generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

smoothing splines regression derivatives and deconvolution abstract the statistical properties of a cubic smoothing spline and its derivative are analyzed it is shown that unless unnatural boundary conditions hold the integrated squared bias is dominated by local effects near the boundary similar effects are shown to occur in the regularized solution of a translation kernel integral equation these results are derived by developing a fourier representation for a smoothing spline author

sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities the goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and in self similar variables the optimal convergence rates to barenblatt self similar profiles and their generalizations it relies on the identification of the optimal constants in some related hardy poincare inequalities and concludes a long series of papers devoted to generalized entropies functional inequalities and rates for nonlinear diffusion equations

rates of decay to equilibria for p laplacian type equations abstract the long time asymptotics for p laplacian type equations t p m div m p 2 m in r n is studied for p 1 and m n p 1 n p 1 the non negative solutions of the equations are shown to behave asymptotically as t like barenblatt type solutions and the explicit rates of decay are established for the convergence of the relative energy the convergence with respect to the wasserstein distances and the convergence with respect to the l 1 norm the rates are proved to be optimal for p 2 the method used is based on mass transportation inequalities

fast diffusion to self similarity complete spectrum long time asymptotics and numerology the complete spectrum is determined for the operator open image in new window on the sobolev space w1 2 rn formed by closing the smooth functions of compact support with respect to the norm open image in new window here the barenblatt profile is the stationary attractor of the rescaled diffusion equation open image in new window in the fast supercritical regime mopen image in new window the same diffusion dynamics represent the steepest descent down an entropy e u on probability measures with respect to the wasserstein distance d2 formally the operator h hess e is the hessian of this entropy at its minimum so the spectral gap h 2 n 1 m found below suggests the sharp rate of asymptotic convergence open image in new window from any centered initial data 0 u 0 x l1 rn with second moments this bound improves various results in the literature and suggests the conjecture that the self similar solution u t x r t n x r t is always slowest to converge the higher eigenfunctions which are polynomials with hypergeometric radial parts and the presence of continuous spectrum yield additional insight into the relations between symmetries of rn and the flow thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass preserving dilation or still better affine image the strange numerology of the spectrum is explained in terms of the number of moments of

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

bias robust estimation in finite populations using nonparametric calibration a standard problem in sample survey inference is that of predicting the finite population total h of a function h y of a random variable y the model based approach to this problem first defines a working model for y and then predicts h by estimating its expectation under conditional on the sample values of y this approach leads to biased predictions if is incorrect we explore an automatic solution to this misspecification bias that uses nonparametric regression to define a robust but inefficient predictor of h and then calibrates this predictor for its bias under an application to prediction of the finite population distribution function of a population of australian beef farms is presented

application of wall functions to generalized nonorthogonal curvilinear coordinate systems a method has been developed for the application of wall functions to generalized curvilinear coordinate systems with nonorthogonal grids two test cases have been computed using this method with the k e turbulence model flow over a flat plate at 0 deg angle of attack using a nonorthogonal grid at the wall and flow over a prolate hemispheroid with a hemispherical nose cap at 0 deg angle of attack all results are compared with experimental data in addition the hemispheroid results are compared with computations using the baldwin lomax algebraic turbulence model and the chien low reynolds number k e turbulence model

phase transitions and symmetry breaking in singular diffusion the long time asymptotics are determined for fast nonlinear diffusion by linearizing otto s gradient descent model at the barenblatt profile the spectrum of the entropy is explicitly determined the dynamics are found to undergo a phase transition in which rotational symmetry is broken as the strength of the nonlinearity is varied

a goodness of fit test for binary regression models based on smoothing methods a new global test statistic for models with continuous covariates and binary response is introduced the test statistic is based on nonparametric kernel methods explicit expressions are given for the mean and variance of the test statistic asymptotic properties are considered and approximate corrections due to parameter estimation are presented properties of the test statistic are studied by simulation the goodness of fit method is illustrated on data from a dutch follow up study on preterm infants recommendations for practitioners are given

a lyapunov functional for the evolution of solutions to the porous medium equation to self similarity ii this article continues the analysis of the preceding article showing that the lyapunov functional introduced by newman can be used to prove the stability of the barenblatt pattle solutions of the porous medium equation

