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

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

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

local empirical likelihood inference for varying coefficient density ratio models based on case control data in this article we develop a varying coefficient density ratio model for case control studies the case and control samples come from two different distributions under the model assumption the ratio of the two densities is related to the linear combination of covariates with varying coefficients through a known function a special case is the exponential tilt model where the log ratio of the two densities is a linear function of covariates we propose a local empirical likelihood el approach to estimate the nonparametric coefficient functions under some regularity assumptions the proposed estimators are shown to be consistent and asymptotically normally distributed the sieve empirical likelihood ratio selr test statistic for detecting whether the varying coefficients are really constant and other related hypotheses is constructed and it follows approximately a chi squared distribution we introduce a modified bootstrap procedure to estimate the null distribution of the selr when sample size is s

reducing variance in univariate smoothing a variance reduction technique in nonparametric smoothing is proposed at each point of estimation form a linear combination of a preliminary estimator evaluated at nearby points with the coefficients specified so that the asymptotic bias remains unchanged the nearby points are chosen to maximize the variance reduction we study in detail the case of univariate local linear regression while the new estimator retains many advantages of the local linear estimator it has appealing asymptotic relative efficiencies bandwidth selection rules are available by a simple constant factor adjustment of those for local linear estimation a simulation study indicates that the finite sample relative efficiency often matches the asymptotic relative efficiency for moderate sample sizes this technique is very general and has a wide range of applications

regularity for solutions of nonlinear evolution equations with nonlinear perturbations abstract the approximate controllability for the nonlinear control system with nonlinear monotone hemicontinuous and the nonlinear perturbations is studied the existence uniqueness and a variation of solutions of the system are also given

parametric approaches to quality adjusted survival analysis international breast cancer study group we present a parametric methodology for performing quality of life adjusted survival analysis using multivariate censored survival data it represents a generalization of the nonparametric q twist method quality adjusted time without symptoms and toxicity the event times correspond to transitions between states of health that differ in terms of quality of life each transition is governed by a competing risks model where the health states are the competing risks overall survival is the sum of the amount of time spent in each health state the first step of the proposed methodology consists of defining a quality function that assigns a score to a life having given health state transitions it is a composite measure of both quantity and quality of life in general the quality function assigns a small value to a short life with poor quality and a high value to a long life with good quality in the second step parametric survival models are fit to the data this is done by repeatedly modeling the conditional cause specific hazard functions given the previous transitions covariates are incorporated by accelerated failure time regression and the model parameters are estimated by maximum likelihood lastly the modeling results are used to estimate the expectation of quality functions standard errors and confidence intervals are computed using the bootstrap and delta methods the results are useful for simultaneously evaluating treatments in terms of quantity and quality of life to demonstrate the proposed methods we perform an analysis of data from the international breast cancer study group trial v which compared short duration chemotherapy versus long duration chemotherapy in the treatment of node positive breast cancer the events studied are 1 the end of treatment toxicity 2 disease recurrence and 3 overall survival

a reexamination of diffusion estimators with applications to financial model validation time homogeneous diffusion models have been widely used for describing the stochastic dynamics of the underlying economic variables recently stanton proposed drift and diffusion estimators based on a higher order approximation scheme and kernel regression method he claimed that higher order approximations must outperform lower order approximations and concluded nonlinearity in the instantaneous return function of short term interest rates to examine the impact of higher order approximations we develop general and explicit formulas for the asymptotic behavior of both drift and diffusion estimators we show that these estimators will reduce the numerical approximation errors in asymptotic biases but their asymptotic variances escalate nearly exponentially with the order of approximation simulation studies also confirm our asymptotic results this variance inflation problem arises not only from nonparametric fitting but also from parametric fitting stanton s work also postulates the interesting qu

