variable window width kernel estimates of probability densities kernel density estimators which allow different amounts of smoothing at different locations are studied modifications of estimators proposed by breiman meisel and purcell 1977 and abramson 1982a which have variable window widths are seen to have very fast rates of convergence these rates have traditionally been obtained using a less natural higher order kernel which has the disadvantage of allowing an estimator which takes on negative values

analytic continuation and boundary continuity of functions of several complex variables this note treats some questions about analytic continuation in several variables the first theorem in effect determines the envelops of holomorphy of certain domains in n the second main result is a continuity theorem if a bounded holomorphic function f on a convex domain in n has boundary values that are continuous on the complement in b of a set of the form int b where is a real hyperplane in n that misses then f is continuous on in addition we obtain what may be regarded as a local version of the theorem in our earlier paper concerning the one dimensional extension property our methods depend on hartogs theorem n 3 and directly on the bochnermartinelli formula n 2

benchmarking optimization software with performance profiles we propose performance profiles distribution functions for a performance metric as a tool for benchmarking and comparing optimization software we show that performance profiles combine the best features of other tools for performance evaluation

transformations to reduce boundary bias in kernel density estimation we consider kernel estimation of a univariate density whose support is a compact interval if the density is non zero at either boundary then the usual kernel estimator can be seriously biased reflection at a boundary removes some bias but unless the first derivative of the density is 0 at the boundary the estimator with reflection can still be much more severely biased at the boundary than in the interior we propose to transform the data to a density that has its first derivative equal to 0 at both boundaries the density of the transformed data is estimated and an estimate of the density of the original data is obtained by a change of variables the transformation is selected from a parametric family which is allowed to be quite general in our theoretical study we propose algorithms where the transformation is either a quartic polynomial a b cumulative density function cdf or a linear combination of a polynomial and a b cdf the last two types of transformation are designed to accommodate possible poles at the boundaries the first two algorithms are tested on simulated data and compared with an adjusted kernel method of rice we find that our proposal performs similarly to rice s for densities with one sided derivatives at the boundaries unlike rice s method our proposal is guaranteed to produce non negative estimates this can be a distinct advantage when the density is 0 at either boundary our algorithm for densities with poles outperforms the rice adjustment when the density has a pole at a boundary

fourier laplace transforms and related questions on certain analytic varieties in c2 abstract for an analytic variety v z 1 z 2 c 2 z 1 z 2 0 defined by a holomorphic function we assume that the point 0 v and that v 0 is smooth in this setting we construct holomorphic differentials on v 0 with prescribed certain of the values of the integrals z 1 k z 2 l z 1 z 2 taken over closed curves on v which surround 0 the construction is quite explicit and is based on a residue process we also study similar questions with specific choices of in which cases we obtain more complete results

root finding by high order iterative methods based on quadratures we discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable these methods are based on newton cotes closed quadrature rules we prove that when a quadrature rule with n 1 nodes is used the resulting iterative method has convergence order at least n 2 starting with the case n 0 which corresponds to the newton s method

mass recentred kernel smoothers the local linear smoother usually gives better performance than the nadaraya watson smoother an exception is the case of data sparsity here we discuss a modification of the nadaraya watson smoother by muller song 1993 based on a horizontal shift of the kernel weights towards the local centre of mass of the design points this gives performance similar to the local linear when that works well and better performance when it does not the new smoother also preserves monotonicity shifting towards the centre of mass is also used to develop a modified kernel density estimate which cancels the well known peak spreading effect

pairwise local fisher and naive bayes improving two standard discriminants abstract the fisher discriminant is probably the best known likelihood discriminant for continuous data another benchmark discriminant is the naive bayes which is based on marginals only in this paper we extend both discriminants by modeling dependence between pairs of variables in the continuous case this is done by local gaussian versions of the fisher discriminant in the discrete case the naive bayes is extended by taking geometric averages of pairwise joint probabilities we also indicate how the two approaches can be combined for mixed continuous and discrete data the new discriminants show promising results in a number of simulation experiments and real data illustrations

