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

local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

global attractors for multivalued random dynamical systems generated by random differential inclusions with multiplicative noise in this paper we consider a stochastic differential inclusion with multiplicative noise it is shown that it generates a multivalued random dynamical system for which there also exists a global random attractor

kernel sections and uniform attractors of multi valued semiprocesses abstract we present the existence of kernel sections which are all compact invariant and pullback attracting of an infinite dimensional general multi valued process constructed by the set valued backward extension of multi valued semiprocesses moreover the structure of the uniform attractors of a family of multi valued semiprocesses and the uniform forward attraction of kernel sections of a family of general multi valued processes are investigated finally we explain our abstract results by considering the mixed wave systems with supercritical exponent and ordinary differential equations

on the multiplicity of equilibrium solutions to a nonlinear diffusion equation on a manifold arising in climatology abstract we analyze the sensitivity of a climatological model with respect to small changes in one of the distinguished parameters the solar constant we start by proving the stabilization of solutions of the evolution model when time tends to infinity later we study the stationary problem and obtain uniqueness or a multiplicity of solutions for different values of the solar constant

local dependence functions for some families of bivariate distributions and total positivity the purpose of this paper is to investigate a very useful application of a certain local dependence function c f x y which was considered recently by holland and wang 20 an interesting property of c f x y is that the underlying joint density f x y is tp 2 that is totally positive of order 2 if and only if c f x y 0 this gives an elegant way to investigate the tp 2 property of any bivariate distribution for the saramanov family the ali mikhail haq family of bivariate distributions and the family of bivariate elliptical distributions we derive the local dependence function and obtain conditions for f x y to be tp 2 these families are quite rich and include many other large classes of bivariate distributions as their special cases similar conditions are obtained for bivariate distributions with exponential conditionals and bivariate distributions with pareto conditionals

a bifurcation theory for a class of discrete time markovian stochastic systems we present a bifurcation theory of smooth stochastic dynamical systems that are governed by everywhere positive transition densities the local dependence structure of the unique strictly stationary evolution of such a system can be expressed by the ratio of joint and marginal probability densities this dependence ratio is a geometric invariant of the system by introducing an equivalence relation defined on these dependence ratios we arrive at a bifurcation theory for which in the compact case the set of stable i e non bifurcating systems is open and dense the theory is illustrated with some simple examples

shared parameter models under random effects misspecification summary a common objective in longitudinal studies is the investigation of the association structure between a longitudinal response process and the time to an event of interest an attractive paradigm for the joint modelling of longitudinal and survival processes is the shared parameter framework where a set of random effects is assumed to induce their interdependence in this work we propose an alternative parameterization for shared parameter models and investigate the effect of misspecifying the random effects distribution in the parameter estimates and their standard errors

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

on the dynamics of nonautonomous periodic general dynamical systems and differential inclusions abstract we establish the existence and stability results for periodic nonautonomous uniform forward attractors of periodic general dynamical systems set valued dynamical systems we also investigate the dynamical behavior of nonautonomous periodic differential inclusion x t f t x t on r m with only upper semi continuous right hand side by applying the abstract results firstly we show that if the system has a compact uniformly attracting set then it has a periodic nonautonomous uniform forward attractor a secondly we prove that a is robust with respect to both internal and external perturbations finally we apply the robustness result to discuss the effects of small time delays to asymptotic stability of the system

open loop optimal control of linear time invariant systems containing the first derivative of the input abstract linear time invariant systems of the form are considered the problem is to find a control vector u t that will drive the state of this system from a given z 0 to some not necessarily fixed final state z t f in some not necessarily fixed time t f while minimizing a cost functional of the form the original system is re written in the form of a descriptor variable system consisting of the equations diag it is shown that there is a matrix such that thus the original optimal control problem is equivalent to the following problem find a control vector u t that drives the descriptor vector x t from a given x 0 to some x t f while minimizing using the results of an earlier paper on optimal control of descriptor systems ibrahim et al 1988 necessary conditions are derived for the existence of minima of the problem of finding sufficient conditions for the existence of minima of is not considered various formulae are obtained for designing open loop controls corresponding to three diffe