a survey on piecewise linear models of regulatory dynamical systems recent developments in understanding the various regulatory systems especially the developments in biology and genomics stimulated an interest in modelling such systems hybrid systems originally developed for process control applications provide advances in modelling such systems a particular class of hybrid systems which are relatively simpler to analyze mathematically but still capable of demonstrating the essential features of many non linear dynamical systems is piecewise linear systems implementation of piecewise linear systems for modelling of regulatory dynamical systems requires different considerations depending on the status of the problem in this work we considered three different cases firstly we consider the inferential modelling problem based on the empirical observations and study the discrete piecewise linear system whose inverse problem is solvable under some assumptions secondly we considered the problem of obtaining some complex regulatory systems by tractable piecewise linear formulations and study the qualitative dynamic features of the systems and their piecewise linear models finally we considered boolean delay equations for building abstract models of regulatory systems which might be the simplest models demonstrating the essential qualitative features of our interest underlying adaption learning and memorization

nonlinear evolution equations with nonmonotonic perturbations soit v un espace de banach reflexif densement et continument plonge dans un espace de hilbert reel h on considere l existence des solutions du probleme aux valeurs initiales du dt au g u f 0 t t u 0 u 0 ou a est un operateur monotone de v dans v g v h est une application continue et f 0 t v est une fonction mesurable

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

a martingale estimating equation for a capture recapture experiment in discrete time we use martingale theory and a method of moments technique to derive a class of estimators for the size of a closed population in a capture recapture experiment in discrete time where the capture probabilities on the successive capture occasions are allowed to be different the same capture probability is assumed to apply to all animals in the population on the same occasion explicit expressions are given for these estimators and their associated standard errors which involve only simple computation an optimal estimator for the population size is found asymptotic results are readily obtained by an application of a martingale central limit theorem two examples are given to compare the results

local polynomial kernel regression for generalized linear models and quasi likelihood functions abstract we investigate the extension of the nonparametric regression technique of local polynomial fitting with a kernel weight to generalized linear models and quasi likelihood contexts in the ordinary regression case local polynomial fitting has been seen to have several appealing features in terms of intuitive and mathematical simplicity one noteworthy feature is the better performance near the boundaries compared to the traditional kernel regression estimators these properties are shown to carry over to generalized linear model and quasi likelihood settings we also derive the asymptotic distributions of the proposed class of estimators that allow for straightforward interpretation and extensions of state of the art bandwidth selection methods

the foundations of finite sample estimation in stochastic processes generally in the literature on stochastic processes estimation is investigated in terms of asymptotic properties in this paper we establish some finite sample optimal estimation let y1 y2 be a discrete stochastic process since we will be discussing results for samples of finite size n we restrict the process to r in the following the term parameter is used in the same broad sense as by godambe thompson 1984 let y be a class of probability distributions f on r and 0 0 f f e jz be a real parameter let hi be a real function of y 1 yi and 0 be such that

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

choice of time scale and its effect on significance of predictors in longitudinal studies time to event regression is a frequent tool in biomedical research in clinical trials this time is usually measured from the beginning of the study the same approach is often adopted in the analysis of longitudinal observational studies however in recent years there has appeared literature making a case for the use of the date of birth as a starting point and thus utilize age as the time to event in this paper we explore different types of age scale models and compare them with time on study models in terms of the estimated regression coefficients they produce we consider six proportional hazards regression models that differ in the choice of time scale and in the method of adjusting for the years before the study by considering the estimating equations of these models as well as numerical simulations we conclude that correct adjustment for the age at entry is crucial in reducing bias of the estimated coefficients the unadjusted age scale model is inferior to any of the five other models considered regardless of their choice of time scale additionally if adjustment for age at entry is made our analyses show very little to suggest that there exists any practically meaningful difference in the estimated regression coefficients depending on the choice of time scale these findings are supported by four practical examples from the framingham heart study

