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

the clebsch gordan coefficients for the two parameter quantum algebra sup q 2 in the lowdin shapiro approach the structure of the irreducible representations of the two parameter quantum algebra sup q 2 is studied the projection operators for this algebra are constructed in the lowdin shapiro form the explicit analytical expressions for the sup q 2 clebsch gordan coefficients are obtained with the help of these projection operators there are clear perspectives to elaborate the theory of tensor operators universal r matrix 6j 9j symbols etc for the sup q 2 algebra using the same tools

a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

the trading time risks of stock investment in stock price drop abstract this article investigates the trading time risk ttr of stock investment in the case of stock price drop of dow jones industrial average dji and hushen300 data csi300 respectively the escape time of stock price from the maximum to minimum in a data window length dwl is employed to measure the absolute ttr the ratio of the escape time to data window length is defined as the relative ttr empirical probability density functions of the absolute and relative ttrs for the dji and csi300 data evidence that i whenever the dwl increases the absolute ttr increases the relative ttr decreases otherwise ii there is the monotonicity or non monotonicity for the stability of the absolute or relative ttr iii there is a peak distribution for shorter trading days and a two peak distribution for longer trading days for the pdf of ratio iv the trading days play an opposite role on the absolute or relative ttr and its stability between dji and csi300 data

quantum algebras on phase space quantum algebras are represented by functions on phase space in terms of product and explicit examples for u sl n 1 r with n 1 are given

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

a fully automated bandwidth selection method for fitting additive models abstract this article describes a fully automated bandwidth selection method for additive models that is applicable to the widely used backfitting algorithm of buja hastie and tibshirani the proposed plug in estimator is an extension of the univariate local linear regression estimator of ruppert sheather and wand and is shown to achieve the same op n 2 7 relative convergence rate for bivariate additive models if more than two covariates are present theoretical justification of the method requires independence of the covariates but simulation experiments show that in practice the method is very robust to violations of this assumption the proposed bandwidth selection method is compared to cross validation through simulation experiments its practical behavior is demonstrated on a real dataset

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

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

on optimal data based bandwidth selection in kernel density estimation a bandwidth selection method is proposed for kernel density estimation this is based on the straightforward idea of plugging estimates into the usual asymptotic representation for the optimal bandwidth but with two important modifications the result is a bandwidth selector with the by nonparametric standards extremely fast asymptotic rate of convergence of n where n denotes sample size comparison is given to other bandwidth selection methods and small sample impact is investigated

the poisson bracket for q deformed systems it is shown that the quantum system of q deformed oscillators with the uq n group symmetry can be obtained by the canonical quantization of a classical system with a modified poisson bracket the modification of the poisson bracket is connected with a non canonical transformation of phase space variables in this approach the deformation parameter turns out to be a function of the planck constant and some dimensional parameters characterizing the classical system

on the efficiency of on line density estimators an on line density estimator may be defined to be one where each update following the arrival of a new data value may be accomplished after no more than a fixed number of calculations this definition should also apply to any empirical bandwidth selection rule for such an estimator recursive estimators comprise only a special case of on line estimators but even there on line bandwidth formulas have not been developed the authors introduce a class of on line estimators and discuss efficiency in this context it is shown that some nonrecursive members of the class achieve greater efficiency than any recursive estimators and that efficiency increases to 100 as the order of the estimated derivative increases on line bandwidth selection rules enabling these high orders of efficiency to be achieved asymptotically are introduced

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

a simple root n bandwidth selector the asymptotically best bandwidth selectors for a kernel density estimator currently require the use of either unappealing higher order kernel pilot estimators or related fourier transform methods the point of this paper is to present a methodology which allows the fastest possible rate of convergence with the use of only nonnegative kernel estimators at all stages of the selection process the essential idea is derived through careful study of factorizations of the pilot bandwidth in terms of the original bandwidth