comparison of model misspecification diagnostics using residuals from least mean of squares and least median of squares fits abstract this article explores model misspecification diagnostics based on least squares and least median of squares fits it shows that in some circumstances least median of squares methods or any other estimator with the exact fit property fail to reveal an incorrectly specified mean function but least squares methods succeed

estimation of a projection pursuit type regression model since the pioneering work of friedman and stuetzle in 1981 projection pursuit algorithms have attracted increasing attention this is mainly due to their potential for overcoming or reducing difficulties arising in nonparametric regression models associated with the so called curse of dimensionality that is the amount of data required to avoid an unacceptably large variance increasing rapidly with dimensionality subsequent work has however uncovered a dependence on dimensionality for projection pursuit regression models here we propose a projection pursuit type estimation scheme with two additional constraints imposed for which the rate of convergence of the estimator is shown to be independent of the dimensionality let mathbf x y be a random vector such that mathbf x x 1 ldots x d t ranges over r d the conditional mean of y given mathbf x mathbf x is assumed to be the sum of no more than d general smooth functions of beta t i mathbf x where beta i in s d 1 the unit sphere in r d centered at the origin a least squares polynomial spline and the final prediction error criterion are used to fit the model to a random sample of size n from the distribution of mathbf x y under appropriate conditions the rate of convergence of the proposed estimator is independent of d

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

correlation curves as local measures of variance explained by regression abstract we call a model for an experiment heterocorrelatious if the strength of the relationship between a response variable y and a covariate x is different in different regions of the covariate space for such experiments we introduce a correlation curve that measures heter ocorrelaticity in terms of the variance explained by regression locally at each covariate value more precisely the squared correlation curve is obtained by first expressing the usual linear model variance explained to total variance formula in terms of the residual variance and the regression slope and then replacing these by the conditional residual variance depending on x and the slope of the conditional mean of y given x x the correlation curve x satisfies the invariance properties of correlation it reduces to the galton pearson correlation in linear models it is between 1 and 1 it is 0 when x and y are independent and it is 1 when y is a function of x we introduce estimates of the correlation curve based on

computation of rotating turbulent flow with an algebraic reynolds stress model la rotation a un effet important sur la structure des ecoulements turbulents les modeles de turbulence bases sur une viscosite tourbillonnaire isotrope ne permettent pas de prevoir de tels ecoulements on propose un modele de contrainte de reynolds algebrique pour modifier la relation de viscosite tourbillonnaire de kolmogorov prandtl de maniere a produire un modele de turbulence anisotrope

information geometry of q gaussian densities and behaviors of solutions to related diffusion equations this paper presents new geometric aspects of the behaviors of solutions to the porous medium equation pme and its associated equation first we discuss thermostatistical structure with information geometry on a manifold of generalized exponential densities a dualistic relation between the two existing formalisms is elucidated next by equipping the manifold of q gaussian densities with such a structure we derive several physically and geometrically interesting properties of the solutions the manifold is proved invariant and attracting for the evolving solutions which play crucial roles in our analysis we demonstrate that the moment conserving projection of a solution coincides with a geodesic curve on the manifold further the evolutional velocities of the second moments and the convergence rate to the manifold are evaluated in terms of the bregman divergence finally we show that the self similar solution is geometrically special in the sense that it is simultaneously geodesic with respect to the mutually dual two affine connections

convex and concave surface curvature effects in wall bounded turbulent flows a fully ellipitic numerical method developed to solve the two dimensional reynolds averaged navier stokes equations for an incompressible fluid is employed in a computational study of several flows involving convex and concave surface curvature the k epsilon turbulence model and variant of it proposed to improve the calculation of flows with streamline curvature are used in conjunction with a one equation near wall model the performance of the method is examined through computations of fully developed curved duct flow and boundary layers on curved surfaces in each case the main features of the flow are consistently reproduced however the computations tend to overpredict the convex wall skin friction and underpredict the concave wall skin friction

nonparametric regression analysis of growth curves in recent years nonparametric curve estimates have been extensively explored in theoretical work there has however been a certain lack of convincing applications in particular involving comparisons with parametric techniques the present investigation deals with the analysis of human height growth where longitudinal measurements were collected for a sample of boys and a sample of girls evidence is presented that kernel estimates of acceleration and velocity of height and of height itself might offer advantages over a parametric fitting via functional models recently introduced for the specific problem considered both approaches are biased but the parametric one shows qualitative and quantitative distortion which both are not easily predictable data analytic problems involved with kernel estimation concern the choice of kernels the choice of the smoothing parameter and also whether the smoothing parameter should be chosen uniformly for all subjects or individually