finite sample variance of local polynomials analysis and solutions abstract fitting local polynomials in nonparametric regression has a number of advantages the attractive theoretical features are in a partial contradiction to variance properties for random design and to practical experience over a broad range of situations no upper bound can be given for the conditional variance the unconditional variance is infinite when using optimal weights with compact support properties are better for gaussian weights we analyze local polynomials for finite sample size both theoretically and numerically it turns out that difficulties arise in sparse regions in the realization of the design when the realization has locally a small variance and or a skew empirical distribution two small sample modifications of local polynomials are presented local increase of bandwidth in sparse regions of the design and local polynomial ridge regression both modifications combine a good finite sample behavior with the asymptotic advantages of local polynomials

a comparison of the bias in four versions of the product limit estimator summary the bias of the product limit estimator depends entirely on the method used to complete the estimator beyond the last observation when it is censored the bias has been investigated for the upper gill 1980 and lower efron 1967 bounds on these methods rather than using a completion method investigators in practice commonly truncate the estimator using data driven rules the bias under two common truncation rules are considered i truncating at the last observation if it is censored and ii truncating at the n l th order statistic a number of inequalities between these biases are derived and properties of the bias functions are proved this investigation shows

theoretical explanation of observed decreasing failure rate pooled data on the times of successive failures of the air conditioning system of a fleet of jet airplanes seemed to indicate that the life distribution had a decreasing failure rate more refined analysis showed that the failure distribution for each airplane separately was exponential but with a different failure rate using the theorem that a mixture of distributions each having a non increasing failure rate e g a mixture of exponential distributions itself has a non increasing failure rate the apparent decreasing failure rate of the pooled air conditioning life distribution was satisfactorily explained the present study has implications in other areas where an observed decreasing failure rate may well be the result of mixing exponential distributions having different parameters

large sample behaviour of the product limit estimator on the whole line weak convergence results are proved for the product limit estimator on the whole line applications are given to confidence band construction estimation of mean lifetime and to the theory of q functions the results are obtained using stochastic calculus and in probability linear bounds for empirical processes

a partially parametric estimator of survival in the presence of randomly censored data many biological or medical experiments have as their goal to estimate the survival function of a specified population of subjects when the time to the specified event may be censored due to loss to follow up the occurrence of another event that precludes the occurrence of the event of interest or the study being terminated before the event of interest occurs this paper suggests an improvement of the kaplan meier product limit estimator when the censoring mechanism is random the proposed estimator treats the uncensored observations nonparametrically and uses a parametric model only for the censored observations one version of this proposed estimator always has a smaller bias and mean squared error than the product limit estimator an example estimating the survival function of patients enrolled in the ohio state university bone marrow transplant program is presented

what price kaplan meier summary the asymptotic efficiency of the kaplan meier product limit estimator relative to the maximum likelihood estimator of a parametric survival function is examined under a random censoring model

semiparametric regression for clustered data we consider estimation in a semiparametric partially generalised linear model for clustered data using estimating equations a marginal model is assumed where the mean of the outcome variable depends on some covariates parametrically and a cluster level covariate nonparametrically a profile kernel method allowing for working correlation matrices is developed we show that the nonparametric part of the model can be estimated using standard nonparametric methods including smoothing parameter estimation and the parametric part of the model can be estimated in a profile fashion the asymptotic distributions of the parameter estimators are derived and the optimal estimators of both the nonparametric and parametric parts are shown to be obtained when the working correlation matrix equals the actual correlation matrix the asymptotic covariance matrix of the parameter estimator is consistently estimated by the sandwich estimator we show that the semiparametric efficient score takes on a simple form and our profile kernel method is semiparametric efficient the results for the case where the nonparametric part of the model is an observation level covariate are noted to be dramatically different

on the small sample performance of efron s and of gill s version of the product limit estimator under nonproportional hazards various proposals have been made to extend the product limit estimator to survival times beyond the largest observation in case that observation is censored two extreme extensions are examined with respect to bias and mean squared error mse their quality depends considerably on the censoring constellation the mse of one extension appears to be robust against a wide variety of nonproportionalities of the hazard rates of the distributions of lifelength and time to censoring