generalized measures of correlation for asymmetry nonlinearity and beyond applicability of pearson s correlation as a measure of explained variance is by now well understood one of its limitations is that it does not account for asymmetry in explained variance aiming to develop broad applicable correlation measures we study a pair of generalized measures of correlation gmc that deals with asymmetries in explained variances and linear or nonlinear relations between random variables we present examples under which the paired measures are identical and they become a symmetric correlation measure that is the same as the squared pearson s correlation coefficient as a result pearson s correlation is a special case of gmc theoretical properties of gmc show that gmc can be applicable in numerous applications and can lead to more meaningful conclusions and improved decision making in statistical inference the joint asymptotics of the kernel based estimators for gmc are derived and are used to test whether or not two random variables are symmetric in explaining variances th

the multivariate skew normal distribution summary the paper extends earlier work on the so called skew normal distribution a family of distributions including the normal but with an extra parameter to regulate skewness the present work introduces a multivariate parametric family such that the marginal densities are scalar skew normal and studies its properties with special emphasis on the bivariate case

on dynamical properties of general dynamical systems and differential inclusions abstract let f be a general dynamical system defined on a complete locally compact metric space x we give a slightly improved version of the lyapunov characterization of asymptotic stability in one of our previous works and provide a short self contained proof for the existence of arbitrarily small positively invariant neighborhoods of compact asymptotically stable sets in the present context based on these results we prove that the uniform attractors of f are connected and are stable with respect to perturbations under appropriate conditions we are also interested in the dynamical properties of differential inclusion x t f x t on r m first we show that if no solutions of the system blow up in finite future time then its reachable mapping f is a general dynamical system then we discuss some asymptotic stability properties of the system in particular we prove that if there exists a nonempty compact connected subset m of r m such that m attracts a neighborhood of itself then the system has a connected uniform attractor a we also prove that a is stable with respect to both internal and external perturbations more precisely we prove that when 0 is sufficiently small the perturbed system x t con f x t b 1 b 1 has a connected uniform attractor a moreover h a a 0 as 0 where h is the hausdorff distance in r m

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

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

attractors for random dynamical systems a criterion for existence of global random attractors for rds is established existence of invariant markov measures supported by the random attractor is proved for spde this yields invariant measures for the associated markov semigroup the results are applied to reation diffusion equations with additive white noise and to navier stokes equations with multiplicative and with additive white noise

on the asymptotic behaviour of solutions of a stochastic energy balance climate model abstract we prove the existence of a random global attractor for the multivalued random dynamical system associated to a nonlinear multivalued parabolic equation with a stochastic term of amplitude of the order of e the equation was initially suggested by north and cahalan following a previous deterministic model proposed by m i budyko for the modeling of some non deterministic variability as for instance the cyclones which can be treated as a fast varying component and are represented as a white noise process in the context of energy balance climate models we also prove the convergence in some sense of the global attractors when e 0 i e the convergence to the global attractor for the associated deterministic case e 0

conditionally specified continuous distributions a distribution is conditionally specified when its model constraints are expressed conditionally for example besag s 1974 spatial model was specified conditioned on the neighbouring states and pseudolikelihood is intended to approximate the likelihood using conditional likelihoods there are three issues of interest existence uniqueness and computation of a joint distribution in the literature most results and proofs are for discrete probabilities here we exclusively study distributions with continuous state space we examine all three issues using the dependence functions derived from decomposition of the conditional densities we show that certain dependence functions of the joint density are shared with its conditional densities therefore two conditional densities involving the same set of variables are compatible if their overlapping dependence functions are identical we prove that the joint density is unique when the set of dependence functions is both compatible and complete in addition a joint density apart from a constant can be computed from the dependence functions in closed form since all of the results are expressed in terms of dependence functions we consider our approach to be dependence based whereas methods in the literature are generally density based applications of the dependence based formulation are discussed

autonomous and non autonomous attractors for differential equations with delays abstract the asymptotic behaviour of some types of retarded differential equations with both variable and distributed delays is analysed in fact the existence of global attractors is established for different situations with and without uniqueness and for both autonomous and non autonomous cases using the classical notion of attractor and the recently new concept of pullback one respectively

estimation in semiparametric spatial regression nonparametric methods have been very popular in the last couple of decades in time series and regression but no such development has taken place for spatial models a rather obvious reason for this is the curse of dimensionality for spatial data on a grid evaluating the conditional mean given its closest neighbors requires a four dimensional nonparametric regression in this paper a semiparametric spatial regression approach is proposed to avoid this problem an estimation procedure based on combining the so called marginal integration technique with local linear kernel estimation is developed in the semiparametric spatial regression setting asymptotic distributions are established under some mild conditions the same convergence rates as in the one dimensional regression case are established an application of the methodology to the classical mercer and hall wheat data set is given and indicates that one directional component appears to be nonlinear which has gone unnoticed in earlier analyses

