nonparametric estimation of conditional quantile functions in the presence of irrelevant covariates abstract allowing for the existence of irrelevant covariates we study the problem of estimating a conditional quantile function nonparametrically with mixed discrete and continuous data we estimate the conditional quantile regression function using the check function based kernel method and suggest a data driven cross validation cv approach to simultaneously determine the optimal smoothing parameters and remove the irrelevant covariates when the number of covariates is large we first use a screening method to remove the irrelevant covariates and then apply the cv criterion to those that survive the screening procedure simulations and an empirical application demonstrate the usefulness of the proposed methods

three new factors of fermat numbers we report the discovery of a new factor for each of the fermat numbers f 13 f 15 f 16 these new factors have 27 33 and 27 decimal digits respectively each factor was found by the elliptic curve method after division by the new factors and previously known factors the remaining cofactors are seen to be composite numbers with 2391 9808 and 19694 decimal digits respectively

corrigenda the problem of sierpinski concerning 2ⁿ 1 math comp 37 1981 no 155 229 231

spectra of weighted lb algebras of entire functions on banach spaces abstract given a decreasing family of continuous weights on a banach space x we consider the weighted inductive limits of spaces of entire functions v h x and v h 0 x motivated by recent research by d carando and p sevilla peris on weighted frechet algebras of entire functions on banach spaces we determine conditions on the family of weights to ensure that the corresponding weighted space is an algebra or has polynomial schauder decompositions we study hormander algebras of entire functions defined on a banach space and we give a description of them in terms of sequence spaces we also focus on algebra homomorphisms between these spaces and obtain a banach stone type theorem for a particular decreasing family of weights finally we study the spectra of these weighted algebras endowing them with an analytic structure

local minima in cross validation functions summary the method of least squares cross validation for choosing the bandwidth of a kernel density estimator has been the object of considerable research through both theoretical analysis and simulation studies the method involves the minimization of a certain function of the bandwidth one of the less attractive features of this method which has been observed in simulation studies but has not previously been understood theoretically is that rather often the cross validation function has multiple local minima the theoretical results of this paper provide an explanation and quantification of this empirical observation through modelling the cross validation function as a gaussian stochastic process asymptotic analysis reveals that the degree of wiggliness of the cross validation function depends on the underlying density through a fairly simple functional but dependence on the kernel function is much more complicated a simulation study explores the extent to which the asymptotic analysis describes the actual situation our techniques may also be used to obtain other related results e g to show that spurious local minima of the cross validation function are more likely to occur at too small values of the bandwidth rather than at too large values

effective nonparametric estimation in the case of severely discretized data often economic data are discretized or rounded to some extent this paper proposes a regression and a density estimator that work especially well when discretization causes conventional kernel based estimators to behave poorly the estimator proposed here is a weighted average of neighboring frequency estimators and the weights are composed of cubic b splines interestingly we show that this estimator can have both a smaller bias and variance than frequency estimators as a means to obtain asymptotic normality and rates of convergence we assume that the discreteness becomes finer as the sample size increases

visual error criteria for qualitative smoothing abstract an important gap between the classical mathematical theory and the practice and implementation of nonparametric curve estimation is due to the fact that the usual norms on function spaces measure something different from what the eye can see visually in a graphical presentation mathematical error criteria that more closely follow visual impression are developed and analyzed from both graphical and mathematical viewpoints examples from wavelet regression and kernel density estimation are considered

spectra of weighted algebras of holomorphic functions we consider weighted algebras of holomorphic functions on a banach space we determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial schauder decompositions we study the spectra of weighted algebras and endow them with an analytic structure we also deal with composition operators and algebra homomorphisms in particular to investigate how their induced mappings act on the analytic structure of the spectrum moreover a banach stone type question is addressed

bootstrap choice of bandwidth for density estimation abstract a bootstrap based choice of bandwidth for kernel density estimation is introduced the method works by estimating the integrated mean squared error imse for any given bandwidth and then minimizing over all bandwidths a straightforward application of the bootstrap method to estimate the imse fails because it does not capture the bias component a smoothed bootstrap method based on an initial density estimate is described that solves this problem it is possible to construct pointwise and simultaneous confidence intervals for the density the simulation study compares cross validation and the bootstrap method over a wide range of densities a long tailed a short tailed an asymmetric and a bimodal among others the bootstrap method uniformly outperforms cross validation the accuracy of the constructed confidence bands improves as the sample size increases