comparison of information and variance maximization strategies for characterizing neural feature selectivity this paper compares several statistical methods for analyzing neural feature selectivity with natural stimuli despite the non gaussian character of correlations in natural stimuli several relevant stimulus dimensions can be found by maximizing either information or as is demonstrated here variance in the case of information the relevance of each dimension is quantified by a kullback leibler divergence between the full input probability distribution and that across inputs associated with positive neural responses both projected onto that dimension we demonstrate that least square matching of the nonlinear prediction based on several dimensions relevant to the recorded spike trains yields an optimization scheme similar to information maximization the relevant dimensions are found as those that capture the most variance in neural response the variance along a stimulus dimension is given by a renyi divergence of order 2 instead of the kullback leibler divergence used for maximizing information statistical errors expected for the two schemes are shown to be similar through both analytical and numerical calculations however in the asymptotic limit of large spike numbers maximizing information results in smaller errors than variance optimization numerical simulations for model cells with different noise levels show that this trend persists and possibly increases when the number of spikes decreases this makes the problem of finding relevant dimensions one of the examples where information theoretic approaches are no more data limited than the variance based measures variance and information optimization also outperform methods based on the spike triggered average for all numbers of spikes and neural noise levels

generalized varying coefficient models with unknown link function summary we propose a new estimation method for generalized varying coefficient models where the link function is specified up to some smoothness conditions consistency and asymptotic normality of the estimated varying coefficient functions are established simulation results and a real data application demonstrate the usefulness of the new method

binomial regression with monotone splines a psychometric application abstract a binomial regression function p x models the probability of rj successes in nj trials as a function of the values of an observed covariate xj and or a latent variable j j 1 j this article explores the use of monotone regression splines to define p and applies them to the representation of test items as functions of examinee ability some illustrative data suggest that the flexibility of monotone splines permits the detection of item characteristics not observable using logistic based or log linear approaches a simulation study indicates that estimates of both item characteristic curves and ability are reasonably precise for numbers of items and examinees typical of large university lectures given a set of such binomial regression functions it can be useful to study the principal components of functional variation the extension of multivariate principal components analysis to permit the analysis of many item characteristic curves is described

prediction of curved channel flow with an extended k epsilon model of turbulence a l aide d une approximation algebrique pour les equations de contraintes de reynolds on calcule une expression plus generale pour le coefficient c de l equation de viscosite turbulente qui rend compte simultanement des effets de courbure de ligne de courant et de la deformation due a la pression dans l ecoulement le dernier incluant les effets d amortissement a la paroi

best matching barenblatt profiles are delayed the growth of the second moments of the solutions of fast diffusion equations is asymptotically governed by the behavior of self similar solutions however at next order there is a correction term which amounts to a delay depending on the nonlinearity and on a distance of the initial data to the set of self similar barenblatt solutions this distance can be measured in terms of a relative entropy to the best matching barenblatt profile this best matching barenblatt function determines a scale in new variables based on this scale which are given by a self similar change of variables if and only if the initial datum is one of the barenblatt profiles the typical scale is monotone and has a limit coming back to original variables the best matching barenblatt profile is delayed compared to the self similar solution with same initial second moment as the initial datum such a delay is a new phenomenon which has to be taken into account for instance when fitting experimental data

generalized linear models a new program for depression is instituted in the hopes of reducing the number of visits each patient makes to the emergency room in the year following treatment predictors include among many others treatment yes no race and drug and alcohol usage indices a common and minimally invasive treatment for jaundice in newborns is exposure to light yet the cost of this is high mainly because of longer hospital stays which are expensive predictors of the cost include race gestational age and birthweight

some diagnostics for binary logistic regression via smoothing summary the paper discusses some diagnostic tools for binary logistic regression which use smoothing techniques smoothed binary data are employed to devise a battery of diagnostic plots from simple procedures which indicate only the adequacy or inadequacy of a maximum likelihood logistic fit to partial residual plots and delta local deviance plots which reveal the type of model inadequacy suggesting perhaps the addition of nonlinear terms or interactions properties of the methods are studied by simulation the diagnostic tools are illustrated on a set of data in which the probability of having telephone service is related to the demographic variables income and education