semiparametric density estimation by local l2 fitting this article examines density estimation by combining a parametric approach with a nonparametric factor the plug in parametric estimator is seen as a crude estimator of the true density and is adjusted by a nonparametric factor the nonparametric factor is derived by a criterion called local l 2 fitting a class of estimators that have multiplicative adjustment is provided including estimators proposed by several authors as special cases and the asymptotic theories are developed theoretical comparison reveals that the estimators in this class are better than or at least competitive with the traditional kernel estimator in a broad class of densities the asymptotically best estimator in this class can be obtained from the elegant feature of the bias function

on interpolation variants of newton s method for functions of several variables a generalization of the variants of newton s method based on interpolation rules of quadrature is obtained in order to solve systems of nonlinear equations under certain conditions convergence order is proved to be 2d 1 where d is the order of the partial derivatives needed to be zero in the solution moreover different numerical tests confirm the theoretical results and allow us to compare these variants with newton s classical method whose convergence order is d 1 under the same conditions

variants of newton s method using fifth order quadrature formulas abstract some variants of newton s method are developed in this work in order to solve nonlinear equations depending on one or several variables based in rules of quadrature of fifth order we prove the third or fifth order of convergence of these methods for dimension one and the second or third order in several variables depending on the behaviour of the second derivative moreover different numeric tests confirm or improve theoretic results and allow us to compare these variants with newton s classical method

on nonparametric kernel density estimates summary the paper introduces the idea of inadmissible kernels and shows that an epanechnikov type kernel is the only admissible kernel an analysis of kernel density estimates leads to two new methods of bias reduction we also discuss a general method of improving kernel density estimates in the sense of having smaller mean squared error finally we provide a theoretical framework for the operational method of using abramson s square root law providing tangible and compact forms for the bias and mean squared error

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

semiparametric pseudo likelihoods in generalized linear models with nonignorable missing data we consider identifiability and estimation in a generalized linear model in which the response variable and some covariates have missing values and the missing data mechanism is nonignorable and unspecified we adopt a pseudo likelihood approach that makes use of an instrumental variable to help identifying unknown parameters in the presence of nonignorable missing data explicit conditions for the identifiability of parameters are given some asymptotic properties of the parameter estimators based on maximizing the pseudo likelihood are established explicit asymptotic covariance matrix and its estimator are also derived in some cases for the numerical maximization of the pseudo likelihood we develop a two step iteration algorithm that decomposes a nonconcave maximization problem into two problems of maximizing concave functions some simulation results and an application to a dataset from cotton factory workers are also presented

variants of newton s method for functions of several variables some variants of newton s method are developed in this work in order to solve systems of nonlinear equations based in trapezoidal and midpoint rules of quadrature we prove the quadratic convergence of one of these methods moreover different numeric tests confirm theoretic results and allow us to compare these variants with newton s classical method

a formula for the derivatives of holomorphic functions in c2 in terms of certain integrals taken on boundaries of analytic varieties abstract we derive a cauchy fantappie type formula which expresses the derivatives of holomorphic functions at a point on a given analytic variety in terms of the values of the function in an arbitrarily small neighborhood of the curve which bounds the variety the formula involves derivatives of functions defined by integrals taken on boundaries of nearby analytic varieties we also apply these formulas to questions related to analytic functionals

kernel density estimation via diffusion we present a new adaptive kernel density estimator based on linear diffusion processes the proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate in addition we propose a new plug in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods we present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability

asymptotics for the transformation kernel density estimator an asymptotic expansion is provided for the transformation kernel density estimator introduced by ruppert and cline let h k be the bandwidth used in the kth iteration k 1 2 t if all bandwidths are of the same order the leading bias term of the lth derivative of the tth iterate of the density estimator has the form b l t x t k 1 h 2 k where the bias factor b t x depends on the second moment of the kernel k as well as on all derivatives of the density f up to order 2t in particular the leading bias term is of the same order as when using an ordinary kernel density estimator with a kernel of order 2t the leading stochastic term involves a kernel of order 2t that depends on k h l and h k f x k 2 t

some quadrature based versions of the generalized newton method for solving nonsmooth equations abstract in this paper modifications of a generalized newton method based on some rules of quadrature are studied the methods considered are newton like iterative schemes for numerical solving systems of nonsmooth equations some mild conditions are given that ensure superlinear convergence to a solution moreover a parameterized version of the midpoint version is presented finally results of numerical tests are established