on fermat s quotient base two this paper extends the search for solutions of the congruence 2p1 0 modp2 to the limit p 6 109 no solution except the well known p 1093 and p 3511 was found in 1969 brillhart tonascia and weinberger 1 reported on the search for solutions of the congruence a 1 modp2 a 2 1 199 for a 2 a special effort was made to consider all p 3 109 only the two known solutions p 1093 and p 3511 were found having the occasional opportunity to use the illiac iv one of the projects decided upon was the recalculation and extension of the above result for a 2 by a somewhat different method the calculation was pushed to twice the above limit that is top 6 109 without finding any further solutions the parallel construction of the illiac iv makes it possible to look for 64 different values of p at the same time the speed of illiac iv is such that a range of 100000 numbers can be searched in one second thus the range for p 6 109 was broken up into 60 runs of 1000 seconds each the program was run as one of a few backlog problems over the past two years let n belong to one of the 64 residue classes that are prime to 240 and let 2m am nbm mod n2 where 0 am n 0 bm n for numbers n as large as 109 the number n2 is a doubly precise integer nevertheless the calculation of am and bm can be accomplished by single precision arithmetic in only o log m operations in fact one uses one or the other of the following two recurrences if m 2h 1 then am 2a2h and bm 2b2h mod n if m 2h and if a 2 rm nqm 0 rm n and if 2ahbh dm mod n then am rm and bm qm dm received march 30 1980 1980 mathematics subject classification primary 10 04 1981 american mathematical society 0025 5718 81 0000 0027 0 1 50 289 this content downloaded from 157 55 39 58 on tue 11 oct 2016 04 26 58 utc all use subject to http about jstor org terms

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

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

exact mean integrated squared error an exact and easily computable expression for the mean integrated squared error mise for the kernel estimator of a general normal mixture density is given for gaussian kernels of arbitrary order this provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis the family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes a number of applications of this method giving important new insights into kernel density estimation are presented among these is the discovery that the usual asymptotic approximations to the mise can be quite inaccurate especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes 1 introduction substantial research has been devoted to kernel density estimation this is because it provides a simple yet appealing context in which to study problems and issues that arise in all types of nonparametric curve estimation this includes regression spectral density and hazard estimation and also a variety of other estimators including histograms splines and orthogonal series three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis simulation and numerical calculation of error criteria each of these methods provides many useful insights into the complicated structure present in the study of curve estimation however each has its limitations as well rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples through general results applying to entire classes of settings the weakness of asymptotics is that they only describe behavior in the limit this is still very useful in many situations because the asymptotics describe the actual situation quite well however it is less useful when the asymptotics have not yet kicked in that is in studying situations where the asymptotically dominant effect has not taken over yet perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring

a new parametric method of estimating the joint probability density abstract we present simple parametric methods that overcome major limitations of the literature on joint marginal density estimation in doing so we do not assume any form of marginal or joint distribution furthermore using our method a multivariate density can be easily estimated if we know only one of the marginal densities we apply our methods to financial data

the twentieth fermat number is composite the twentieth fermat number f20 2220 1 has been proven composite by machine computation the fermat numbers are the numbers fn 224 1 orginally conjectured by fermat to be prime for all n in fact only for n equal to 0 through 4 are they known to be prime and small factors of f9 fl1 fl2 fl5 fl6 have been known for some time as part of a long term test of the hardware reliability of the cray 2 supercomputer at the supercomputing research center the authors proved that f20 2220 1 which had been the smallest fermat number of unknown character is composite the test for compositeness was the standard technique of pepin 5 for n 1 fn is prime if and only if 3 f 2 1 mod fn this test for compositeness does not of course produce factors of the number but was the test used for proving the compositeness of f7 f8 flo f13 f14 2 3 4 6 7 the result of the computation on the cray 2 has been verified by performing the same computation on a cray x mp belonging to cray research the total computation time on the cray 2 was about 10 cpu days the time on the cray xmp was 82 hours both programs ran as single processor programs on any available cpu of the respective machines the ability of either computer to run in parallel on multiple cpu s was not used the time needed to test fn for n in the range 10 through 20 is just slightly more than four times the time needed to test fn the number of multiplications doubles in incrementing n and the time required for each multiplication doubles being dependent almost entirely on the length of the operands our programs would thus determine the character of f22 which is now the smallest fermat number of unknown character in a little more than 16 times the time needed for our computation on f20 the table below summarizes what is now known about the fermat numbers for n less than or equal to 22 a status list for larger n appears in 1 this computation roughly one million squarings modulo a one million bit number would be impossible to do even on supercomputers without fast fourier transform techniques for integer multiplication since one reason for performing this computation was to verify hardware reliablity and not to minimize the execution time the program was written entirely in cray fortran and called cray library functions for the fft s the program itself was very simple and only about 200 lines long much of which was used for checkpointing and restarting the program the program was called into execution every time the cray 2 was restarted and so received april 6 1987 revised june 5 1987 1980 mathematics subject classification 1985 revnsion primary ilyii lla51 11 04 d1988 american mathematical society 0025 5718 88 1 00 25 per page