empirical likelihood ratio test for median and mean residual lifetime prompted by several recent papers on the inference on median and mean residual life time we note that the testing involving the mean or median residual life function in censored survival data can be obtained by an easy application of the general empirical likelihood ratio test this approach has several advantages 1 there is no need to estimate the variance covariance at all which may become prohibitively complicated for other procedures that require the estimation of such 2 when inverting the tests to obtain confidence regions intervals this procedure inherits all the good properties of a likelihood ratio test 3 free software implementation of the test is readily available

adaptively varying coefficient spatiotemporal models we propose an adaptive varying coefficient spatiotemporal model for data that are observed irregularly over space and regularly in time the model is capable of catching possible non linearity both in space and in time and non stationarity in space by allowing the auto regressive coefficients to vary with both spatial location and an unknown index variable we suggest a two step procedure to estimate both the coefficient functions and the index variable which is readily implemented and can be computed even for large spatiotemporal data sets our theoretical results indicate that in the presence of the so called nugget effect the errors in the estimation may be reduced via the spatial smoothing the second step in the estimation procedure proposed the simulation results reinforce this finding as an illustration we apply the methodology to a data set of sea level pressure in the north sea

summarising censored survival data using the mean residual life function the mean residual life function provides a clear and simple summary of the effect of a treatment or a risk factor in units of time avoiding hazard ratios or probability scales which require careful interpretation estimation of the mean residual life is complicated by the upper tail of the survival distribution not being observed as for example patients may still be alive at the end of the follow up period various approaches have been developed to estimate the mean residual life in the presence of such right censoring in this work a novel semi parametric method that combines existing non parametric methods and an extreme value tail model is presented where the limited sample information in the tail prior to study termination is used to estimate the upper tail behaviour this approach will be demonstrated with simulated and real life examples copyright 2015 john wiley sons ltd

maximum likelihood estimation of a survival function with a change point for truncated and interval censored data this paper considers estimation of a survival function when there exists a change point and the survival time of interest is defined as elapsed time between two related events furthermore there exists censoring on observations on the occurrences of both events and truncation on observations on the occurrence of the second event and thus the survival time of interest to obtain the maximum likelihood estimator of a survival function an em algorithm is developed when the survival function is completely unknown before the change point and known up to a vector of unknown parameters after the change point the idea is a generalization of that discussed in moeschberger and klein simulations and an example are used to evaluate and illustrate the algorithm

small sample results for the kaplan meier estimator abstract whereas much is known about the asymptotic properties of the kaplan meier 1958 estimator kme of a survival function exact results for small samples have been difficult to obtain in this article we obtain an exact expression for the th moment 0 of the kme under a model of proportional hazards this enables us under proportional hazards to a study the bias of the kme and b compare the exact variance of the kme to its asymptotic variance

a heavy censoring limit theorem for the product limit estimator a key identity for the product limit estimator due to aalen and johansen 1978 and gill 1980 is shown to be a consequence of the exponential formula of doleans dade 1970 the basic counting processes in the censored data problem are shown to converge jointly to poisson processes under heavy censoring g n rightarrow d delta 0 but n 1 g n rightarrow alpha where g n is the censoring distribution the poisson limit theorem for counting processes implies poisson type limit theorems under heavy censoring for the cumulative hazard function estimator and product limit estimator the latter in combination with the key identity of aalen johansen and gill and martingale properties of the limit processes yields a new approximate variance formula for the product limit estimator which is compared numerically with recent finite sample calculations for the case of proportional hazard censoring due to chen hollander and langberg 1982

kkm fan s lemma for solving nonlinear vector evolution equations with nonmonotonic perturbations abstract in this paper we are interested in the existence of solutions of the following initial value problem du dt a u g u f on 0 t with u 0 u 0 where a v v is a monotone operator g v v is a nonlinear nonmonotone operator and f 0 t v is a measurable function by means of a recent generalization of the famous kkm fan s lemma

