on variable bandwidth selection in local polynomial regression the performances of data driven bandwidth selection procedures in local polynomial regression are investigated by using asymptotic methods and simulation the bandwidth selection procedures considered are based on minimizing prelimit approximations to the conditional mean squared error mse when the mse is considered as a function of the bandwidth h we first consider approximations to the mse that are based on taylor expansions around h 0 of the bias part of the mse these approximations lead to estimators of the mse that are accurate only for small bandwidths h we also consider a bias estimator which instead of using small h approximations to bias naively estimates bias as the difference of two local polynomial estimators of different order and we show that this estimator performs well only for moderate to large h we next define a hybrid bias estimator which equals the taylor expansion based estimator for small h and the difference estimator for moderate to large h we find that the mse estimator based on this hybrid bias estimator leads to a bandwidth selection procedure with good asymptotic and for our monte carlo examples finite sample properties

log canonical thresholds of certain fano hypersurfaces we study log canonical thresholds on quartic threefolds quintic fourfolds and double spaces as an important application we show that they have kahler einstein metrics if they are general

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

best possible constant for bandwidth selection for the data based choice of the bandwidth of a kernel density estimator several methods have recently been proposed which have a very fast asymptotic rate of convergence to the optimal bandwidth in particular the relative rate of convergence is the square root of the sample size which is known to be the best possible the point of this paper is to show how semiparametric arguments can be employed to calculate the best possible constant coefficient that is an analog of the usual fisher information in this convergence this establishes an important benchmark as to how well bandwidth selection methods can ever hope to perform it is seen that some existing methods attain the bound others do not

an effective bandwidth selector for local least squares regression abstract local least squares kernel regression provides an appealing solution to the nonparametric regression or scatterplot smoothing problem as demonstrated by fan for example the practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter in this article we apply the ideas of plug in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators our results are applicable to odd degree local polynomial fits and can be extended to other settings such as derivative estimation and multiple nonparametric regression an implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice a by product of our work is the development of a class of nonparametric variance estimators based on local least squares ideas and plug in rules for their implementation

density estimation by wavelet thresholding density estimation is a commonly used test case for nonparametric estimation methods we explore the asymptotic properties of estimators based on thresholding of empirical wavelet coefficients minimax rates of convergence are studied over a large range of besov function classes b pq and for a range of global l p error measures 1 p p some form of nonlinearity is essential since the minimax linear estimators are suboptimal by polynomial powers of n a second approach using an approximation of a gaussian white noise model in a mallows metric is used to attain exactly optimal rates of convergence for quadratic error p 2

the degrees of plane curves with prescribed log canonical threshold abstract we obtain a bound on the degree of curves with fixed log canonical threshold at very general points on p 2

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

bandwidth selection for kernel density estimation the problem of automatic bandwidth selection for a kernel density estimator is considered it is well recognized that the bandwidth estimate selected by the least squares cross validation is subject to large sample variation this difficulty limits the application of the cross validation estimate based on characteristic functions an important expression for the cross validation bandwidth estimate is obtained the expression clearly points out the source of variation to stabilize the variation a simple bandwidth selection procedure is proposed it is shown that the stabilized bandwidth selector gives a strongly consistent estimate of the optimal bandwidth under commonly used smoothness conditions the stabilized bandwidth estimate has a faster convergence rate than the convergence rate of the cross validation estimate for sufficiently smooth density functions it is shown that the stabilized bandwidth estimate is asymptotically normal with a relative convergence rate n 1 2 instead of the rate n 1 10 of the cross validation estimate a plug in estimate and an adjusted plug in estimate are also proposed and their asymptotic distributions are obtained it is noted that the plug in estimate is asymptotically efficient the adjusted plug in bandwidth estimate and the stabilized bandwidth estimate are shown to be asymptotically equivalent the simulation results verify that the proposed procedures perform much better than the cross validation for finite samples

existence and asymptotic behavior of pade approximants to the korteweg de vries multisoliton solutions the summation procedure of the pade type is applied to the perturbation expansion of the solution of the potential korteweg de vries equation k d v introduced by rosales for the n soliton solution without background the n 1 n pade approximants are shown to exist for n n their asymptotic behavior is investigated and it is found that it corresponds to a system of n solitons with the leading velocity parameters the analogous results for the k d v then follow in agreement with some previous numerical observations