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

sizer for exploration of structures in curves abstract in the use of smoothing methods in data analysis an important question is which observed features are really there as opposed to being spurious sampling artifacts an approach is described based on scale space ideas originally developed in the computer vision literature assessment of significant zero crossings of derivatives results in the sizer map a graphical device for display of significance of features with respect to both location and scale here scale means level of resolution that is bandwidth

factorization of the eighth fermat number we describe a monte carlo factorization algorithm which was used to factorize the fermat number f8 2256 1 previously f8 was known to be composite but its factors were unknown comments only the abstract is given here the full paper appeared as 2 for a succinct proof of primality of the larger factor q8 of f8 see 3 at the time of this paper lenstra s elliptic curve method ecm had not been invented thus a modification of pollard s rho method 1 was used to factor f8 and several mersenne numbers 4 for the factorization of a larger fermat number by ecm see 5 the smaller factor of f8 is 1238926361552897 the epigram i am now entirely persuaded to employ the method a handy trick on gigantic composite numbers may appeal to readers who wish to memorize this factor

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

the maximal smoothing principle in density estimation abstract we propose a widely applicable method for choosing the smoothing parameters for nonparametric density estimators it has come to be realized in recent years e g see hall and marron 1987 scott and terrell 1987 that cross validation methods for finding reasonable smoothing parameters from raw data are of very limited practical value their sampling variability is simply too large the alternative discussed here the maximal smoothing principle suggests that we consider using the most smoothing that is consistent with the estimated scale of our data this greatly generalizes and exploits a phenomenon noted in terrell and scott 1985 that measures of scale tend to place upper bounds on the smoothing parameters that minimize asymptotic mean integrated squared error of density estimates such as histograms and frequency polygons the method avoids the extreme sampling variability of cross validation by using ordinary scale estimators such as the standard deviation and interquartile range which

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

the factorization of the ninth fermat number in this paper we exhibit the full prime factorization of the ninth fermat number f9 2 512 1 it is the product of three prime factors that have 7 49 and 99 decimal digits we found the two largest prime factors by means of the number field sieve which is a factoring algorithm that depends on arithmetic in an algebraic number field in the present case the number field used was q fifth root 2 the calculations were done on approximately 700 workstations scattered around the world and in one of the final stages a supercomputer was used the entire factorization took four months

a reliable data based bandwidth selection method for kernel density estimation we present a new method for data based selection of the bandwidth in kernel density estimation which has excellent properties it improves on a recent procedure of park and marron which itself is a good method in various ways first the new method has superior theoretical performance second it also has a computational advantage third the new method has reliably good performance for smooth densities in simulations performance that is second to none in the existing literature these methods are based on choosing the bandwidth to approximately minimize good quality estimates of the mean integrated squared error the key to the success of the current procedure is the reintroduction of a non stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

homomorphisms and composition operators on algebras of analytic functions of bounded type abstract let u and v be convex and balanced open subsets of the banach spaces x and y respectively in this paper we study the following question given two frechet algebras of holomorphic functions of bounded type on u and v respectively that are algebra isomorphic can we deduce that x and y or x and y are isomorphic we prove that if x or y has the approximation property and h wu u and h wu v are topologically algebra isomorphic then x and y are isomorphic the converse being true when u and v are the whole space we get analogous results for h b u and h b v giving conditions under which an algebra isomorphism between h b x and h b y is equivalent to an isomorphism between x and y we also obtain characterizations of different algebra homomorphisms as composition operators study the structure of the spectrum of the algebras under consideration and show the existence of homomorphisms on h b x with pathological behaviors

