large character sums pretentious characters and the polya vinogradov theorem in 1918 polya and vinogradov gave an upper bound for the maximal size of character sums which still remains the best known general estimate one of the main results of this paper provides a substantial improvement of the polya vinogradov bound for characters of odd bounded order in 1977 montgomery and vaughan showed how the polya vinogradov inequality may be sharpened assuming the generalized riemann hypothesis we give a simple proof of their estimate and provide an improvement for characters of odd bounded order the paper also gives characterizations of the characters for which the maximal character sum is large and it finds a hidden structure among these characters

autoregressive and cross lagged model for bivariate non commensurate outcomes autoregressive and cross lagged models have been widely used to understand the relationship between bivariate commensurate outcomes in social and behavioral sciences but not much work has been carried out in modeling bivariate non commensurate e g mixed binary and continuous outcomes simultaneously we develop a likelihood based methodology combining ordinary autoregressive and cross lagged models with a shared subject specific random effect in the mixed model framework to model two correlated longitudinal non commensurate outcomes the estimates of the cross lagged and the autoregressive effects from our model are shown to be consistent with smaller mean squared error than the estimates from the univariate generalized linear models inclusion of the subject specific random effects in the proposed model accounts for between subject variability arising from the omitted and or unobservable but possibly explanatory subject level predictors our model is not restricted to the case with equal number of events per subject and it can be extended to different types of bivariate outcomes we apply our model to an ecological momentary assessment study with complex dependence and sampling data structures specifically we study the dependence between the condom use and sexual satisfaction based on the data reported in a longitudinal study of sexually transmitted infections we find negative cross lagged effect between these two outcomes and positive autoregressive effect within each outcome copyright 2017 john wiley sons ltd

bayesian latent variable models for clustered mixed outcomes a general framework is proposed for modelling clustered mixed outcomes a mixture of generalized linear models is used to describe the joint distribution of a set of underlying variables and an arbitrary function relates the underlying variables to be observed outcomes the model accommodates multilevel data structures general covariate effects and distinct link functions and error distributions for each underlying variable within the framework proposed novel models are developed for clustered multiple binary unordered categorical and joint discrete and continuous outcomes a markov chain monte carlo sampling algorithm is described for estimating the posterior distributions of the parameters and latent variables because of the flexibility of the modelling framework and estimation procedure extensions to ordered categorical outcomes and more complex data structures are straightforward the methods are illustrated by using data from a reproductive toxicity study

response models for mixed binary and quantitative variables summary a number of special representations are considered for the joint distribution of qualitative mostly binary and quantitative variables in addition to the conditional gaussian models and to conditional gaussian regression chain models some emphasis is placed on models derived from an underlying multivariate normal distribution and on models in which discrete probabilities are specified linearly in terms of unknown parameters the possibilities for choosing between the models empirically are examined as well as the testing of independence and conditional independence and the estimation of parameters often the testing of independence is exactly or nearly the same for a number of different models

preconditioners for pseudodifferential equations on the sphere with radial basis functions in a previous paper a preconditioning strategy based on overlapping domain decomposition was applied to the galerkin approximation of elliptic partial differential equations on the sphere in this paper the methods are extended to more general pseudodifferential equations on the sphere using as before spherical radial basis functions for the approximation space and again preconditioning the ill conditioned linear systems of the galerkin approximation by the additive schwarz method numerical results are presented for the case of hypersingular and weakly singular integral operators on the sphere mathbb s 2

strongly elliptic pseudodifferential equations on the sphere with radial basis functions spherical radial basis functions are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere these equations arise from geodesy the approximate solutions are found by the galerkin and collocation methods a salient feature of the paper is a unified theory for error analysis of both approximation methods

joint models for the association of longitudinal binary and continuous processes with application to a smoking cessation trial joint models for the association of a longitudinal binary and a longitudinal continuous process are proposed for situations in which their association is of direct interest the models are parameterized such that the dependence between the two processes is characterized by unconstrained regression coefficients bayesian variable selection techniques are used to parsimoniously model these coefficients a markov chain monte carlo mcmc sampling algorithm is developed for sampling from the posterior distribution using data augmentation steps to handle missing data several technical issues are addressed to implement the mcmc algorithm efficiently the models are motivated by and are used for the analysis of a smoking cessation clinical trial in which an important question of interest was the effect of the exercise treatment on the relationship between smoking cessation and weight gain