manifesto for a growth econometrics while the last three decades have seen remarkable advances in the econometric analysis of many areas of microeconomics and macroeconomics growth economics has not experienced anything close to such progress this is so despite the veritable explosion of theoretical and empirical studies of economic growth that has occurred in the last 15 years a sign of the need for a growth econometrics is made clear when one re ects on how little the empirical and theoretical literatures on economic growth make contact for example while spillover e ects and attendant nonlinearities continue to be hallmarks of the new endogenous growth theory empirical practice still largely focuses on linear models whose speci cation is suggested by the neoclassical exogenous growth model of solow and whose empirical implementation is often asserted to provide evidence in favor of that model cf barro 1991 it is no exaggeration to say that the theoretical and empirical growth literatures are evolving with little interaction it is only through an econometrics that can link theory to data analysis and hypothesis testing that a synergy can be achieved empirical analyses of growth generally follow a common strategy a crosssection or panel of countries is employed in which per capita output growth rates are assumed to depend over a common time horizon on two sets of variables one set of variables consists of initial per capita output savings and population growth rates variables that are suggested by the solow growth model the second set of variables consists of control variables that correspond to whatever additional determinants of growth a researcher wishes to examine in a survey published last year danny quah and i durlauf and quah 1999 found that over 90 di erent additional variables had appeared in the literature as of 1998 the current number is certainly much larger despite the fact that only about 120 countries are available for analysis in the standard data sets a result of this paucity of data relative to theories is the vast number of nonoverlapping studies that have appeared a number of the problems that plague empirical growth analysis represent variants of traditional problems with the interpretation of econometric models

local divergence and association summary in applications of differential geometry to problems of parametric statistical inference the notion of divergence is often used to measure the separation between two parametric densities in this paper we move away from the parametric framework of much of statistical differential geometry and instead employ divergence in a covariate setting many data analyses involve investigation of how the conditional distribution f y i x x of a random variable y changes with covariate values x we propose using local divergence between conditional distributions as a measure of this change and thus as a general measure of association between y and the covariates x under certain regularity conditions we define a class of divergences which are locally the rao distance lr the limiting lr divergence is bounded below by the signal to noise ratio with equality holding if and only if the conditional density comes from the natural exponential family the correlation curve and in particular the correlation coefficient are simply transformations of the signal to noise ratio and thus are transformations of local divergence we therefore obtain a differential geometric interpretation of standard ideas of association the class of lr divergences is broad and includes the kullback leibler divergence and renyi oc information measures we can therefore interpret local association as local utility or information gain of order cx we obtain comparable results without regularity conditions

local gaussian correlation a new measure of dependence abstract it is a common view among finance analysts and econometricians that the correlation between financial objects becomes stronger as the market is going down and that it approaches one when the market crashes having the effect of destroying the benefit of diversification the purpose of this paper is to introduce a local dependence measure that gives a precise mathematical description and interpretation of such phenomena we propose a new local dependence measure a local correlation function based on approximating a bivariate density locally by a family of bivariate gaussian densities using local likelihood at each point the correlation coefficient of the approximating gaussian distribution is taken as the local correlation existence uniqueness and limit results are established a number of properties of the local gaussian correlation and its estimate are given along with examples from both simulated and real data this new method of modelling carries with it the prospect of being able to do locally for a general density what can be done globally for the gaussian density in a sense it extends gaussian analysis from a linear to a non linear environment

estimation of heteroscedasticity in regression analysis consider the regression model y i g t i varepsilon i 1 leq i leq n with nonrandom design variables t i and measurements y i for the unknown regression function g cdot we assume that the data are heteroscedastic i e e varepsilon 2 i sigma 2 i not equiv operatorname const and investigate how to estimate sigma 2 i if sigma 2 i sigma 2 t i with a smooth function sigma 2 cdot initial estimators tilde sigma 2 i can be improved by kernel smoothers and the resulting class of estimators is shown to be uniformly consistent these estimates can be used to improve the estimation of the regression function g itself in parametric and nonparametric models further applications are suggested