factors of fermat numbers and large primes of the form 2ⁿ 1 a new factor is given for each of the fermat numbers f52 f931 f6835 and f9448 in addition a factor of f75 discovered by gary gostin is presented the current status for all f is shown in a table primes of the form k 2 1 k odd are listed for 31 k 149 1500 ii 4000 and for 151 k 199 1000 ii 4000 some primes for even larger values of ii are included the largest one being 5 213165 1 also a survey of several related questions is given in particular values of k such that k 2 i is composite for every ii are considered as well as odd values of h such that 3h 2 1 never yields a twin prime pair

oversmoothed nonparametric density estimates abstract the optimal histogram for a sample of size n from a density defined on the entire real line requires at least 2n 1 3 bins under mild smoothness conditions similar bounds exist for the frequency polygon and kernel estimators values near these bounds give nearly optimal results for a variety of smooth densities providing good quick density estimates

bivariate frequency analysis of floods using a diffusion based kernel density estimator recent focus of flood frequency analysis ffa studies has been on development of methods to model joint distributions of variables such as peak flow volume and duration that characterize a flood event as comprehensive knowledge of flood event is often necessary in hydrological applications diffusion process based adaptive kernel d kernel is suggested in this paper for this purpose it is data driven flexible and unlike most kernel density estimators always yields a bona fide probability density function it overcomes shortcomings associated with the use of conventional kernel density estimators in ffa such as boundary leakage problem and normal reference rule the potential of the d kernel is demonstrated by application to synthetic samples of various sizes drawn from known unimodal and bimodal populations and five typical peak flow records from different parts of the world it is shown to be effective when compared to conventional gaussian kernel and the best of seven commonly used copulas gumbel hougaard frank clayton joe normal plackett and student s t in estimating joint distribution of peak flow characteristics and extrapolating beyond historical maxima selection of optimum number of bins is found to be critical in modeling with d kernel

multivariate bandwidth selection for local linear regression the existence and properties of optimal bandwidths for multivariate local linear regression are established using either a scalar bandwidth for all regressors or a diagonal bandwidth vector that has a different bandwidth for each regressor both involve functionals of the derivatives of the unknown multivariate regression function estimating these functionals is difficult primarily because they contain multivariate derivatives in this paper an estimator of the multivariate second derivative is obtained via local cubic regression with most cross terms left out this estimator has the optimal rate of convergence but is simpler and uses much less computing time than the full local estimator using this as a pilot estimator we obtain plug in formulae for the optimal bandwidth both scalar and diagonal for multivariate local linear regression as a simpler alternative we also provide rule of thumb bandwidth selectors all these bandwidths have satisfactory performance in our simulation study

distributed fusion with multi bernoulli filter based on generalized covariance intersection in this paper we propose a distributed multiobject tracking algorithm through the use of multi bernoulli mb filter based on generalized covariance intersection g ci our analyses show that the g ci fusion with two mb posterior distributions does not admit an accurate closed form expression to solve this problem we first approximate the fused posterior as the unlabeled version of delta generalized labeled mb distribution referred to as generalized mb gmb distribution then to allow the subsequent fusion with another mb posterior distribution e g fusion with a third sensor node in the sensor network or fusion in the feedback working mode we further approximate the fused gmb posterior distribution as an mb distribution which matches its first order statistical moment the proposed fusion algorithm is implemented using sequential monte carlo technique and its performance is highlighted by numerical results

factors of generalized fermat numbers generalized fermat numbers have the form f b m b 2m 1 their odd prime factors are of the form k 2 n 1 k odd n m it is shown that each prime is a factor of some f b m for approximately 1 k bases b independent of n divisors of generalized fermat numbers of base 6 base 10 and base 12 are tabulated three new factors of standard fermat numbers are included

a flexible and fast method for automatic smoothing abstract the choice of smoothing parameter or bandwidth is crucial when applying nonparametric regression estimators such as kernel estimators the optimal choice depends on the data at hand a data driven bandwidth selection close to the optimal one would make these curve estimators objective more reliable and easier to use minimizing residual mean squared error criteria such as cross validation have been frequently proposed to estimate the optimal smoothing parameter empirical and theoretical evidence indicates that cross validation rules and related methods lead to rather variable estimated optimal bandwidths the method presented here builds on estimating the asymptotically optimal bandwidth from the data since estimators for the residual variance and for an asymptotic expression for the bias are plugged into the asymptotic formula such selection rules are called plug in estimators the functional that quantifies bias is approximated by the integrated squared second derivative of the regre

rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density in this paper we investigate the asymptotic properties of nonparametric bayesian mixtures of betas for estimating a smooth density on 0 1 we consider a parametrization of beta distributions in terms of mean and scale parameters and construct a mixture of these betas in the mean parameter while putting a prior on this scaling parameter we prove that such bayesian nonparametric models have good frequentist asymptotic properties we determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a holder class with regularity for all positive leading to a minimax adaptive estimating procedure of the density we also believe that the approximating results obtained on these mixtures of beta densities can be of interest in a frequentist framework

factorization of the tenth fermat number we describe the complete factorization of the tenth fermat number f10 by the elliptic curve method ecm f10 is a product of four prime factors with 8 10 40 and 252 decimal digits the 40 digit factor was found after about 140 mflop years of computation we also discuss the complete factorization of other fermat numbers by ecm and summarize the factorizations

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

bootstrap choice of the smoothing parameter in kernel density estimation summary cross validation based on integrated squared error has already been applied to the choice of smoothing parameter in the kernel method of density estimation in this paper an alternative resampling plan based on the bootstrap is proposed in an attempt to estimate mean integrated squared error this leads to a further data based choice of smoothing parameter the two methods are compared and some simulations and examples demonstrate the relative merits for large samples the bootstrap performs better than cross validation for many distributions

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

intrinsic regression models for medial representation of subcortical structures the aim of this article is to develop a semiparametric model to describe the variability of the medial representation of subcortical structures which belongs to a riemannian manifold and establish its association with covariates of interest such as diagnostic status age and gender we develop a two stage estimation procedure to calculate the parameter estimates the first stage is to calculate an intrinsic least squares estimator of the parameter vector using the annealing evolutionary stochastic approximation monte carlo algorithm and then the second stage is to construct a set of estimating equations to obtain a more efficient estimate with the intrinsic least squares estimate as the starting point we use wald statistics to test linear hypotheses of unknown parameters and establish their limiting distributions simulation studies are used to evaluate the accuracy of our parameter estimates and the finite sample performance of the wald statistics we apply our methods to the detection of the diffe

the amount of noise inherent in bandwidth selection for a kernel density estimator abstract any practical method of constructing a bandwidth must depend only on a statistical sample and should produce some sort of estimate of this bandwidth the purpose of this paper is to show that there a well defined limits to the accuracy of all data driven bandwidth estimates put another way there is an unbridgeable gap between the minimum integrated square error attained using a optimal bandwidth and the minimum achievable integrated square error using a data driven bandwidth estimate additional keywords stochastic processes cross validation and random variables

on local likelihood density estimation this paper considers a class of local likelihood methods introduced by eguchi and copas unified asymptotic results are presented in the usual smoothing context of the bandwidth h tending to zero as the sample size tends to infinity we present our results pointwise in the univariate case but then go on to extend them to global properties and to indicate how to cope with the multivariate case specific members of the class due to copas and hjort and jones are seen to be members of a subset of the whole class with the same and best small h behavior further comparisons between members of the class are alluded to based on the complementary large h asymptotic results of eguchi and copas

particle based smoothed marginal map estimation for general state space models we consider the smoothing problem for a general state space system using sequential monte carlo smc methods the marginal smoother is assumed to be available in the form of weighted random particles from the smc output new algorithms are developed to extract the smoothed marginal maximum a posteriori map estimate of the state from the existing marginal particle smoother our method does not need any kernel fitting to obtain the posterior density from the particle smoother the proposed estimator is then successfully applied to find the unknown initial state of a dynamical system and to address the issue of parameter estimation problem in state space models

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

succinct proofs of primality for the factors of some fermat numbers we give short and easily verified proofs of primality for the factors of the fermat numbers f5 f6 f7 and f8

adaptive bayesian multivariate density estimation with dirichlet mixtures summary we show that rate adaptive multivariate density estimation can be performed using bayesian methods based on dirichlet mixtures of normal kernels with a prior distribution on the kernel s covariance matrix parameter we derive sufficient conditions on the prior specification that guarantee convergence to a true density at a rate that is minimax optimal for the smoothness class to which the true density belongs no prior knowledge of smoothness is assumed the sufficient conditions are shown to hold for the dirichlet location mixture of normals prior with a gaussian base measure and an inverse wishart prior on the covariance matrix parameter locally h older

error rates of non bayes classification rules and the robustness of fisher s linear discriminant function summary we call a classification procedure non bayes if it does not converge to the bayes classification procedure an asymptotic expansion is found for the expected error rate of such a classification rule this is used to compare the estimates of fisher s linear discriminant rule f and the quadratic discriminant rule q under departures from the equal variance matrices assumption it is found that f is quite robust to departures from the equal variances assumption