optimal smoothing in single index models single index models generalize linear regression they have applications to a variety of fields such as discrete choice analysis in econometrics and dose response models in biometrics where high dimensional regression models are often employed single index models are similar to the first step of projection pursuit regression a dimension reduction method in both cases the orientation vector can be estimated root n consistently even if the unknown univariate function or nonparametric link function is assumed to come from a large smoothness class however as we show in the present paper the similarities end there in particular the amount of smoothing necessary for root n consistent orientation estimation is very different in the two cases we suggest a simple empirical rule for selecting the bandwidth appropriate to single index models this rule is studied in a small simulation study and an application in binary response models

on the use of nonparametric regression for model checking summary the use of nonparametric regression is explored to check the fit of a parametric regression model the principal aim is to check the validity of the regression curve rather than necessarily to detect outliers a pseudo likelihood ratio test is developed to provide a global assessment of fit and simulation bands are used to indicate the nature of departures from the model the types of data considered include discrete response variables where standard diagnostic techniques are often not appropriate and first order autoregressive series several numerical examples are given nonparametric regression can be used in an informal graphical way to assess the relationship between a response and an explanatory variable in this paper we aim to develop more formal methods of assessing the assumptions of a parametric model in particular when regression diagnostics of the type developed for normal linear models are not readily available the principal aim is to check the validity of the systematic part of the model by comparing a nonparametric estimate of the regression curve with a parametric one such a comparison may also identify outliers although the distinction between outliers and model inadequacy is not always easy two techniques are used to assess the fit of a parametric model in 2 confidence bands are constructed around the fitted regression curve by simulation a comparison of these with the nonparametric curve gives an indication of the nature of any departures from the model in 3 a pseudo likelihood ratio test is developed this provides a quantitative global assessment of fit in applying these ideas special emphasis is given to discrete data and notably logistic regression because of the difficulty in applying standard residual based model checking techniques to this type of response variable a poisson regression example is discussed in 4 however the underlying ideas have wider applications autoregressive time series of order 1 are discussed in 6 sections 5 and 7 discuss general issues we first discuss the context of binary regression with a single covariate and the difficulties caused by the discreteness of the response variable the observed data are assumed to be of the form xi yi ni where xi is a covariate value and yi has a binomial

penalized spline estimation for partially linear single index models single index models are potentially important tools for multivariate nonparametric regression they generalize linear regression by replacing the linear combination 0tx with a nonparametric component 0 0tx where 0 is an unknown univariate link function by reducing the dimensionality from that of a general covariate vector x to a univariate index 0tx single index models avoid the so called curse of dimensionality we propose penalized spline p spline estimation of 0 in partially linear single index models where the mean function has the form 0 0tx 0tz the p spline approach offers a number of advantages over other fitting methods for single index models all parameters in the p spline single index model can be estimated simultaneously by penalized nonlinear least squares as a direct least squares fitting method our approach is rapid and computationally stable standard nonlinear least squares software can be used moreover joint inference for 0 0 and 0 is possible by

a lyapunov functional for the evolution of solutions to the porous medium equation to self similarity i we employ the transformation properties and the integral invariants of the porous medium equation in m dimensions to construct a lyapunov functional which characterizes the evolution of the profile to its self similar solution

estimation in generalised varying coefficient models with unspecified link functions abstract in this paper we study the generalised varying coefficient models where the link function is unspecified and the response variable can be either continuous or discrete as the link function is unspecified the models under investigation become unidentifiable in this paper we derive an identification condition for the generalised varying coefficient models which is much weaker and more reasonable than that given by kuruwita et al 2011 whose model can be seen as a special case of our modelling framework under the identification condition we introduce a nonparametric iterative procedure to estimate the functional coefficient with its direction and norm as well as the unspecified link function and then establish the asymptotic properties of the resulting nonparametric estimators furthermore a weighted least squares based algorithm is provided to implement the iterative estimation procedure the simulation studies and empirical application show that our estimation methodology works quite well in both small and median sample cases