local polynomial quantile regression with parametric features we propose a new approach to conditional quantile function estimation that combines both parametric and nonparametric techniques at each design point a global possibly incorrect pilot parametric model is locally adjusted through a kernel smoothing fit the resulting quantile regression estimator behaves like a parametric estimator when the latter is correct and converges to the nonparametric solution as the parametric start deviates from the true underlying model we give a bahadur type representation of the proposed estimator from which consistency and asymptotic normality are derived under an mixing assumption we also propose a practical bandwidth selector based on the plug in principle and discuss the numerical implementation of the new estimator finally we investigate the performance of the proposed method via simulations and illustrate the methodology with a data example

on the effect of bias estimation on coverage accuracy in nonparametric inference abstractnonparametric methods play a central role in modern empirical work while they provide inference procedures that are more robust to parametric misspecification bias they may be quite sensitive to tuning parameter choices we study the effects of bias correction on confidence interval coverage in the context of kernel density and local polynomial regression estimation and prove that bias correction can be preferred to undersmoothing for minimizing coverage error and increasing robustness to tuning parameter choice this is achieved using a novel yet simple studentization which leads to a new way of constructing kernel based bias corrected confidence intervals in addition for practical cases we derive coverage error optimal bandwidths and discuss easy to implement bandwidth selectors for interior points we show that the mse optimal bandwidth for the original point estimator before bias correction delivers the fastest coverage error decay rate after bias correction when second order equi

on certain integrals associated to cr functions we construct explicit 0 primitives of cauchy fantappie type kernels on analytic subvarieties of domains in cn outside the zero set of a holomorphic map defined on the variety then we use these primitives to derive by means of a process of passing to a residue integral formulas for cr functions defined on subsets of the boundary of a variety certain geometric restrictions on the varieties are imposed these primitives apply in the particular case of the bochner martinelli kernel in domains in cn and we use these primitives in this case to prove a criterion for holomorphic extendability of cr functions defined on certain subsets of the boundary of a domain in cn introduction let d g c cn be bounded domains with smooth boundary and d c g let h g c be a holomorphic map and set z h c e g h o in 2 we constructed kernels ah 4 z and h so that if f is a holomorphic function on z h and z e z h n d then 1 f z f 4c a 4 z a fh c jez h n od the differential forms a h and a h were explicitly defined in terms of h a hefer decomposition hi of h i e holomorphic functions hij c z in gx g with n hi 4 hi z e hij c z j zj i 1 p c z e g x g j 1 and in terms of a function y od x d cn so that n e yj c z cj zj 1 for c z e od x d j l certain geometric restrictions transversality conditions were imposed which guarantee that z h n 3d is a smooth manifold in this paper we pursue further these constructions by proving an integral formula satisfied by cr functions which are defined on certain parts of the boundary of a variety first we construct another kernel i h z which is a received by the editors february 15 1988 1980 mathematics subject classification 1985 revision primary 32a25 32a27 32d1 5 1989 american mathematical society 0002 9947 89 1 00 25 per page

on the bias of variable bandwidth curve estimators summary a major difficulty in understanding the properties of variable bandwidth methods breiman meisel purcell 1977 abramson 1982 is that extremely lengthy and complex algebra is needed to assess the influence of bias indeed the complexity is so great that it has forced investigators to use computer algebraic manipulation to determine formulae for bias in this note we completely eliminate these algebraic obstacles by presenting a simple easy to use formula which gives explicitly the bias of a variable bandwidth estimator to arbitrarily high order in very general problems