estimation of the size of an open population from capture recapture data using weighted martingale methods summary a weighted martingale method akin to a moving average is proposed to allow the use of modified closed population methods in the estimation of the size of a smoothly changing open population when there are frequent capture occasions we concentrate here on modifications to martingale estimating functions for model mt but a wide range of closed population estimators may be modified in this fashion the method is motivated by and applied to weekly capture recapture data from the mai po bird sanctuary in hong kong simulations show that the weighted martingale estimator compared well with the jolly seber estimator when the conditions for the latter to be valid are met and it performed far better when individuals were allowed to leave and reenter the population expressions are derived for the asymptotic bias and variance of the estimator in an appendix

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

local linear regression on manifolds and its geometric interpretation high dimensional data analysis has been an active area and the main focus areas have been variable selection and dimension reduction in practice it occurs often that the variables are located on an unknown lower dimensional nonlinear manifold under this manifold assumption one purpose of this article is regression and gradient estimation on the manifold and another is developing a new tool for manifold learning as regards the first aim we suggest directly reducing the dimensionality to the intrinsic dimension d of the manifold and performing the popular local linear regression llr on a tangent plane estimate an immediate consequence is a dramatic reduction in the computational time when the ambient space dimension p d we provide rigorous theoretical justification of the convergence of the proposed regression and gradient estimators by carefully analyzing the curvature boundary and nonuniform sampling effects we propose a bandwidth selector that can handle heteroscedastic errors with re

regression smoothing parameters that are not far from their optimum abstract it is well known that data driven regression smoothing parameters based on cross validation and related methods exhibit a slow rate of convergence to their optimum in an earlier article we showed that this rate can be as slow as n 1 10 that is for a bandwidth 0 optimizing the averaged squared error n 1 10 0 0 tends to an asymptotic normal distribution in this article we consider mean averaged squared error optimal bandwidths h 0 this nonrandom smoothing parameter can be approximated much faster we use the technique of double smoothing to show that there is an such that under certain conditions n 1 2 h 0 h 0 tends to an asymptotic normal distribution

adaptive varying coefficient linear models varying coefficient linear models arise from multivariate nonparametric regression nonlinear time series modelling and forecasting functional data analysis longitudinal data analysis and others it has been a common practice to assume that the vary coefficients are functions of a given variable which is often called an index a frequently asked question is which variable should be used as the index in this paper we explore the class of the varying coefficient linear models in which the index is unknown and is estimated as a linear combination of regression and or other variables this will enlarge the modelling capacity substantially we search for the index such that the derived varying coefficient model provides the best approximation to the underlying unknown multi dimensional regression function in the least square sense the search is implemented through the newly proposed hybrid backfitting algorithm the core of the algorithm is the alternative iteration between estimating the index through a one step scheme and estimating coefficient functions through a one dimensional local linear smoothing the generalised cross validation method for choosing bandwidth is efficiently incorporated into the algorithm the locally significant variables are selected in terms of the combined use of t statistic and akaike information criterion we further extend the algorithm for the models with two indices simulation shows that the proposed methodology has appreciable flexibility to model complex multivariate nonlinear structure and is practically feasible with average modern computers the methods are further illustrated through the canadian mink muskrat data in 1925 1994 and the pound dollar exchange rates in 1974 1983

projection as a method for increasing sensitivity and eliminating nuisance parameters l article analyse des methodes statistiques basees sur les fonctions d estimation qui sont des fonctions de parametres et de donnees a la fois destinees a permettre une inference sur un parametre inconnu une methode de projection peut etre utilisee pour la reduction de ces fonctions afin d accroitre la puissance d un test et l efficacite d un estimateur ponctuel ou bien de rendre une fonction d inference insensible a la valeur d un parametre de nuisance