limitations of the near wall k epsilon turbulence model comparisons are given at two different reynolds numbers between the measured turbulent kinetic energy in channel flow and the predictions of several near wall variants of the k e closure included among these are new calculations based on a form of the e equation that is consistent with the physically correct boundary condition it is observed that all of the approaches fail to account for the large peak k value in the wall region that is evident in the experimental data by contrasting the computed energy budget with its measured values it is shown that this defect may be attributed to a fundamental inconsistency in the commonly used model for the pressure diffu sion term in the k equation near the boundary he k e closure is widely utilized today in the prediction of the mean properties of turbulent flowfields in recent years extensions of its basic form have been developed to permit treatment of complex fluid motions such as those found in curved pipes 1 compressible boundary layers 2 com bustion chambers 3 etc a strong incentive for the continued development and application of the k e model has been the remarkable success it has achieved in the prediction of mean velocity fields in wall bounded shear flows such as occur in channels pipes and boundary layers 4 6 the success of the k e closure in these relatively simple flows is not complete however since the high accuracy obtained for mean velocity is not duplicated in the case of the turbulent kinetic energy k a notable defect4 7 9 is the 25 or more underprediction of the large peak k value shown by clark 10 and kreplin and eckelmann11 to occur at y 15 in a channel flow for ap plications where predictions of turbulence levels are impor tant such as in internal combustion engine simulations 12 this failure can be of great practical significance it may be expected that major errors in the prediction of k stem from deficiencies in the low turbulent reynolds number forms of the k e equations which have been developed for the region adjacent to boundaries jones and launder4 for mulated the first such model that permitted k and e to be calculated down to a solid wall without the use of wall func tions several more recent studies including those of lam and bremhorst 7 hassid and poreh 8 and chien 9 have at tempted to improve upon the jones and launder approach but these authors report only modest gains in the accuracy of k the object of the present study is to elucidate the cause of this discrepancy by making direct comparisons between the terms in the exact and modeled energy equations to ac complish this a set of new calculations was performed in which e was required to satisfy the physically correct bound ary condition at the wall it was revealed that the failure of the k e model to predict the correct peak k value is most likely due to reliance on a flawed model for the pressure velocity correlation in the k equation this relation will be shown to be inconsistent with the experimentally measured values of the exact term as found by kreplin and eckelmann 11 and may be directly implicated in suppressing the peak k value removal of this limitation should allow for better predictions of k near solid boundaries

quasi likelihood regression with unknown link and variance functions abstract we consider the multiple regression model e y g x t var y sgrave 2 with predictors x link function g and variance function sgrave 2 the aim is to reduce the assumptions in a fully parametric generalized linear model or a quasi likelihood model by allowing the link and the variance functions to be unknown but smooth these functions are then estimated nonparametrically and the estimates are substituted into the quasi likelihood function we propose a three stage approach to identify this semiparametric model by estimating the link function the variance function and the vector of regression coefficients in the linear predictor of the model consistency results for the link and the variance function estimators as well as the asymptotic limiting distribution of the regression coefficients are obtained we show that the resulting parameter estimates are asymptotically efficient as compared to the quasi likelihood parameter estimates obtained for the case where link an

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

a law of the iterated logarithm for kernel estimators of regression functions for fixed design regression data kernel estimation is widely used to estimate v t thev th derivative of t denoting such an estimator by hat mu nv t this paper is concerned with the almost sure convergence of hat mu nv t e hat mu nv t it is shown that under several inserting procedures a law of iterated logarithm holds

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

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

robustness of design in dose response studies summary we construct experimental designs for dose response studies the designs are robust against possibly misspecified link functions for this they minimize the maximum meansquared error of the estimated dose required to attain a response in 100p of the target population here p might be one particular value p d 0 5 corresponds to ed50 estimation or it might range over an interval of values of interest the maximum of the mean squared error is evaluated over a kolmogorov neighbourhood of the fitted link both the maximum and the minimum must be evaluated numerically the former is carried out by quadratic programming and the latter by simulated annealing