latent variable models for mixed discrete and continuous outcomes we propose a latent variable model for mixed discrete and continuous outcomes the model accommodates any mixture of outcomes from an exponential family and allows for arbitrary covariate effects as well as direct modelling of covariates on the latent variable an em algorithm is proposed for parameter estimation and estimates of the latent variables are produced as a by product of the analysis a generalized likelihood ratio test can be used to test the significance of covariates affecting the latent outcomes this method is applied to birth defects data where the outcomes of interest are continuous measures of size and binary indicators of minor physical anomalies infants who were exposed in utero to anticonvulsant medications are compared with controls

correlated bivariate continuous and binary outcomes issues and applications increasingly multiple outcomes are collected in order to characterize treatment effectiveness or to evaluate the impact of large policy initiatives often the multiple outcomes are non commensurate e g measured on different scales the common approach to inference is to model each outcome separately ignoring the potential correlation among the responses we describe and contrast several full likelihood and quasi likelihood multivariate methods for non commensurate outcomes we present a new multivariate model to analyze binary and continuous correlated outcomes using a latent variable we study the efficiency gains of the multivariate methods relative to the univariate approach for complete data all approaches yield consistent parameter estimates when the mean structure of all outcomes depends on the same set of covariates efficiency gains by adopting a multivariate approach are negligible in contrast when the mean outcomes depend on different covariate sets large efficiency gains are realized three real examples illustrate the different approaches

association models for clustered data with binary and continuous responses we consider analysis of clustered data with mixed bivariate responses i e where each member of the cluster has a binary and a continuous outcome we propose a new bivariate random effects model that induces associations among the binary outcomes within a cluster among the continuous outcomes within a cluster between a binary outcome and a continuous outcome from different subjects within a cluster as well as the direct association between the binary and continuous outcomes within the same subject for the ease of interpretations of the regression effects the marginal model of the binary response probability integrated over the random effects preserves the logistic form and the marginal expectation of the continuous response preserves the linear form we implement maximum likelihood estimation of our model parameters using standard software such as proc nlmixed of sas our simulation study demonstrates the robustness of our method with respect to the misspecification of the regression model as well as the random effects model we illustrate our methodology by analyzing a developmental toxicity study of ethylene glycol in mice

meshless galerkin methods using radial basis functions we combine the theory of radial basis functions with the field of galerkin methods to solve partial differential equations after a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions

optimal dose finding designs with correlated continuous and discrete responses in dose finding clinical studies it is common that multiple endpoints are of interest for instance in phase i ii studies efficacy and toxicity are often the primary endpoints which are observed simultaneously and which need to be evaluated together motivated by this we confine ourselves to bivariate responses and focus on the most analytically difficult case a mixture of continuous and categorical responses we adopt the bivariate probit dose response model and quantify our goal by a utility function we study locally optimal designs two stage optimal designs and fully adaptive designs under different ethical and cost constraints in the experiments we assess the performance of two stage designs and fully adaptive designs via simulations our simulations suggest that the two stage designs are as efficient as and may be more efficient than the fully adaptive designs if there is a moderate sample size in the initial stage in addition two stage designs are easier to construct and implement and thus can be a useful approach in practice copyright 2011 john wiley sons ltd

on the asymptotic convergence of collocation methods we prove quasioptimal and optimal order estimates in various sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines the analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications equations to which the analysis is applied include fredholm integral equations of the second kind certain first kind fredholm equations singular integral equations involving cauchy kernels a variety of integro differential equations and two point boundary value problems for ordinary differential equations the error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard galerkin methods we compare the collocation method with a standard galerkin method using splines of the same degree showing that the galerkin method is quasioptimal in a sobolev space of lower index and furnishes optimal order approximation for a range of sobolev indices containing and extending below that for the collocation method and so the standard galerkin method achieves higher rates of convergence

moments and distribution of central l values of quadratic twists of elliptic curves we show that if one can compute a little more than a particular moment for some family of l functions then one has upper bounds of the conjectured order of magnitude for all smaller positive real moments and a one sided central limit theorem holds we illustrate our method for the family of quadratic twists of an elliptic curve obtaining sharp upper bounds for all moments below the first we also establish a one sided central limit theorem supporting a conjecture of keating and snaith our work leads to a conjecture on the distribution of the order of the tate shafarevich group for rank zero quadratic twists of an elliptic curve and establishes the upper bound part of this conjecture assuming the birch swinnerton dyer conjecture