convergence of a semi lagrangian scheme for the one dimensional vlasov poisson system a semi lagrangian scheme is proposed for solving the periodic one dimensional vlasov poisson system in phase space on unstructured meshes the distribution function f t x v and the electric field e t x are shown to converge to the exact solution values in the l infty norm the rate of convergence is in o h4 3

do validation for kernel density estimation bandwidth selection in kernel density estimation is one of the fundamental model selection problems of mathematical statistics the study of this problem took major steps forward with the articles of hall and marron 1987 and hall and johnstone 1992 since then the focus seems to have been on various versions of implementing the so called plug in method aimed at estimating the minimum mean integrated squared error mise the most successful of these efforts still seems to be the plug in method of sheather and jones 1991 or park and marron 1990 that we also use as a benchmark in this article in this article we derive a new theorem deriving the asymptotic theory for linear combinations of bandwidths obtained from different selectors as for example direct and indirect cross validation and plug in where we take advantage of recent advances in the study of indirect cross validation see hart and yi 1998 hart and lee 2005 and savchuk hart and sheather 2008 2010 we conclude that the slow

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

dynamic behaviors and measurements of financial market crash rate abstract this paper proposes the conditional crash rate ccr to study the degree of financial market crisis and evaluate the risk of financial market crash based on maximum drawdown and escape rate it is found that the ccr of stock price in financial market is equivalent to the escape rate of the drawdown time series corresponding to price series based on the mean escape rate the analytical solution of ccr of financial market of stock price described by black scholes model is obtained for the ccr empirical researches show that the theoretical results of the heston model are in good agreement with the results of the real financial data of the sp ii we can find critical phenomena resonance and inverse resonance behaviors in the function of ccr vs mean reversion of volatility ii the parameter variation induces the appearance of noise enhancement stability in the function of ccr vs amplitude of volatility fluctuation

the convergence theory of particle in cell methods for multidimensional vlasov poisson systems in this work a convergence theory of particle in cell methods for multidimensional vlasov poisson systems is provided such methods replace the continuous distribution of masses or charges by disc

a multirate dsp model for estimation of discrete probability density functions the problem of estimating a probability density function pdf from measurements has been widely studied by many researchers even though much work has been done in the area of pdf estimation most of it was focused on the continuous case we propose a new model based approach for modeling and estimating discrete probability density functions or probability mass functions this approach is based on multirate signal processing theory and it has several advantages over the conventional histogram method we illustrate the pdf estimation procedure and analyze the statistical properties of the pdf estimates based on this model a novel scheme is introduced that can be used for estimating the pdf in the presence of noise furthermore the proposed ideas are extended to the more general case of estimating multivariate pdfs finally we also consider practical issues such as optimizing the coefficients of a digital filter which is an integral part of the model this allows us to apply the proposed model to solve real world problems simulation results are given where appropriate in order to demonstrate the ideas

wavelet methods for fast resolution of elliptic problems this paper shows that the use of wavelets to discretize an elliptic problem with dirichlet or neumann boundary conditions has two advantages an explicit diagonal preconditioning makes the condition number of the corresponding matrix become bounded by a constant and the order of approximation is locally of spectral type in contrast with classical methods using a conjugate gradient method one thus obtains fast numerical algorithms of resolution a comparison is also drawn between wavelet and classical methods

convergence of a vortex in cell method for the two dimensional euler equations on decrit une methode des tourbillons dans les cellules pour les equations d euler bidimensionnelles on etablit une estimation d erreur