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

on improving density estimators which are not bona fide functions in order to improve the rate of decrease of the imse for nonparametric kernel density estimators with nonrandom bandwidth beyond o n 4 5 all current methods must relax the constraint that the density estimate be a bona fide function that is be nonnegative and integrate to one in this paper we show how to achieve similar improvement without relaxing any of these constraints the method can also be applied for orthogonal series adaptive orthogonal series spline jackknife and other density estimators and assures an improvement of the imse for each sample size

kernel smoothing for nested estimation with application to portfolio risk measurement nested estimation involves estimating an expectation of a function of a conditional expectation via simulation this problem has of late received increasing attention amongst researchers due to its broad applicability particularly in portfolio risk measurement and in pricing complex derivatives in this paper we study a kernel smoothing approach we analyze its asymptotic properties and present efficient algorithms for practical implementation while asymptotic results suggest that the kernel smoothing approach is preferable over nested simulation only for low dimensional problems we propose a decomposition technique for portfolio risk measurement through which a high dimensional problem may be decomposed into low dimensional ones that allow an efficient use of the kernel smoothing approach numerical studies show that with the decomposition technique the kernel smoothing approach works well for a reasonably large portfolio with 200 risk factors this suggests that the proposed methodology may serve

on the effect of the multidimensional weight functions on the stability of iterative processes abstract in this work we start from a family of iterative methods for solving nonlinear multidimensional problems designed using the inclusion of a weight function on its iterative expression a deep dynamical study of the family is carried out on polynomial systems by selecting different weight functions and comparing the results obtained in each case this study shows the applicability of the multidimensional dynamical analysis in order to select the methods of the family with the best stability properties

new methods for bias correction at endpoints and boundaries we suggest two new translation based methods for estimating and correcting for bias when estimating the edge of a distribution the first uses an empirical translation applied to the argument of the kernel in order to remove the main effects of the asymmetries that are inherent when constructing estimators at boundaries placing the translation inside the kernel is in marked contrast to traditional approaches such as the use of high order kernels which are related to the jackknife and in effect apply the translation outside the kernel our approach has the advantage of producing bias estimators that while enjoying a high order of accuracy are guaranteed to respect the sign of bias our second method is a new bootstrap technique it involves translating an initial boundary estimate toward the body of the dataset constructing repeated boundary estimates from data that lie below the respective translations and employing averages of the resulting empirical bias approximations to estimate the bias of the original estimator the first of the two methods is most appropriate in univariate cases and is studied there the second approach may be used to bias correct estimates of boundaries of multivariate distributions and is explored in the bivariate case

using specially designed exponential families for density estimation i this paper have y being portions of the real line or of the plane but the methodology applies just as well to higher dimensionalities and to more complicated spaces estimates of g y are traditionally constructed in two quite different ways by maximum likelihood fitting within some parametric family such as the normal or by nonparametric methods such as kernel density estimation these two methods can be combined by putting an exponential family through a nonparametric estimator the resulting hybrid estimators are the specially designed exponential families of the title figure 1 shows a simple example of this methodology the y are pain i scores for n s 67 women each obtained by averaging the results from a questionnaire administered after an operation the scale runs from y s 0 s w x

pseudocomposition a technique to design predictor corrector methods for systems of nonlinear equations abstract a new technique for designing iterative methods for solving nonlinear systems is presented this procedure called pseudocomposition uses a known method as a predictor and the gaussian quadrature as a corrector the order of convergence of the resulting scheme depends among other factors on the order of the last two steps of the predictor we also introduce a new iterative algorithm of order six and apply the mentioned technique to generate a new method of order ten finally some numerical test are shown

an efficient high order iterative scheme for large nonlinear systems with dynamics abstract this study suggests a new general scheme of high convergence order for approximating the solutions of nonlinear systems the proposed scheme is the extension of an earlier study of parhi and gupta this method requires two vector function two jacobian matrices two inverse matrices and one frozen inverse matrix per iteration convergence error computational efficiency and numerical experiments are performed to verify the applicability and validity of the proposed methods compared with existing methods finally we discuss the strange fixed points and conjugacy functions on a particular case of our scheme the basins of attractions also demonstrate the dynamical behavior of this special case in the neighborhood of required roots and also assert the theoretical outcomes