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

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

a study of least squares and maximum likelihood for image reconstruction in positron emission tomography positron emission tomography pet is a radiologic tool offering a unique capability for measuring tissue metabolism in vivo a number of biological and physical factors limit the resolution of pet so often the statistical aspects of image reconstruction have an appreciable effect on the quality of information obtained from a study to a first approximation reconstruction involves the solution of a linear inverse problem with a line integral radon type transform standard filtered back projection reconstruction is based on the method of least squares although computationally efficient the method does not enforce positivity constraints leading to undesirable negative artifacts in the results maximum likelihood based approaches to reconstruction do not suffer from this problem but their computational complexity has limited the ability to determine quantitatively the improvements in image quality in this paper asymptotic approximations and numerical simulations are used to examine the least squares and maximum likelihood methods in some detail the studies are carried out for idealized representations of conventional and time of flight tomographs the asymptotic analysis indicates that for a range of sobolev norms the rates of estimation of least squares and maximum likelihood reconstructions are of the same order this is borne out by numerical studies however in these studies maximum likelihood is found to be more efficient than least squares on a conventional distance angle tomograph the root mean square error is on the order of 10 20 smaller for maximum likelihood reconstructions the corresponding reduction in the root mean square error on a time of flight tomograph is in the 5 range similar results are found using more complex region of interest based measures of reconstruction accuracy in addition it is found that much of the effect of maximum likelihood can apparently be realized by postprocessing least squares solutions in an ad hoc manner to reduce negativity artifacts since the postprocessing scheme is computationally fast this suggests that in pet it may be possible to realize the benefits of maximum likelihood without the substantial computational overhead of the em algorithm

cross validation of multivariate densities abstract in recent years the focus of study in smoothing parameter selection for kernel density estimation has been on the univariate case while multivariate kernel density estimation has been largely neglected in part this may be due to the perception that calibrating multivariate densities is substantially more difficult in this article we explicitly derive and compare multivariate versions of the bootstrap method of taylor the least squares cross validation method developed by bowman and rudemo and a biased cross validation method similar to that of scott and terrell for multivariate kernel estimation using the product kernel estimator the theoretical behavior of these cross validation algorithms is shown to improve surprisingly as the dimension increases approaching the best rate of o n 1 2 simulation studies suggest that the new biased cross validation method performs quite well and with reasonable variability as compared to the other two methods bivariate examples with heart disease

brst cohomology for uq sl 2 representations a nilpotent brst operator qq is constructed for the q deformation of the universal enveloping algebra of sl 2 an associated operator rq such that qq rq cq is introduced where cq is the q deformed casimir invariant cq fe 1 2h q 1 2h 1 q the related operator qq conjugate to qq is also considered the resulting brst cohomology of uq sl 2 representations is discussed

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

comparison of data driven bandwidth selectors abstract this article compares several promising data driven methods for selecting the bandwidth of a kernel density estimator the methods compared are least squares cross validation biased cross validation and a plug in rule the comparison is done by asymptotic rate of convergence to the optimum and a simulation study it is seen that the plug in bandwidth is usually most efficient when the underlying density is sufficiently smooth but is less robust when there is not enough smoothness present we believe the plug in rule is the best of those currently available but there is still room for improvement

extent to which least squares cross validation minimises integrated square error in nonparametric density estimation let ho o and c be the windows which minimise mean integrated square error integrated square error and the least squares cross validatory criterion respectively for kernel density estimates it is argued that o not ho should be the benchmark for comparing different data driven approaches to the determination of window size asymptotic properties of ho o and c o and of differences between integrated square errors evaluated at these windows are derived it is shown that in comparison to the benchmark o the observable window c performs as well as the so called optimal but unattainable window ho to both first and second order

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

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

biometrika centenary nonparametrics summary contributions in biometrika to nonparametric statistics are surveyed with particular emphasis on more recent work it is argued that even in the journal s very early years it had interests in smoothing methods for density estimation and in techniques for nonparametric regression in addition to its more obvious focus on classical nonparametric statistics such as ranking methods biometrika s early involvement with permutation testing has links to its contemporary work on bootstrap methods emphases on different types of nonparametric statistics have evolved over time however a rapidly increasing proportion of the journal s pages is devoted to papers that include some use of nonparametric ideas or methods

the brst operator of uq sl 2 and real forms in a previous letter a nilpotent brst operator qq was constructed for the q deformation of the universal enveloping algebra of sl 2 the present addendum resolves issues of non uniqueness in the construction of the brst operator by imposing additional structure associated with real forms and corrects a misprint in the previous work