valuations and multiplier ideals this article is the third of a series of work on a new approach to the study of singularities of various objects in a local two dimensional setting our focus in the present paper is on multiplier ideals and singularity exponents in the discussion below we fix an equicharacteristic zero two dimensional regular local ring r m with algebraically closed residue field an important example is the ring r do o holomorphic germs at the origin in c2 in fj1 we introduced the space v consisting of all r u 00 valued valuations on r centered at m and normalized by m 1 this space is naturally a tree it is a union of uncountably many real segments patched together in such a way that v remains homotopic to a point it is also an r tree in the classical sense for a natural metric we hence call v the valuative tree it encodes in a natural way all possible blowups of r centered at m and therefore gives a way of measuring quite precisely singularities of different kinds of objects the points in v that are not ends form the subtree vqm of quasimonomial valuations these valuations which can alternatively be characterized as abhyankar valuations of rank 1 or as

empirical functionals and efficient smoothing parameter selection a striking feature of curve estimation is that the smoothing parameter h 0 which minimizes the squared error of a kernel or smoothing spline estimator is very difficult to estimate this is manifest both in slow rates of convergence and in high variability of standard methods such as cross validation we quantify this difficulty by describing nonparametric information bounds and exhibit asymptotically efficient estimators of h 0 that attain the bounds the efficient estimators are substantially less variable than cross validation and other current procedures and simulations suggest that they may offer improvements at moderate sample sizes at least in terms of minimizing the squared error

convergence of a high order semi lagrangian scheme with propagation of gradients for the one dimensional vlasov poisson system in this paper we give a proof of convergence of a new numerical method introduced in n besse and e sonnendrucker j comput phys 191 2003 pp 341 376 for the vlasov equation the numerical method is based on the semi lagrangian principle and the transport of the gradient of the statistical distribution function in order to get a high order and stable reconstruction these kinds of new schemes have been successfully implemented on unstructured meshes of four dimensional phase space cf n besse etude mathematique et numerique de l equation de vlasov sur des maillages non structures de l espace des phases these de l universite louis pasteur strasbourg france 2003 n besse j segre and e sonnendrucker transport theory statist phys 34 2005 pp 311 332 in order to make a rigorous proof of convergence of this method and simplify the convergence analysis we have considered the periodic one dimensional vlasov poisson system in phase space on a grid the distribution f t x v and the electric field are shown to converge to the exact solution values in h 1 norm the rate of convergence is of mathcal o delta t 2 frac delta x 4 alpha delta t frac delta v 4 alpha delta t alpha in n 2 alpha leq 1

root n bandwidths selectors in multivariate kernel density estimation based on a random sample of size n from an unknown d dimensional density f the problem of selecting the bandwidths in kernel estimation of f is investigated the optimal root n relative convergence rate for bandwidth selection is established and the information bounds in this convergence are given and a stabilized bandwidth selector sbs is proposed it is known that for all d the bandwidths selected by the least squares cross validation lscv have large sample variations the proposed sbs as an improvement of lscv will reduce the variation of lscv without significantly inflating its bias the key idea of the sbs is to modify the d dimensional sample characteristic function beyond some cut off frequency in estimating the integrated squared bias it is shown that for all d and sufficiently smooth f and kernel if the bandwidth in each coordinate direction varies freely then the multivariate sbs is asymptotically normal with the optimal root n relative convergence rate and achieves the conjectured lower bound on the covariance matrix

the numerical analysis of random particle methods applied to vlasov poisson fokker planck kinetic equations we devise and study a random particle method for approximating vlasov poisson fokker planck systems such a proposed scheme takes into account the fact that the trajectories of a particle undergoing brownian motion due to collisions with the medium or background particles can be obtained as the solutions of stochastic differential equations i e the langevin equations these equations are the precise analogs of the deterministic hamiltonian system in the collisionless model the particle approximation in particular simulates the action of viscosity by the use of independent wiener processes brownian motions the error analysis generalizes that of ganguly and victory siam j numer anal 26 1989 pp 249 288 by incorporating into the consistency estimates the sampling errors generated by the random motion of the particles these particular errors are the dominant component of the consistency errors and are estimated by applying bennett s inequality for estimating tail probabilities for standard

convolution on the n sphere with application to pdf modeling in this paper we derive an explicit form of the convolution theorem for functions on an n sphere our motivation comes from the design of a probability density estimator for n dimensional random vectors we propose a probability density function pdf estimation method that uses the derived convolution result on sn random samples are mapped onto the n sphere and estimation is performed in the new domain by convolving the samples with the smoothing kernel density the convolution is carried out in the spectral domain samples are mapped between the n sphere and the n dimensional euclidean space by the generalized stereographic projection we apply the proposed model to several synthetic and real world data sets and discuss the results