a copula based model for multivariate non normal longitudinal data analysis of a dose titration safety study on a new antidepressant a new model for multivariate non normal longitudinal data is proposed in a first step each longitudinal series of data corresponding to a given response is modelled separately using a copula to relate the marginal distributions of the response at each time of observation in a second step at each observation time the conditional on the past distributions of each response are related using another copula describing the relationship between the corresponding variables note that there is no need to consider the same family of distributions for these response variables the technique is illustrated in a dose titration safety study on a new antidepressant the haemodynamic effect on diastolic blood pressure systolic blood pressure and heart rate is studied these three responses are measured repeatedly over time on ten healthy volunteers during the dose escalation the available covariates are sex and the concentration of drug in the plasma at time of measurement

regression models for a bivariate discrete and continuous outcome with clustering abstract developmental toxicity studies of laboratory animals play a crucial role in the testing and regulation of chemicals and pharmaceutical compounds exposure to developmental toxicants typically causes a variety of adverse effects such as fetal malformations and reduced fetal weight at term in this article we discuss regression methods for jointly analyzing bivariate discrete and continuous outcomes that are motivated by the statistical problems that arise in analyzing data from developmental toxicity studies we focus on marginal regression models that is models in which the marginal expectation of the bivariate response vector is related to a set of covariates by some known link functions in these models the regression parameters for the marginal expectation are of primary scientific interest whereas the association between the binary and continuous response is considered to be a nuisance characteristic of the data we describe a likelihood based approach based on the general location mode

joint response graphs and separation induced by triangular systems summary we consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions the independences are captured by missing edges in a directed graph a matrix form of such a graph called the generating edge matrix is triangular so the distributions that are generated over such graphs are called triangular systems we study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models i e for alternative recursive factorizations of the given density using joint conditional distributions for this we introduce families of linear triangular equations which do not require assumptions of distributional form the strength of the associations that are implied by such linear families for chain graph models is derived the edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix it is shown how induced independences and dependences can be studied by graphs by edge matrix calculations and via the properties of densities some ways of using the results are illustrated

moderating effects of subgroups in linear models summary possibilities for moderating effects of a subgrouping variable on strength or direction of an association have been much discussed by social scientists but have not been given satisfactory statistical formulations the results concern directed measures of associations in linear models containing just three variables some key words analysis of covariance analysis of variance cg distribution conditional independence graphical chain model parallel regressions yule simpson paradox 1 introduction linear models are commonly used as a framework to estimate and test how a continuous response variable depends on potential influencing variables this paper is concerned with the situation in which two influences are random variables as well which may be discrete or continuous the continuous random variables are to stand for quantitative properties of observational units while the discrete variables either capture qualitative properties or they represent subgroups of the population to which the observational units belong to think of all variables as random variables is usually appropriate for data obtained from observational studies where one cannot control which levels or values of the influencing variables are to be observed the linear model permits an analysis conditional on fixed levels and values of the influences but frequently the association structure among the influences is in social science applications of interest itself and as it will turn out it is important for a correct interpretation of how the response depends on the influences awareness of such problems in interpretation is widespread among social scientists as is documented by extensive discussions of moderating effects see for example saunders 1956 zedeck 1971 and baron kenny 1986 there a variable is called a moderator if its presence changes the strength or direction of an association however methods recommended to social scientists for identifying moderating effects given in the above literature or by cohen cohen 1983 pp 310 4 have been shown to be seriously deficient wermuth 1988 thus there is need for clarification we treat the case of a potential discrete moderator variable and call its categories or levels the subgroups in the linear model a moderating effect is closely tied to the notion of consistent results on an association wermuth 1987 in the present context results are said to be weakly consistent if the associations within subgroups coincide in direction and strongly consistent if the associ ations within subgroups coincide in direction and strength this notion depends typically on the chosen measure of association in linear models we look at directed measures of