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

direction estimation in single index regressions we propose a general dimension reduction method that combines the ideas of likelihood correlation inverse regression and information theory we do not require that the dependence be confined to particular conditional moments nor do we place restrictions on the predictors or on the regression that are necessary for methods like ordinary least squares and sliced inverse regression although we focus on single index regressions the underlying idea is applicable more generally illustrative examples are presented

semiparametric analysis of heterogeneous data using varying scale generalized linear models this article describes a class of heteroscedastic generalized linear regression models in which a subset of the regression parameters are rescaled nonparametrically and develops efficient semiparametric inferences for the parametric components of the models such models provide a means to adapt for heterogeneity in the data due to varying exposures varying levels of aggregation and so on the class of models considered includes generalized partially linear models and nonparametrically scaled link function models as special cases we present an algorithm to estimate the scale function nonparametrically and obtain asymptotic distribution theory for regression parameter estimates in particular we establish that the asymptotic covariance of the semiparametric estimator for the parametric part of the model achieves the semiparametric lower bound we also describe bootstrap based goodness of scale test we illustrate the methodology with simulations published data and data from collaborative research on ultrasound safety

nonparametric quasi likelihood the quasi likelihood function proposed by wedderburn broadened the scope of generalized linear models by specifying the variance function instead of the entire distribution however complete specification of variance functions in the quasi likelihood approach may not be realistic we define a nonparametric quasi likelihood by replacing the specified variance function in the conventional quasi likelihood with a nonparametric variance function estimate this nonparametric variance function estimate is based on squared residuals from an initial model fit the rate of convergence of the nonparametric variance function estimator is derived it is shown that the asymptotic limiting distribution of the vector of regression parameter estimates is the same as for the quasi likelihood estimates obtained under correct specification of the variance function thus establishing the asymptotic efficiency of the nonparametric quasi likelihood estimates we propose bandwidth selection strategies based on deviance and pearson s chi square statistic it is demonstrated in simulations that for finite samples the proposed nonparametric quasi likelihood method can improve upon extended quasi likelihood or pseudo likelihood methods where the variance function is assumed to fall into a parametric class with unknown parameters we illustrate the proposed methods with applications to dental data and cherry tree data

on singular diffusion equations with applications to self organized criticality we consider solutions of the singular diffusion equation t um 1ux x m 0 associated with the flux boundary condition limx um 1ux x 0 the evolutions defined by this problem depend on both m and we prove existence and stability of traveling wave solutions parameterized by each traveling wave is stable in its appropriate evolution these traveling waves are in l1 for 1 m 0 but have non integrable tails for m 1 we also show that these traveling waves are the same as those for the scalar conservation law ut f u x uxx for a particular nonlinear convection term f u f u m 1993 john wiley sons inc

logistic regression diagnostics a maximum likelihood fit of a logistic regression model and other similar models is extremely sensitive to outlying responses and extreme points in the design space we develop diagnostic measures to aid the analyst in detecting such observations and in quantifying their effect on various aspects of the maximum likelihood fit the elements of the fitting process which constitute the usual output parameter estimates standard errors residuals etc will be used for this purpose with a properly designed computing package for fitting the usual maximum likelihood model the diagnostics are essentially free for the asking in particular good data analysis for logistic regression models need not be expensive or time consuming

functional regression analysis of fluorescence curves summary fluorescence curves are useful for monitoring changes in photosynthesis activity various summary measures have been used to quantify differences among fluorescence curves corresponding to different treatments but these approaches may forfeit valuable information as each individual fluorescence curve is a functional observation it is natural to consider a functional regression model the proposed model consists of a nonparametric component capturing the general form of the curves and a semiparametric component describing the differences among treatments and allowing comparisons of treatments several graphical model checking approaches are introduced both approximate asymptotic confidence intervals as well as simulation based confidence intervals are available analysis of data from a crop experiment using the proposed model shows that the salient features in the fluorescence curves are captured adequately the proposed functional regression model is useful for analysis of high throughput fluorescence curve data from regular monitoring or screening of plant growth

an improved estimator of the density function at the boundary abstract we propose a new method of boundary correction for kernel density estimation the technique is a kind of generalized reflection method involving reflecting a transformation of the data the transformation depends on a pilot estimate of the logarithmic derivative of the density at the boundary in simulations the new method is seen to clearly outperform an earlier generalized reflection idea it also has overall advantages over boundary kernel methods and a nonnegative adaptation thereof although the latter are competitive in some situations we also present the theory underlying the new methodology