a crossvalidation method for estimating conditional densities summary we extend the idea of crossvalidation to choose the smoothing parameters of the double kernel local linear regression for estimating a conditional density our selection rule optimises the estimated conditional density function by minimising the integrated squared error we also discuss three other bandwidth selection rules an ad hoc method used by fan et al 1996 a bootstrap method of hall et al 1999 for bandwidth selection in the estimation of conditional distribution functions modified by bashtannyk hyndman 2001 to cover conditional density functions and finally a simple approach proposed by hyndman yao 2002 the performance of the new approach is compared with these three methods by simulation studies and our method performs outstandingly well the method is illustrated by an application to estimating the transition density and the value at risk of treasury bill data

langevin equation involving three fractional orders in this paper the existence and uniqueness of initial value problems for nonlinear langevin equation involving three fractional orders are discussed we use a new norm that is convenient for the fractional and singular differential equations this norm and the contraction mapping principles are the main tools for investigating the existence and uniqueness of the desired issue the fractional derivatives are described in caputo sense

the kernel estimate of a regression function in likelihood based models abstract smoothing splines have a penalized likelihood motivation good and gaskins 1971 allowing direct application to nonparametric regression in likelihood based models the notion of a weighted likelihood for the nonparametric kernel estimation of a regression function is proposed generalizing the local likelihood theory of tibshirani and hastie 1987 let the data be of the form xi yi i 1 n where xi 0 1 d are lattice points and the yi are independent random variables from a family of distributions with parameter i g xi with g having continuous partial derivatives of order k 2 the goal is to arrive at a nonparametric estimate o of o g xo for a fixed point xo 0 1 d we consider the estimator o that maximizes the weighted likelihood function w n i 1 w xo xi b log f yi with f the density of yi w a symmetric kernel with compact support and b the bandwidth that controls the degree of smoothing sufficient conditions for consistency and asympt

solutions of nonlinear evolution inclusions abstract in this paper the solution of the nonlinear evolution inclusion problem of the form u t b t u t f t is studied in this problem the operators are of type m or type s which are different from those of pseudo monotone operators that had been studied by many authors at the same time we study the perturbation problem in fact many kinds of evolution equations can be generalized by this problem the former results are improved and generalized by our conclusions and we will give more applications

low order approximations in deconvolution and regression with errors in variables summary we suggest two new methods which are applicable to both deconvolution and regression with errors in explanatory variables for nonparametric inference the two approaches involve kernel or orthogonal series methods they are based on defining a low order approximation to the problem at hand and proceed by constructing relatively accurate estimators of that quantity rather than attempting to estimate the true target functions consistently of course both techniques could be employed to construct consistent estimators but in many contexts of importance e g those where the errors are gaussian consistency is from a practical viewpoint an unattainable goal we rephrase the problem in a form where an explicit interpretable low order approximation is available the information that we require about the error distribution the error in variables distribution in the case of regression is only in the form of low order moments and so is readily obtainable by a rudimentary analysis of indirect measurements of errors e g through repeated measurements in particular we do not need to estimate a function such as a characteristic function which expresses detailed properties of the error distribution this feature of our methods coupled with the fact that all our estimators are explicitly defined in terms of readily computable averages means that the methods are particularly economical in computing time

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

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

estimation of the size of an open population using local estimating equations ii a partially parametric approach des methodes de lissage par noyau sont utilisees pour etendre une modification de l approche de lloyd et yip 1991 developpee pour des populations fermees a des populations ouvertes pour lesquelles les occasions de captures sont frequentes cette approche complete les methodes non parametriques precedentes et les simulations montrent que lorsque les hypotheses parametriques sont confirmees elle fournit une erreur quadratique moyenne integree plus petite que la methode non parametrique complete cette methode est appliquee a des donnees annuelles de capture recapture de puffin a bec grele sur une periode de 48 ans