brst cohomology in classical mechanics the classical non quantum cohomology of the becchi rouet stora tyutin brst symmetry in phase space is defined and worked out no group action for the gauge transformations is assumed the results cover therefore the general case of an open algebra and are valid off shell each cohomology class contains all brst invariant functions with fixed ghost number an integer which differ from each other by a brst variation if the ghost number is negative there is only the trivial class whose elements are equivalent to zero if the ghost number is positive or zero there is a bijective correspondence between the brst classes and those of the exterior derivative along the gauge orbits these gauge orbits lie in the phase space surface on which the gauge generators are constrained to vanish the brst invariant functions of ghost numberp are then related to closedp forms along the orbits the addition of a brst variation corresponds to the addition of an exact form some comments about the quantum case are included the physical meaning of the classes with ghost number greater than zero is not discussed

suq h cross to 0 2 and suq h cross 2 the classical and quantum q deformations of the su 2 algebra ii the hopf algebra the yang baxter equation and multi deformed algebraic structures for pt i see ibid vol 23 p 4185 1990 the suq 2 algebra is realized by means of both the poisson brackets in classical mechanics and commutators in quantum mechanics in a system with q deformed oscillators of two different types the structures of the hopf algebra and the quantum yang baxter equation are also discussed on a quantum level a set of j representations of the quantum algebra suq 2 is constructed based on the type ii q oscillators multi deformations of the oscillators of the two types and multi deformed algebras expressed in poisson brackets as well as in lie brackets are proposed

a brief survey of bandwidth selection for density estimation abstract there has been major progress in recent years in data based bandwidth selection for kernel density estimation some second generation methods including plug in and smoothed bootstrap techniques have been developed that are far superior to well known first generation methods such as rules of thumb least squares cross validation and biased cross validation we recommend a solve the equation plug in bandwidth selector as being most reliable in terms of overall performance this article is intended to provide easy accessibility to the main ideas for nonexperts

classification using kernel density estimates multiscale analysis and visualization the use of kernel density estimates in discriminant analysis is quite well known among scientists and engineers interested in statistical pattern recognition using a kernel density estimate involves properly selecting the scale of smoothing namely the bandwidth parameter the bandwidth that is optimum for the mean integrated square error of a class density estimator may not always be good for discriminant analysis where the main emphasis is on the minimization of misclassification rates on the other hand cross validation based methods for bandwidth selection which try to minimize estimated misclassification rates may require huge computation when there are several competing populations besides such methods usually allow only one bandwidth for each population density estimate whereas in a classification problem the optimum bandwidth for a class density estimate may vary significantly depending on its competing class densities and their prior probabilities therefore in a multiclass problem it

peter hall s contributions to nonparametric function estimation and modeling peter hall made wide ranging and far reaching contributions to nonparametric modeling he was one of the leading figures in the developments of nonparametric techniques with over 300 published papers in the field alone this article gives a selective overview on the contributions of peter hall to nonparametric function estimation and modeling the focuses are on density estimation nonparametric regression bandwidth selection boundary corrections inference under shape constraints estimation of residual variances analysis of wavelet estimators multivariate regression and applications of nonparametric methods

lower bounds for bandwidth selection in density estimation this paper establishes asymptotic lower bounds which specify in a variety of contexts how well in terms of relative rate of convergence one may select the bandwidth of a kernel density estimator these results provide important new insights concerning how the bandwidth selection problem should be considered in particular it is shown that if the error criterion is integrated squared error ise then even under very strong assumptions on the underlying density relative error of the selected bandwidth cannot be reduced below ordern 1 10 as the sample size grows this very large error indicates that any technique which aims specifically to minimize ise will be subject to serious practical difficulties arising from sampling fluctuations cross validation exhibits this very slow convergence rate and does suffer from unacceptably large sampling variation on the other hand if the error criterion is mean integrated squared error mise then relative error of bandwidth selection can be reduced to ordern 1 2 when enough smoothness is assumed therefore bandwidth selection techniques which aim to minimize mise can be much more stable and less sensitive to small sampling fluctuations than those which try to minimize ise we feel this indicates that performance in minimizing mise rather than ise should become the benchmark for measuring performance of bandwidth selection methods