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

extended convergence results for the newton kantorovich iteration we present new semilocal and local convergence results for the newton kantorovich method these new results extend the applicability of the newton kantorovich method on approximate zeros by improving the convergence domain and ratio given in earlier studies by argyros 2003 cianciaruso 2007 smale 1986 and wang 1999 these advantages are also obtained under the same computational cost numerical examples where the old sufficient convergence criteria are not satisfied but the new convergence criteria are satisfied are also presented in this study

semiparametric estimation of binary response models with endogenous regressors abstract in this paper we propose a two step semiparametric maximum likelihood sml estimator for the coefficients of a single index binary choice model with endogenous regressors when identification is achieved via a control function approach the first step consists of estimating a reduced form equation for the endogenous regressors and extracting the corresponding residuals in the second step the latter are added as control variates to the outcome equation which is in turn estimated by sml we establish the estimator s n consistency and asymptotic normality in a simulation study we compare the properties of our estimator with those of existing alternatives highlighting the advantages of our approach

the application of newton s method in vector form for solving nonlinear scalar equations where the classical newton method fails in this paper we propose a strategy to obtain the solutions of a scalar equation f x 0 through solving an associated system of two equations in this way the solutions of the associated system lying on the identity line provide solutions of the given equation in most cases the associated system cannot be solved exactly solving this system with the newton method for systems may result more efficient than the application of the scalar newton method to the simpler equation in some pathological cases in which the scalar newton method does not provide solution this strategy works appropriately some examples are given to illustrate the performance of the proposed strategy

improving the domain of parameters for newton s method with applications we present a new technique to improve the convergence domain for newton s method both in the semilocal and local case it turns out that with the new technique the sufficient convergence conditions for newton s method are weaker the error bounds are tighter and the information on the location of the solution is at least as precise as in earlier studies numerical examples are given showing that our results apply to solve nonlinear equations in cases where the old results cannot apply

high order data sharpening for density estimation it is shown that data sharpening can be used to produce density estimators that enjoy arbitrarily high orders of bias reduction practical advantages of this approach relative to competing methods are demonstrated they include the sheer simplicity of the estimators which makes code for computing them particularly easy to write very good mean squared error performance reduced wiggliness of estimates and greater robustness against undersmoothing

relative efficiencies of kernel and local likelihood density estimators summary local likelihood methods enjoy advantageous properties such as good performance in the presence of edge effects that are rarely found in other approaches to nonparametric density estimation however as we argue in this paper standard kernel methods can have distinct advantages when edge effects are not present we show that whereas the integrated variances of the two methods are virtually identical the integrated squared bias of a conventional kernel estimator is less than that of a local log linear estimator by as much as a factor of 4 moreover the greatest bias improvements offered by kernel methods occur when they are needed most i e when the effect of bias is particularly high similar comparisons can also be made when high degree local log polynomial fits are assessed against high order kernel methods for example although as is well known high degree local polynomial fits offer potentially infinite efficiency gains relative to their kernel competitors the converse is also true indeed the asymptotic value of the integrated squared bias of a local log quadratic estimator can exceed any given constant multiple of that for the competing kernel method in all cases the densities that suffer problems in the context of local log likelihood methods can be chosen to be symmetric either unimodal or bimodal either infinitely or compactly supported and to have arbitrarily many derivatives as functions on the real line they are not pathological however our results reveal quantitative differences between global performances of local log polynomial estimators applied to unimodal or multimodal distributions

asymptotic bias and variance for a general class of varying bandwidth density estimators we consider a general class of varying bandwidth estimators of a probability density function the class includes the abramson estimator transformation kernel density estimator tkde jones transformation kernel density estimator jtkde nearest neighbour type estimator nn jones linton nielsen estimator jln taylor series approximations of tkde ttkde and simpson s formula approximations of tkde stkde each of these estimators needs a pilot estimator starting with an ordinary kernel estimator hat f 1 it is possible to iterate and compute a sequence of estimates hat f 2 hat f t using each estimate as a pilot estimator in the next step the first main result is a formula for the bias order if the bandwidths used in different steps have a common orderh h n the bias of hat f k is of orderh2k m k 1 t herehm is the bias order of the ideal estimator defined by using the unknownf as pilot the second main result is a recursive formula for the leading bias and stochastic terms in an asymptotic expansion of the density estimates ifm it is possible to make hat f t asymptotically equivalent to the ideal estimator

a unified model for solving a system of nonlinear equations in this paper we suggest and analyze a unified model of two step iterative method for solving a system of nonlinear equations we prove that the unified model achieves fourth order convergence several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods from the comparison with some known methods it is observed that the present methods are of practical utility

