decay rate of the green function in a random potential on the bethe lattice and a criterion for localization it is shown that in the tight binding anderson model on the bethe lattice the exponential decay rate of the green function can be obtained for arbitrary energies and arbitrary disorder analytical results in the case of the lorentzian distribution of site energies are presented as an application of these results a criterion for the localized region on the corresponding regular lattice on the level of the bethe peierls approximation is proposed our criterion yields exact results in the one dimensional limit and yields correct band edges for the hypercubic lattice in the vanishing limit of disorder the mobility edge trajectory obtained by our criterion is given by an elliptic curve in the case of the lorentzian distribution and its shape is found to be in qualitatively good agreement with that obtained by the finite size scaling method in the three dimensional system

exploring regression structure using nonparametric functional estimation abstract average derivative functionals of regression are proposed for nonparametric model selection and diagnostics the functionals are of the integral type which under certain conditions allows their estimation at the usual parametric rate of n 1 2 we analyze asymptotic properties of the estimators of these functionals based on kernel regression these estimators can then be used for assessing the validity of various restrictions imposed on the form of regression in particular we show how they could be used to reduce the dimensionality of the model assess the relative importance of predictors measure the extent of nonlinearity and nonadditivity and under certain conditions help identify projection directions in projection pursuit models and decide on the number of these directions

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

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

gelation in coagulating systems abstract a review of recent achievements in the theory of sol gel transitions is given this paper outlines the basic ideas of a stochastic approach and describes its possible application as a basis for a theory of gelation in coagulating systems the kinetics of coagulation is described in terms of the probability w to find in the system a given set of monomers dimers etc at time t this approach allows for a consideration of finite coagulating systems i e those containing a finite number of monomeric units the evolution equation for the generating functional of w is formulated and solved exactly for the coagulation kernel proportional to the product of masses of coagulating particles for which the traditional smoluchowski s scheme leads to a paradox related to the loss of the total mass of the coagulating particles this paradox is resolved within the scheme described in this paper it is shown that a single giant object a gel invisible in the thermodynamic limit arises after a critical time and begins to consume the mass of smaller but higher populated fraction the sol this approach is then applied for considering the time evolution of random graphs the distribution of the linked components of a random graph over their orders and numbers of edges is found exactly the paper is concluded with a short consideration of the sol gel transition in systems with instantaneous sinks at large particle mass it is demonstrated how the sol gel transition manifests itself in such systems

parametric and nonparametric methods for understanding the relationship between carcinogen induced dna adduct levels in distal and proximal regions of the colon an important problem in studying the etiology of colon cancer is understanding the relationship between dna adduct levels broadly dna damage in cells within colonic crypts in distal and proximal parts of the colon following treatment with a carcinogen and different types of diet in particular it is important to understand whether rats who have elevated adduct levels in particular positions in distal region crypts also have elevated levels in the same positions of the crypts in proximal regions and whether this relationship depends on diet we cast this problem as estimating the correlation function of two responses as a function of a covariate for studies where both responses are measured on the same experimental units but not the same subsampling units parametric and nonparametric methods are developed and applied to a dataset from an ongoing study leading to potentially important and surprising biological results theoretical calculations suggest that the nonparametric method based on nonparam

exact solutions for the coagulation fragmentation equation the authors present new exact equilibrium and time dependent solutions to the two main versions of the coagulation fragmentation equation equilibria are first found for the pure coagulation and pure fragmentation equations equilibrium solutions for the full coagulation fragmentation equation are then constructed when the kernels governing the reaction rates are chosen to be in a particular form time dependent solutions are also derived for the pure fragmentation equation

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

self similar solutions to a coagulation equation with multiplicative kernel abstract existence of self similar solutions to the oort hulst safronov coagulation equation with multiplicative coagulation kernel is established these solutions are given by s t y s t for t y 0 t 0 where t is some arbitrary positive real number s t 3 t t 1 3 and the parameter ranges in a given interval c 3 in addition the second moment of these self similar solutions blows up at time t as for the profile it belongs to l 1 0 y 2 d y for each c 3 but its behaviour for small and large y varies with the parameter

numerical simulation of the smoluchowski coagulation equation in this paper we develop a numerical scheme for the smoluchowski coagulation equation which relies on a conservative formulation and a finite volume approach several numerical simulations are performed to test the validity of the scheme and the expected behavior of the model in particular the gelation phenomenon and the long time behavior of the solution are numerically studied

quantile association regression models it is often important to study the association between two continuous variables in this work we propose a novel regression framework for assessing conditional associations on quantiles we develop general methodology which permits covariate effects on both the marginal quantile models for the two variables and their quantile associations the proposed quantile copula models have straightforward interpretation facilitating a comprehensive view of association structure which is much richer than that based on standard product moment and rank correlations we show that the resulting estimators are uniformly consistent and weakly convergent as a process of the quantile index simple variance estimators are presented which perform well in numerical studies extensive simulations and a real data example demonstrate the practical utility of the methodology supplementary materials for this article are available online

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

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

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

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

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

mathematical analysis of finite volume preserving scheme for nonlinear smoluchowski equation abstract this study presents the convergence analysis of the recently developed finite volume preserving scheme forestier coste and mancini 2012 for approximating a coalescence or smoluchowski equation the idea of the finite volume scheme is to preserve the total volume in the system by modifying the coalescence kernel using the notion of overlapping bins cells the consistency of the finite volume scheme is examined thoroughly in order to prove second order convergence on uniform non uniform smooth and locally uniform grids independently of the aggregation kernel the theoretical observations of order of convergence is verified using the experimental order of convergence for analytically tractable kernels

optimal kernel weights under a power criterion abstract we develop an approach to choosing optimal kernel weights for nonparametric estimation of the slope of the regression function which uses maximization of power rather than minimization of integrated mean squared error imse as its optimality criterion this power criterion leads to optimal kernel weights whose derivation is simpler than under other criteria and which provides an intuitive understanding of the nature of the optimality

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