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

sinh arcsinh distributions we introduce the sinh arcsinh transformation and hence by applying it to a generating distribution with no parameters other than location and scale usually the normal a new family of sinh arcsinh distributions this four parameter family has symmetric and skewed members and allows for tailweights that are both heavier and lighter than those of the generating distribution the central place of the normal distribution in this family affords likelihood ratio tests of normality that are superior to the state of the art in normality testing because of the range of alternatives against which they are very powerful likelihood ratio tests of symmetry are also available and are very successful three parameter symmetric and asymmetric subfamilies of the full family are also of interest heavy tailed symmetric sinh arcsinh distributions behave like johnson su distributions while their light tailed counterparts behave like sinh normal distributions the sinh arcsinh family allowing a seamless transition between the two via the normal controlled by a single parameter the sinh arcsinh family is very tractable and many properties are explored likelihood inference is pursued including an attractive reparameterization illustrative examples are given a multivariate version is considered options and extensions are discussed

regularity of trajectory attractor and upper semicontinuity of global attractor for a 2d non newtonian fluid abstract there are two results within this paper the one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non newtonian fluid on 2d bounded domains for which the solution to each initial value could be non unique the other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from m to r l l where m m 1 is an expanding sequence of simply connected bounded and smooth subdomains of such that m as m that is let a and a m be the global attractors of the fluid corresponding to and m respectively we establish that for any neighborhood o a of a the global attractor a m enters o a if m is large enough

variance function estimation in regression the effect of estimating the mean abstract the authors consider estimation of a variance function g in regression problems such estimation requires simultaneous estimation of the mean function f we obtain sharp results on the extent to which the smoothness of f influences best rates of convergence for estimating g for example in nonparametric regression with two derivatives on g classical rates of convergence are possible if and only if the unknown f satisfies a lipschitz condition of order 1 3 or more if a parametric model is known for g then g may be estimated n 1 2 consistently if and only if f is lipschitz of order 1 2 or more optimal rates of convergence are attained by kernel estimators

analysis of variance coefficient of determination and f test for local polynomial regression this paper provides anova inference for nonparametric local polynomial regression lpr in analogy with anova tools for the classical linear regression model a surprisingly simple and exact local anova decomposition is established and a local r squared quantity is defined to measure the proportion of local variation explained by fitting lpr a global anova decomposition is obtained by integrating local counterparts and a global r squared and a symmetric projection matrix are defined we show that the proposed projection matrix is asymptotically idempotent and asymptotically orthogonal to its complement naturally leading to an f test for testing for no effect a by product result is that the asymptotic bias of the projected response based on local linear regression is of quartic order of the bandwidth numerical results illustrate the behaviors of the proposed r squared and f test the anova methodology is also extended to varying coefficient models

attractors for differential equations with unbounded delays we prove the existence of attractors for some types of dierential problems contain ing in nite delays applications and examples are provided to illustrate the theory which is valid for both cases with and without explicit dependence of time and with or without uniqueness of solutions as well

pullback attractor for differential evolution inclusions with infinite delays we analyze the existence of pullback attractor for non autonomous differential inclusions with infinite delays in banach spaces by using measures of noncompactness the obtained results can be applied to control systems driven by semilinear partial differential equations and multivalued feedbacks

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

global attractor for a non autonomous integro differential equation in materials with memory abstract the long time behavior of an integro differential parabolic equation of diffusion type with memory terms expressed by convolution integrals involving infinite delays and by a forcing term with bounded delay is investigated in this paper the assumptions imposed on the coefficients are weak in the sense that uniqueness of solutions of the corresponding initial value problems cannot be guaranteed then it is proved that the model generates a multivalued non autonomous dynamical system which possesses a pullback attractor first the analysis is carried out with an abstract parabolic equation then the theory is applied to the particular integro differential equation which is the objective of this paper the general results obtained in the paper are also valid for other types of parabolic equations with memory

morse decompositions for general dynamical systems and differential inclusions with applications to control systems the morse decomposition theory for single valued dynamical systems is extended to general dynamical systems and differential inclusions relations between chain recurrent sets chain recurrent components chain control sets and morse decompositions for open loop systems are discussed the robustness of feedback laws of closed loop systems with respect to small time delays and sample hold controls is also addressed

free final time optimal control of descriptor systems linear time invariant systems of the form e dx dt ax bu are considered where e is a square matrix that may be singular and x is a column vector called the descriptor vector of the system it is assumed that for any admissible initial descriptor vector x 0 any control vector u t yields one and only one descriptor vector x t the problem is this find a control vector u t that will drive the descriptor vector of the system from a fixed vector x 0 to a not necessarily fixed final descriptor vector x tf while together with some not a priori fixed final time tf minimizing a cost functional j 1 2 1 0 xt qx ut ru dt using elementary matrix and variational techniques necessary conditions are derived for the existence of minima of j the problem of finding sufficient conditions for the existence of minima of j is not considered the general results are applied to the special case e diag in m 0 b o im the problem of choosing the matrix q of the cost functional is investigated