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

uniform in bandwidth consistency of kernel type function estimators we introduce a general method to prove uniform in bandwidth consistency of kernel type function estimators examples include the kernel density estimator the nadaraya watson regression estimator and the conditional empirical process our results may be useful to establish uniform consistency of data driven bandwidth kernel type function estimators

multivariate density estimation by bayesian sequential partitioning consider a class of densities that are piecewise constant functions over partitions of the sample space defined by sequential coordinate partitioning we introduce a prior distribution for a density in this function class and derive in closed form the marginal posterior distribution of the corresponding partition a computationally efficient method based on sequential importance sampling is presented for the inference of the partition from this posterior distribution compared to traditional approaches such as the kernel method or the histogram the bayesian sequential partitioning bsp method proposed here is capable of providing much more accurate estimates when the sample space is of moderate to high dimension we illustrate this by simulated as well as real data examples the examples also demonstrate how bsp can be used to design new classification methods competitive with the state of the art

an automatic bandwidth selector for kernel density estimation summary to select a proper bandwidth is a critical step in kernel density estimation it is well known that the bandwidth selected by cross validation has a large variability this difficulty limits the applicability of cross validation to reduce the variability we suggested modifying the sample characteristic function beyond some cut off frequency in estimating the bias term of the mean integrated squared error it is proposed to select the cut off frequency by a generalization of cross validation for smooth density functions the asymptotic distribution of the bandwidth estimator based on the estimated cut off frequency is obtained the proposed bandwidth estimator has a relative convergence rate n which is much faster than the rate n 1110 for the bandwidth estimate selected by cross validation a modification which reduces the chance of selecting a large cut off frequency is also suggested in simulation studies the advantages of the proposed procedures are clearly demonstrated the procedures are also applied to some data sets

six new factors of fermat numbers in the past few years several investigators 1 11 have found many new prime factors of the fermat numbers fm 22 1 the increased availability and speed of computing equipment have allowed the search limits for factors of these numbers to be extended significantly in tables below we include the known search limits and all of the results that have been obtained since h c williams 1978 paper 12 was published using the method outlined by hallyburton and brillhart 8 we tested integers of the form k 2r 1 k odd as factors of the fermat numbers fm 9 m n 2 for the ranges shown in table 1 during our search the following new prime factors were found 1120049 23 1jf29

novel and simple non parametric methods of estimating the joint and marginal densities we introduce very simple non parametric methods that overcome key limitations of the existing literature on both the joint and marginal density estimation in doing so we do not assume any form of the marginal distribution or joint distribution a priori furthermore our method circumvents the bandwidth selection problems we compare our method to the kernel density method

data driven optimal transport the problem of optimal transport between two distributions x and y is extended to situations where the distributions are only known through a finite number of samples xi and yj a weak formulation is proposed based on the dual of the kantorovich formulation with two main modifications replacing the expected values in the objective function by their empirical means over the xi and yj and restricting the dual variables u x and v y to a suitable set of test functions adapted to the local availability of sample points a procedure is proposed and tested for the numerical solution of this problem based on a fluidlike flow in phase space where the sample points play the role of active lagrangian markers 2016 wiley periodicals inc

differences of composition operators between weighted type spaces of holomorphic functions on the unit ball of cn we consider differences of composition operators between weighted type or little weighted type spaces of analytic functions on the unit ball with weighted sup norms and give estimates for the distance of these differences to the space of compact operators we also study the boundedness and compactness of these operators

envelopes of open sets and extending holomorphic functions on dual banach spaces abstract we investigate certain envelopes of open sets in dual banach spaces which are related to extending holomorphic functions we give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible we also establish connections to the study of iterated weak sequential closures of convex sets in the dual of separable spaces

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

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

from finite sample to asymptotics a geometric bridge for selection criteria in spline regression this paper studies under the setting of spline regression the connection between finite sample properties of selection criteria and their asymptotic counterparts focusing on bridging the gap between the two we introduce a bias variance decomposition of the prediction error using which it is shown that in the asymptotics the bias term dominates the variability term providing an explanation of the gap a geometric exposition is provided for intuitive understanding the theoretical and geometric results are illustrated through a numerical example

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