estimating survival curves with left truncated and interval censored data under monotone hazards we show that the nonparametric maximum likelihood estimator npmle of a survival function may severely underestimate the survival probabilities at very early times for left truncated and interval censored data as an alternative we propose to compute the nonparametric mle under a nondecreasing hazard assumption the monotone mle by a gradient projection algorithm when the assumption holds the projection step is accomplished via an isotonic regression algorithm the pool adjacent violators algorithm this gradient projection algorithm is computationally efficient and converges globally monte carlo simulations show superior performance of the monotone mle over that of the npmle in terms of either bias or variance even for large samples the methodology is illustrated with the application to the wisconsin epidemiological study of diabetic retinopathy data to estimate the probability of incidence of retinopathy

existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators abstract in this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a banach space with locally monotone operators which is a generalization of the classical result for monotone operators in particular we show that local monotonicity implies pseudo monotonicity the main results are applied to pde of various types such as porous medium equations reaction diffusion equations the generalized burgers equation the navier stokes equation the 3d leray model and the p laplace equation with non monotone perturbations

adaptive bandwidth choice for kernel regression abstract a data based procedure is introduced for local bandwidth selection for kernel estimation of a regression function at a point the estimated bandwidth is shown to be consistent and asymptotically normal as an estimator of the asymptotic optimal value for minimum mean square estimation simulation studies indicate satisfactory behavior of the new bandwidth estimator in finite samples the findings are improvements over a global bandwidth estimator the same methodology works for local linear regression and extends easily to weighted local polynomial fits

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

semiparametric regression in likelihood based models abstract a weighted likelihood is used to estimate the parameters in a semiparametric model involving two covariates and allowing an association between the covariates the development is for arbitrary but specified densities of the observations the estimators are consistent and asymptotically normal hypothesis testing of the parametric component can be performed using a wald test simulations and analysis of data with bernoulli observations demonstrate the estimators application speckman developed kernel estimators where the conditional density of the observations is normal with p parametric covariates speckman s estimators and the new estimators are asymptotically equivalent with the bias of speckman s estimators being smaller as an example we study the relationship between a binary response indicating the occurrence of an intraoperative cardiac complication icc in vascular surgery patients and two risk factors duration of the operation or and asa score which is an evaluation of the patien

efficient local estimation for time varying coefficients in deterministic dynamic models with applications to hiv 1 dynamics recently deterministic dynamic models have become very popular in biomedical research and other scientific areas examples include modeling human immunodeficiency virus hiv dynamics pharmacokinetic pharmacodynamic analysis tumor cell kinetics and genetic network modeling in this article we propose estimation methods for the time varying coefficients in deterministic dynamic systems that are usually described by a set of differential equations three two stage local polynomial estimators are proposed and their asymptotic normality is established an alternative approach a discretization method that is widely used in stochastic diffusion models is also investigated we show that the discretization method that uses the simple euler discretization approach for the deterministic dynamic model does not achieve the optimal convergence rate compared with the proposed two stage estimators we use monte carlo simulations to study the finite sample performance and use a real data application to hiv dynamic

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

efficiency of martingale methods in recapture studies summary the asymptotic variances of various estimators of the population size from a recapture study are calculated the estimation methods compared are maximum likelihood and a class of natural martingale estimators with both time variation and behavioural response it is found that i the martingale methods appropriate for time variation or behavioural response are fully efficient ii there is no loss of efficiency at all in allowing for time variation iii there is large loss of efficiency in allowing for behavioural response iv there is a natural martingale method that allows for both time variation and behavioural response

statistical inference procedure for a hypergeometric model for capture recapture experiments abstract we use martingale theory and a method of moments technique to derive a class of estimators for a multihypergeometric model for a capture recapture experiment certain choices of weight function lead to some standard estimators explicit expression for the estimators and their associated standard errors of the parameter population size is obtained optimal estimator among the class of estimators is derived which gives the tightest confidence interval for the population size three examples are given a simulation study is given to assess the performance of the proposed estimators with other existing ones