bandwidth selection for indirect density estimation based on corrupted histogram data abstract motivated by a number of practical applications we consider a class of indirect nonparametric density estimation problems in which the observed data consist of a histogram of binned empirically corrected counts due to variability in the process of correction the histogram cannot be modeled in terms of simple scaled poisson statistics this departure necessitates the development of a new methodology for bandwidth selection a variant of the method of unbiased risk estimation is proposed the methodology is studied using numerical simulations and asymptotic analysis tools in the context of a class of idealized density deconvolution problems the methodology is adapted for application to the practical reconstruction problem of positron emission tomography pet realistic numerical simulations and physical phantom data are presented to validate the approach in this setting some illustrations with cerebral glucose utilization and myocardial blood flow studies from some actual patient data sets

multivariate online kernel density estimation with gaussian kernels we propose a novel approach to online estimation of probability density functions which is based on kernel density estimation kde the method maintains and updates a non parametric model of the observed data from which the kde can be calculated we propose an online bandwidth estimation approach and a compression revitalization scheme which maintains the kde s complexity low we compare the proposed online kde to the state of the art approaches on examples of estimating stationary and non stationary distributions and on examples of classification the results show that the online kde outperforms or achieves a comparable performance to the state of the art and produces models with a significantly lower complexity while allowing online adaptation

slq 2 realizations for kepler and oscillator potentials and q canonical transformations the realizations of the lie algebra corresponding to the dynamical symmetry group so 2 1 of the kepler and oscillator potentials are q deformed the q canonical transformation connecting two realizations is given and a general definition for the q canonical transformation is deduced a q schrodinger equation for a kepler like potential is obtained from the q oscillator schrodinger equation the energy spectrum and the ground state wavefunction are calculated

suq h 0 2 and suq h 2 the classical and quantum q deformations of the su 2 algebra by means of classical harmonic oscillators and their canonical quantization we show that the q deformation of the su 2 lie algebra can be realized at a classical level in the sense of poisson brackets denoted by suq inf not 0 2 and that the canonical quantization of the system leads to the quantum q deformation of the su 2 algebra denoted by suq 2 this means that the q deformation and the h cross quantization are in general two different concepts

bfv brst analysis of the classical and quantum q deformations of the sl 2 algebra bfv brst charge for quantum algebras is not unique different constructions of it in classical and quantum phase space for the universal enveloping algebra of the q deformed sl 2 are discussed in the quantum framework a positive definite scalar product is used to introduce a co bfv brst charge to study the cohomology problem by means of the techniques of the non deformed case moreover deformation of the phase space without deforming the generators of sl 2 is considered h cross q deformation of the phase space is shown to yield the witten s second deformation for sl 2 to study the bfv brst cohomology problem when both the quantum phase space and the group are deformed a two parameter deformation of sl 2 is proposed and its bfv brst charge is given

necklace solid on solid models of interfaces and surfaces using the exact generating functions for solid on solid models the authors construct necklace partition functions for wetting and prewetting at a sticky wall in two bulk dimensions three component generalisations of these models constructed to study interfacial wetting are introduced and solved based on these models they discuss the types of interfacial critical behaviour to be expected in three component systems in two dimensions

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

two sample quantile tests under general conditions summary a simple nonparametric two sample test for equality of a given collection of quantiles is developed which can be applied to a variety of empirical distribution functions including the kaplan meier estimator a self consistent estimator for doubly censored data and an estimator for repeated measures data the null hypothesis tested is that the quantiles are equal but other aspects of the distributions may diver between the two samples this procedure can also be applied to quantile testing in group sequential clinical trials with staggered patient entry a simple simulation study demonstrates that the moderate sample size properties of this procedure are reasonable

biased and unbiased cross validation in density estimation abstract nonparametric density estimation requires the specification of smoothing parameters the demands of statistical objectivity make it highly desirable to base the choice on properties of the data set in this article we introduce some biased cross validation criteria for selection of smoothing parameters for kernel and histogram density estimators closely related to one investigated in scott and factor 1981 these criteria are obtained by estimating l 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared l 2 error or mean integrated squared error these criteria are roughly the analog of wahba s 1981 generalized cross validation procedure for orthogonal series density estimators we present the relationship of the biased cross validation procedure to the least squares cross validation procedure which provides unbiased estimates of the mean integrated squared error both methods are shown to be based on u statistics we compare the two metho