the probability integrals of bivariate normal distributions a contingency table approach summary after partitioning the plane into fine rectangular regions the bivariate normal probability integrals over all the rectangular regions can be considered as a population contingency table the row and column marginals of this contingency table are two discretized univariate normal distributions the logarithm of the local cross product ratios are approximately linear given the correlation coefficient of the bivariate normal a contingency table with discretized normal marginals and proper cross product ratios can be constructed which represents a table of probability integrals for the bivariate normal

on the regularization effect of space time white noise on quasi linear parabolic partial differential equations we show that an existence and uniqueness and a comparison theorem hold if we add a space time white noise to a quasi linear parabolic equation in one space dimension even if the nonlinearity is only measurable and not even locally bounded

asymptotically optimal difference based estimation of variance in nonparametric regression summary we define and compute asymptotically optimal difference sequences for estimating error variance in homoscedastic nonparametric regression our optimal difference sequences do not depend on unknowns such as the mean function and provide substantial improvements over the suboptimal sequences commonly used in practice for example in the case of normal data the usual variance estimator based on symmetric second order differences is only 64 efficient relative to the estimator based on optimal second order differences the efficiency of an optimal mth order difference estimator relative to the error sample variance is 2m 2m 1 again this is for normal data and increases as the tails of the error distribution become heavier

uniform attractors for non autonomous random dynamical systems abstract this paper is devoted to establishing a random uniform attractor theory for non autonomous random dynamical systems nrds the uniform attractor is defined as the minimal compact uniformly pullback attracting random set nevertheless the uniform pullback attraction in fact implies a uniform forward attraction in probability and implies also an almost uniform pullback attraction for discrete time sequences though no invariance is required by definition the uniform attractor can have a negative semi invariance under certain conditions several existence criteria for uniform attractors are given and the relationship between uniform and cocycle attractors is carefully studied to overcome the measurability difficulty the symbol space is required to be polish which is shown fulfilled by the hulls of l l o c p r l r functions p r 1 moreover uniform attractors for continuous nrds are shown determined by uniformly attracting deterministic compact sets finally the uniform attractor for a stochastic reaction diffusion equation with translation bounded external forcing are studied as applications

robust estimation in heteroscedastic linear models we consider a heteroscedastic linear model in which the variances are given by a parametric function of the mean responses and a parameter theta we propose robust estimates for the regression parameter beta and show that as long as a reasonable starting estimate of theta is available our estimates of beta are asymptotically equivalent to the natural estimate obtained with known variances a particular method for estimating theta is proposed and shown by monte carlo to work quite well especially in power and exponential models for the variances we also briefly discuss a feedback estimate of beta

the local dependence function in the context of recent interest in extending scalar association measures to local association functions attention is drawn to the local dependence function the mixed partial derivative of the log density proposed by holland wang 1987 we provide an alternative motivation to that provided by holland wang we also describe a few basic properties of the local dependence function and indicate how it may be estimated from data

correlation curves measures of association as functions of covariate values for experiments where the strength of association between a response variable y and a covariate x is different over different regions of values for the covariate x we propose local nonparametric dependence functions which measure the strength of association between y and x as a function of x x our dependence functions are extensions of galton s idea of strength of co relation from the bivariate normal case to the nonparametric case in particular a dependence function is obtained by expressing the usual galton pearson correlation coefficient in terms of the regression line slope 3 and the residual variance o 2 and then replacing 3 and 72 by a nonparametric regression slope 13 x and a nonparametric residual variance o 2 x var yix respectively our local dependence functions are standardized nonparametric regression curves which provide universal scale free measures of the strength of the relationship between variables in nonlinear models they share most of the properties of the correlation coefficient and they reduce to the usual correlation coefficient in the bivariate normal case for this reason we call them correlation curves we show that in a certain sense they quantify lehmann s notion of regression dependence finally the correlation curve concept is illustrated using data from a study of the relationship between cholesterol levels x and triglyceride concentrations y of heart patients

s shaped bifurcation branch in a quasilinear multivalued model arising in climatology abstract in this paper we show the existence of a continuous and unbounded connected s shaped set q u where q is the solar constant and u satisfies a quasilinear eventually multivalued stationary equation on a riemannian manifold without boundary arising as a stationary energy balance model for the earth surface temperature