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

a one dimensional heat equation with mixed boundary conditions abstract we study a nonlinear one dimensional heat equation with nonmonotone perturbation and with mixed boundary conditions that can even be discontinuous we show that we can balance these two main difficulties in order to obtain existence of globally defined strong solutions for this class of problems the main tools used are the theory of monotone operators and the galerkin method

extremes and local dependence in stationary sequences extensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction a a weak mixing condition restricting long range dependence b a local condition restricting the clustering of high level exceedances

kriging with nonparametric variance function estimation a method for fitting regression models to data that exhibit spatial correlation and heteroskedasticity is proposed it is well known that ignoring a nonconstant variance does not bias least squares estimates of regression parameters thus data analysts are easily lead to the false belief that moderate heteroskedasticity can generally be ignored unfortunately ignoring nonconstant variance when fitting variograms can seriously bias estimated correlation functions by modeling heteroskedasticity and standardizing by estimated standard deviations our approach eliminates this bias in the correlations a combination of parametric and nonparametric regression techniques is used to iteratively estimate the various components of the model the approach is demonstrated on a large data set of predicted nitrogen runoff from agricultural lands in the midwest and northern plains regions of the u s a for this data set the model comprises three main components 1 the mean function which includes farming practice variables local soil and climate characteristics and the nitrogen application treatment is assumed to be linear in the parameters and is fitted by generalized least squares 2 the variance function which contains a local and a spatial component whose shapes are left unspecified is estimated by local linear regression and 3 the spatial correlation function is estimated by fitting a parametric variogram model to the standardized residuals with the standardization adjusting the variogram for the presence of heteroskedasticity the fitting of these three components is iterated until convergence the model provides an improved fit to the data compared with a previous model that ignored the heteroskedasticity and the spatial correlation

semiparametric estimation of animal abundance using capture recapture data from open populations a semiparametric partially linear model for the size of an open population is proposed and inference is conducted using weighted martingale estimating equations this extends a previous nonparametric approach to modeling capture recapture data for open populations with frequent capture occasions analytic expressions for the large sample variances are derived and these are confirmed in a simulation study the method is illustrated on monthly penguin banding data collected over 6 years

on the optimal control of strongly nonlinear evolution equations abstract in this paper we study optimal control problems governed by nonlinear evolution equations with nonmonotone nonlinearities in the state first we prove an existence theorem then using the penalty method we obtain necessary conditions for optimality from them we derive a bang bang characterization of optimal controls finally an example of a nonlinear parabolic optimal control system is worked out in detail

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

local estimating equations abstract estimating equations have found wide popularity recently in parametric problems yielding consistent estimators with asymptotically valid inferences obtained via the sandwich formula motivated by a problem in nutritional epidemiology we use estimating equations to derive nonparametric estimators of a parameter depending on a predictor the nonparametric component is estimated via local polynomials with loess or kernel weighting asymptotic theory is derived for the latter in keeping with the estimating equation paradigm variances of the nonparametric function estimate are estimated using the sandwich method in an automatic fashion without the need typical in the literature to derive asymptotic formulas and plug in an estimate of a density function the same philosophy is used in estimating the bias of the nonparametric function that is an empirical method is used without deriving asymptotic theory on a case by case basis the methods are applied to a series of examples the applicatio

a comparison of several methods of estimating the survival function when there is extreme right censoring summary when there is extreme censoring on the right the kaplan meier product limit estimator is known to be a biased estimator of the survival function several modifications of the kaplan meier estimator are examined and compared with respect to bias and mean squared error

empirical estimation of a distribution function with truncated and doubly interval censored data and its application to aids studies in this paper we discuss the non parametric estimation of a distribution function based on incomplete data for which the measurement origin of a survival time or the date of enrollment in a study is known only to belong to an interval also the survival time of interest itself is observed from a truncated distribution and is known only to lie in an interval to estimate the distribution function a simple self consistency algorithm a generalization of turnbull s 1976 journal of the royal statistical association series b 38 290 295 self consistency algorithm is proposed this method is then used to analyze two aids cohort studies for which direct use of the em algorithm dempster laird and rubin 1976 journal of the royal statistical association series b 39 1 38 which is computationally complicated has previously been the usual method of the analysis