local likelihood based on kernel censoring by drawing an analogy with likelihood for censored data a local likelihood function is proposed which gives more weight to observations near a region of interest in the sample space resulting methods can be used for assessing local departures from a parametric model and for semiparametric density estimation some theory and three examples is given

a simple bias reduction method for density estimation a new method for bias reduction in nonparametric density estimation is proposed the method is a simple two stage multiplicative bias correction its theoretical properties are investigated and simulations indicate its practical potential the method is easy to compute and to analyse and extends simply to multivariate and other estimation problems

locally parametric nonparametric density estimation this paper develops a nonparametric density estimator with parametric overtones suppose f x is some family of densities indexed by a vector of parameters we define a local kernel smoothed likelihood function which for each x can be used to estimate the best local parametric approximant to the true density this leads to a new density estimator of the form f x o x thus inserting the best local parameter estimate for each new value of x when the bandwidth used is large this amounts to ordinary full likelihood parametric density estimation while for moderate and small bandwidths the method is essentially nonparametric using only local properties of data and the model alternative ways more general than via the local likelihood are also described the methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class properties of this new semiparametric estimator are investigated our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias the new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework the methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression

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

a comparison of higher order bias kernel density estimators abstract we consider many kernel based density estimators all theoretically improving bias from o h 2 as the smoothing parameter h 0 to o h 4 examples include higher order kernels variable kernel methods and transformation and multiplicative bias correction approaches we stress the similarities between what appear to be disparate approaches in particular we show how the mean squared errors of all methods have the same form our main practical contribution is a comparative simulation study that isolates the most promising approaches it remains debatable however as to whether even the best methods give worthwhile improvements at least for small to moderate sample exploratory purposes

discretized and interpolated kernel density estimates abstract the kernel estimate of a probability density function inherits its smoothness properties from the kernel density chosen by the investigator nevertheless for computational and especially graphical reasons exact kernel density estimates are often presented in piecewise constant discretized or piecewise linear interpolated form using the exact estimate only at some grid of points the asymptotic integrated mean squared error properties of these modifications are studied it is seen that discretization may adversely affect the order of magnitude of this risk criterion so care needs to be taken in practice to limit the degree of discretization employed this roughness effect essentially disappears when interpolation is used instead then it takes a remarkably coarse grid to result in more than a negligible deterioration in the estimate s performance data discretization prior to applying the kernel density estimation prescription is also addressed such prebinning does not affect the smoot

local likelihood density estimation local likelihood was introduced by tibshirani and hastie as a method of smoothing by local polynomials in non gaussian regression models in this paper an extension of these methods to density estimation is discussed and comparison with other methods of density estimation presented the local likelihood method has particularly strong advantages over kernel methods when estimating tails of densities and in multivariate settings suppose constraints are incorporated in a simple manner asymptotic properties of the estimate are discussed a method for computing the estimate is outlined c code to implement the estimation procedure described in this paper together with s interfaces for graphical display of results are available at http cm bell labs com stat project locfit index html

boosting kernel density estimates a bias reduction technique summary this paper proposes an algorithm for boosting kernel density estimates we show that boosting is closely linked to a previously proposed method of bias reduction and indicate how it should enjoy similar properties numerical examples and simulations are used to illustrate the findings and we also suggest further areas of research

a new approach based on the newton s method to solve systems of nonlinear equations an alternative strategy for solving systems of nonlinear equations when the classical newton s method fails is presented the proposed strategy is an extension for systems based on the idea presented in the article ramos and vigo aguiar 2015 it relies in obtaining the approximate solutions of a given system of equations through solving an associated system obtained through the theory underlying the newton s method in this way the solutions of the associated system that are not 2 cycles of the newton iteration function provide solutions of the original system as it is usual in most cases the associated system cannot be solved exactly and some iterative procedure must be used for particular starting values solving the associated system with the newton s method results to be more efficient than the application of the newton s method to the original system some examples are given to illustrate the performance of the proposed strategy performance profiles are evaluated in terms of number of iterations error and cpu time