a note on asymptotic methods for jump phenomena traditional singular perturbation methods are employed to develop a solution to a differential equation considered by reiss siam j appl math 39 1980 pp 440 455 which models an elementary chemical process the results are compared with those found by reiss who used a novel asymptotic method to construct solutions which exhibit rapid transient behavior it is shown that reiss jump solution corresponds to the asymptotic large time representation of the more complete solution found from a formal matched asymptotic expansion procedure a comparison of results in the rapid transition region obtained from the exact solution from those found by reiss and from the matched asymptotic expansion solution developed here show that the last is far more accurate than the second

wavelet based density estimation for noise reduction in plasma simulations using particles for given computational resources the accuracy of plasma simulations using particles is mainly limited by the noise due to limited statistical sampling in the reconstruction of the particle distribution function a method based on wavelet analysis is proposed and tested to reduce this noise the method known as wavelet based density estimation wbde was previously introduced in the statistical literature to estimate probability densities given a finite number of independent measurements its novel application to plasma simulations can be viewed as a natural extension of the finite size particles fsp approach with the advantage of estimating more accurately distribution functions that have localized sharp features the proposed method preserves the moments of the particle distribution function to a good level of accuracy has no constraints on the dimensionality of the system does not require an a priori selection of a global smoothing scale and its able to adapt locally to the smoothness of the density based on the given discrete particle data moreover the computational cost of the denoising stage is of the same order as one time step of a fsp simulation the method is compared with a recently proposed proper orthogonal decomposition based method and it is tested with three particle data sets involving different levels of collisionality and interaction with external and self consistent fields

the convergence analysis of fully discretized particle methods for solving vlasov poisson systems error estimates for fully discretized particle or simulation methods for multidimensional vlasov poisson systems are provided the treatment requires stability results which generalize to the vlasov poisson context those of j t beale and a majda math comp 39 1982 pp 29 52 and those of c anderson and c greengard siam j numer anal 22 1985 pp 413 440 for studying the convergence of vortex methods with these stability results the total error at each timestep can be studied this approach produces estimates better than those which would be obtained by considering the errors in the semidiscrete approximation siam j numer anal 26 1989 pp 249 288 separate from the errors incurred in the time discretization of the mollified hamiltonian trajectory equations

a formula for jumping numbers in a two dimensional regular local ring abstract in this article we give an explicit formula for the jumping numbers of an ideal of finite colength in a two dimensional regular local ring with an algebraically closed residue field for this purpose we associate a certain numerical semigroup to each vertex of the dual graph of a log resolution of the ideal

on log canonical thresholds of reducible plane curves l etf x y be an irreducible power series j igusa found a formula for the log canonical threshold of f in terms of the first pair of puiseux exponents in this paper we give an alternate proof for the result using shokurov s adjunction theory and the classification of log terminal pairs the method also gives a generalization of igusa s result to the case of reducible curves

adaptive root n estimates of integrated squared density derivatives based on a random sample of size n from an unknown density f on the real line the nonparametric estimation of k f k x 2 dx k 0 1 is considered these functionals are important in a number of contexts the proposed estimates of k is constructed in the frequency domain by using the sample characteristic function it is known that the sample characteristic function at high frequency is dominated by sample variation and does not contain much information about f hence the variation of the estimate can be reduced by modifying the sample characteristic function beyond some cutoff frequency it is proposed to select adaptively the cutoff frequency by a generalization of the smoothed cross validation the exact convergence rate of the proposed estimate to k is established it depends solely on the smoothness of f for sufficiently smooth f it is shown that the proposed estimate is asymptotically normal attains the optimal o p n 1 2 rate and achieves the information bound finally to improve the performance of the proposed estimate at small to moderately large n two modifications are proposed one modification is for estimating 0 it reduces bias of the estimate the other modification is for estimating k k 1 it reduces sample variation of the estimate in simulation studies the superior performance of the proposed procedures is clearly demonstrated