adapting for the missing link we consider the fitting of generalized linear models in wich the link function is assumed to be unknown and propose the following computational method first estimate regression coefficients using the canonical link then estimate the link via a kernel smoother treating the direction in the predictor space determined by the regression coefficients as known then reestimate the direction using the estimated link and alternate between these two steps we show that under fairly general conditions n 1 2 consistent estimates of the direction are obtained a small monte carlo study is presented

polynomial spline estimation and inference of proportional hazards regression models with flexible relative risk form the cox proportional hazards model usually assumes an exponential form for the dependence of the hazard function on covariate variables however in practice this assumption may be violated and other relative risk forms may be more appropriate in this article we consider the proportional hazards model with an unknown relative risk form issues in model interpretation are addressed we propose a method to estimate the relative risk form and the regression parameters simultaneously by first approximating the logarithm of the relative risk form by a spline and then employing the maximum partial likelihood estimation an iterative alternating optimization procedure is developed for efficient implementation statistical inference of the regression coefficients and of the relative risk form based on parametric asymptotic theory is discussed the proposed methods are illustrated using simulation and an application to the veteran s administration lung cancer data

nonlinear diffusions extremal properties of barenblatt profiles best matching and delays abstract in this paper we consider functionals based on moments and nonlinear entropies which have a linear growth in time in case of source type solutions to the fast diffusion or porous medium equations that are also known as barenblatt solutions as functions of time these functionals have convexity properties for generic solutions so that their asymptotic slopes are extremal for barenblatt profiles the method relies on scaling properties of the evolution equations and provides a simple and direct proof of sharp gagliardo nirenberg sobolev inequalities in scale invariant form the method also gives refined estimates of the growth of the second moment and as a consequence establishes the monotonicity of the delay corresponding to the best matching barenblatt solution compared to the barenblatt solution with same initial second moment here the notion of best matching is defined in terms of a relative entropy

change points in nonparametric regression analysis an otherwise smooth regression model are proposed the assumptions needed are much weaker than those made in parametric models the proposed estimators apply as well to the detection of discontinuities in derivatives and therefore to the detection of change points of slope and of higher order curvature the proposed estimators are based on a comparison of left and right one sided kernel smoothers weak convergence of a stochastic process in local differences to a gaussian process is established for properly scaled versions of estimators of the location of a change point the continuous mapping theorem can then be invoked to obtain asymptotic distributions and corresponding rates of convergence for change point estimators these rates are typically faster than n 1 2 rates of global lp convergence of curve estimates with appropriate kernel modifications adapting to estimated change points are derived as a consequence it is shown that these rates of convergence are the same as if the location of the change point was known the methods are illustrated by means of the well known data on the annual flow volume of the nile river between 1871 and 1970 1 introduction nonparametric regression methods are usually applied in order to obtain a smooth fit of a regression curve without having to specify a parametric class of regression functions sometimes a generally smooth curve might contain an isolated discontinuity or change point in the curve or in a possibly higher order derivative and in many cases interest focuses on the occurrence of such change points in parametric approaches to the regression change point problem simple linear regressions before and after a possible change point are assumed and then the possibility of a discontinuity in the form of a jump or of a jump in the first derivative or equivalently a slope change is incorporated into the model see for instance hinkley 1969 and brown durbin and evans 1975 the analysis of change points which describe sudden localized changes typically occurring in economics medicine and the physical sciences has recently found increasing interest general smoothness assumptions allowing for a large class of regression functions to be considered seem to be more appropriate in a variety of applied problems than parametric modelling an

long time asymptotics for the porous medium equation the spectrum of the linearized operator we compute the complete spectrum of the displacement hessian operator which is obtained from the confined porous medium equation by linearization around its stationary attractor the barenblatt profile on a formal level the operator is conjugate to the hessian of the entropy via similarity transformation we show that the displacement hessian can be understood as a self adjoint operator and find that its spectrum is purely discrete the knowledge of the complete spectrum and the explicit information about the corresponding eigenfunctions give new insights on the convergence and higher order asymptotics of solutions to the porous medium equation towards its attractor more precisely the inspection of the eigenfunctions allows to identify symmetries in rn with flows whose rates of convergence are faster than the uniform translation governed bound the present work complements the analogous study of denzler mccann for the fast diffusion equation