on partial local smoothing rules for curve estimation summary we compare the performances of local and global rules for smoothing parameter choice in terms of asymptotic mean squared errors of the resulting estimators in some instances there is surprisingly little to choose between local and global approaches our analysis identifies contexts where the differences are small or large this work motivates development of smoothing rules that form a half way house between local and global smoothing there interpolation provides a basis for partial local smoothing a key result shows that interpolation on even a coarse grid can produce a very good approximation to full local smoothing our theoretical and numerical results lead us to suggest linear interpolation of a bandwidth obtained by integral approximations on discrete intervals

on bandwidth variation in kernel estimates a square root law we consider kernel estimation of a smooth density f at a point but depart from the usual approach in admitting an adaptive dependence of the sharpness of the kernels on the underlying density proportionally varying the bandwidths like f 1 2 at the contributing readings lowers the bias to a vanishing fraction of the usual value and makes for performance seen in well known estimators that force moment conditions on the kernel and so sacrifice positivity of the curve estimate issues of equivariance and variance stabilitization are treated

monte carlo state space likelihoods by weighted posterior kernel density estimation maximum likelihood estimation and likelihood ratio tests for nonlinear non gaussian state space models require numerical integration for likelihood calculations several methods including monte carlo mc expectation maximization mc likelihood ratios direct mc integration and particle filter likelihoods are inefficient for the motivating problem of stage structured population dynamics models in experimental settings an mc kernel likelihood mckl method is presented that estimates classical likelihoods up to a constant by weighted kernel density estimates of bayesian posteriors mckl is derived by using bayesian posteriors as importance sampling densities for unnormalized kernel smoothing integrals mc error and mode bias due to kernel smoothing are discussed and two methods for reducing mode bias are proposed zooming in on the maximum likelihood parameters using a focused prior based on an initial estimate and using a posterior cumulant based approximation of mode bias a simulated example show

on plug in rules for local smoothing of density estimators optimal local smoothing of a curve estimator requires knowledge of various derivatives of the curve in the neighbourhood of the point at which estimation is being conducted one empirical approach to selecting the amount of smoothing is to employ pilot estimators to approximate those derivatives and substitute the approximate values into an analytical formula for the desired local bandwidth in the present paper we study how bandwidth choice for the pilot estimators affects the performance of the final estimator our conclusions are rather curious depending on circumstance the pilot estimators should be substantially oversmoothed or undersmoothed relative to the amount of smoothing that would be optimal if they were to be employed themselves for point estimation occasionally the optimal amount of undersmoothing is so extreme as to render the pilot estimators inconsistent here the resulting local bandwidth is asymptotically random it is not asymptotic to a sequence of constants

improved variable window kernel estimates of probability densities variable window width kernel density estimators with the width varying proportionally to the square root of the density have been thought to have superior asymptotic properties the rate of convergence has been claimed to be as good as those typical for higher order kernels which makes the variable width estimators more attractive because no adjustment is needed to handle the negativity usually entailed by the latter however in a recent paper terrell and scott show that these results can fail in important cases in this paper we characterize situations where the fast rate is valid and also give rates for a variety of cases where they are slower in addition a modification of the usual variable window width estimator is proposed which does have the earlier claimed rates of convergence

derivative free iterative methods for nonlinear systems in this work we introduce a new operator of divided differences that preserves the convergence order when it is used for approximating the jacobian matrix in iterative method for solving nonlinear systems we obtain derivative free iterative methods with lower computational cost than the corresponding ones with different operators of divided differences we also study the global convergence of these methods by analyzing their dynamical behavior

some iterative methods for solving a system of nonlinear equations in this paper we suggest and analyze two new two step iterative methods for solving the system of nonlinear equations using quadrature formulas we prove that these new methods have cubic convergence several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods these new iterative methods may be viewed as an extension and generalizations of the existing methods for solving the system of nonlinear equations

estimating and depicting the structure of a distribution of random functions summary we suggest a nonparametric approach to making inference about the structure of distributions in a potentially infinite dimensional space for example a function space and displaying information about that structure it is suggested that the simplest way of presenting the structure is through modes and density ascent lines the latter being the projections into the sample space of the curves of steepest ascent up the surface of a functionaldata density modes are always points in the sample space and ascent lines are always one parameter structures even when the sample space is determined by an infinite number of parameters they are therefore relatively easily depicted our methodology is based on a functional form of an iterative data sharpening algorithm