

transformations in density estimation abstract for the density estimation problem the global window width kernel density estimator does not perform well when the underlying density has features that require different amounts of smoothing at different locations in this article we propose to transform the data with the intention that a global window width is more appropriate for the density of the transformed data the density estimate of the original data is the back transform by change of variables of the global window width estimate of the transformed data s density we explore choosing the transformation from suitable parametric families data based selection rules for the choice of transformations and the window width are discussed application to real and simulated data demonstrates the usefulness of our proposals

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

on convergence properties of pocket algorithm the problem of finding optimal weights for a single threshold neuron starting from a general training set is considered among the variety of possible learning techniques the pocket algorithm has a proper convergence theorem which asserts its optimality however the original proof ensures the asymptotic achievement of an optimal weight vector only if the inputs in the training set are integer or rational this limitation is overcome in this paper by introducing a different approach that leads to the general result furthermore a modified version of the learning method considered called pocket algorithm with ratchet is shown to obtain an optimal configuration within a finite number of iterations independently of the given training set

wavelet transforms versus fourier transforms this note is a very basic introduction to wavelets it starts with an orthogonal basis of piecewise constant functions constructed by dilation and translation the wavelet transform maps each f x to its coefficients with respect to this basis the mathematics is simple and the transform is fast faster than the fast fourier transform which we briefly explain but approximation by piecewise constants is poor to improve this first wavelet we are led to dilation equations and their unusual solutions higher order wavelets are constructed and it is surprisingly quick to compute with them always indirectly and recursively

on overline gl 4 mathbb r covariant extensions of the dirac equation infinite component generalizations of both massless and massive dirac equations are constructed which are covariant with respect to the double covering of the general linear group in four dimensions these generalized dirac equations can be made covariant with respect to the full diffeomorphism group of the spacetime manifold by replacing ordinary derivatives by covariant derivatives in the usual way

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

on locally adaptive density estimation abstract theoretical and practical aspects of the sample point adaptive positive kernel density estimator are examined a closed form expression for the mean integrated squared error is obtained through the device of preprocessing the data by binning with this expression the exact behavior of the optimally adaptive smoothing parameter function is studied for the first time the approach differs from most earlier techniques in that bias of the adaptive estimator remains o h 2 and is not improved to the rate o h 4 a practical algorithm is constructed using a modification of least squares cross validation simulated and real examples are presented including comparisons with a fixed bandwidth estimator and a fully automatic version of abramson s adaptive estimator the results are very promising

algebra and physics of the unitary multiplicity free representations of sl 4 r the systematics of the multiplicity free unitary irreducible representations of sl 4 r are restudied and an amended list is presented an automorphism essential to the physical application for particles and fields in minkowski space is described

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

distribution of the length of the longest significance run on a bernoulli net and its applications we consider the length of the longest significance run in a two dimensional bernoulli net and derive its asymptotic limit distribution our results can be considered as generalizations of known theorems in significance runs we give three types of theoretical results 1 reliability style lower and upper bounds 2 erdos renyi law and 3 the asymptotic limit distribution to understand the rate of convergence to the asymptotic distributions we carry out numerical simulations the convergence rates in a variety of situations are presented to understand the relation between the length of the longest significance run s and the success probability p we propose a dynamic programming algorithm to implement simultaneous simulations insights from numerical studies are important for choosing the values of design parameters in a particular application which motivates this article the distribution of the length of the longest significance run in a bernoulli net is critical in applying a multiscale metho

structural analysis and elementary representations of sl 4 r and gl 4 r and their covering groups the structure of the groups sl 4 r and gl 4 r and their universal cover ing groups sl 4 r and gl 4 r respectively and lie algebras sl 4 r and gl 4 r respectively are studied the parabolic subgroups and subalgebras are identified and the cuspidal parabolic subgroups singled out the iwasawa and bruhat decompositions are given explicitly all elementary representations er of sl 4 r are explicitly given in two equivalent realizations using the preceding detailed structural analysis the sl 4 r constructions are used for the explicit realization of all er of sl 4 r gl 4 r and gl 4 r the results shall be applied among other things elsewhere for the construction of all irreducible representations of the above groups

formulae for mean integrated squared error of nonlinear wavelet based density estimators we provide an asymptotic formula for the mean integrated squared error mise of nonlinear wavelet based density estimators we show that unlike the analogous situation for kernel density estimators this mise formula is relatively unaffected by assumptions of continuity in particular it is available for densities which are smooth in only a piecewise sense another difference is that in the wavelet case the classical mise formula is valid only for sufficiently small values of the bandwidth for larger bandwidths mise assumes a very different form and hardly varies at all with changing bandwidth this remarkable property guarantees a high level of robustness against oversmoothing not encountered in the context of kernel methods we also use the mise formula to describe an asymptotically optimal empirical bandwidth selection rule