on sharp transitions in making squares in the fastest performing integer factoring algorithms one creates a sequence of integers in a pseudo random way and wishes to rapidly determine a subsequence whose product is a square in 1994 pomerance stated the following problem which encapsulates all of the key issues select integers a1 a2 at random from the interval 1 x until some nonempty subsequence has product equal to a square find a good estimate for the expected stopping time of this process a good solution should allow one to determine the optimal choice of parameters in many factoring algorithms pomerance 1994 using an idea of schroeppel 1985 showed that with probability 1 o 1 the rst subsequence whose product equals a square

a density theorem on automorphic l functions and some applications we establish a density theorem of automorphic l functions and give some applications on the extreme values of these l functions at s 1 and the distribution of the hecke eigenvalue of holomorphic cusp forms contents

rigidity theorems for multiplicative functions we establish several results concerning the expected general phenomenon that given a multiplicative function f mathbb n rightarrow mathbb c the values of f n and f n a are generally independent unless f is of a special form first we classify all bounded completely multiplicative functions having uniformly large gaps between its consecutive values this implies the solution of the following folklore conjecture for any completely multiplicative function f mathbb n rightarrow mathbb t we have begin aligned liminf n rightarrow infty f n 1 f n 0 end aligned second we settle an old conjecture due to chudakov on the generalized characters in actes du congres international des mathematiciens nice 1970 tome 1 p 487 gauthier villars paris that states that any completely multiplicative function f mathbb n rightarrow mathbb c that a takes only finitely many values b vanishes at only finitely many primes and c has bounded discrepancy is a dirichlet character this generalizes previous work of tao on the erd s discrepancy problem finally we show that if many of the binary correlations of a 1 bounded multiplicative function are asymptotically equal to those of a dirichlet character chi mod q then f n chi n n it for all n where chi is a dirichlet character modulo q and t in mathbb r this establishes a variant of a conjecture of h cohn for multiplicative arithmetic functions the main ingredients include the work of tao on logarithmic elliott conjecture correlation formulas for pretentious multiplicative functions developed earlier by the first author and szemeredi s theorem for long arithmetic progressions

solving non strongly elliptic pseudodifferential equations on a sphere using radial basis functions non strongly elliptic pseudodifferential equations on the unit sphere arise from geodesy an example of equations of this type is the boundary integral reformulation of a boundary value problem with the laplace equation in the interior domain of the unit sphere and a robin boundary condition approximate solutions with spherical radial basis functions are found by the galerkin and collocation methods the paper presents a unified theory for error analysis of both approximation methods the theoretical results are corroborated by numerical experiments it is noted that the stiffness matrix arising from the galerkin method for this problem resembles that arising from a least squares method

copula based regression models for a bivariate mixed discrete and continuous outcome this paper is concerned with regression models for correlated mixed discrete and continuous outcomes constructed using copulas our approach entails specifying marginal regression models for the outcomes and combining them via a copula to form a joint model specifically we propose marginal regression models e g generalized linear models to link the outcomes marginal means to covariates to account for associations between outcomes we adopt the gaussian copula to indirectly specify their joint distributions our approach has two advantages over current methods one regression parameters in models for both outcomes are marginally meaningful and two the association is margin free in the sense that it is characterized by the copula alone by assuming a latent variable framework to describe discrete outcomes the copula used still uniquely determines the joint distribution in addition association measures between outcomes can be interpreted in the usual way we report results of simulations concerning the bias and efficiency of two likelihood based estimation methods for the model finally we illustrate the model using data on burn injuries

a posteriori error estimation for the laplace beltrami equation on spheres with spherical splines abstract we prove a posteriori upper and lower bounds for the error estimates when solving the laplace beltrami equation on the unit sphere by using the galerkin method with spherical splines adaptive mesh refinements based on these a posteriori error estimates are used to reduce complexity and computational cost of the corresponding discrete problems the theoretical results are corroborated by numerical experiments

mixed multivariate generalized linear models for assessing lower limb arterial stenoses experiments and observational studies often involve gathering information on several response variables enabling us to model their dependence on observable predictor variables despite associations between the response variables they are often analysed separately using general and generalized linear models this paper investigates applications of multivariate regression analysis to improve the accuracy of predictions and decisions in the specific context of diagnosing arterial stenoses in human legs two basic models are developed for this application using i four binary responses and ii a mixture of two binary and two normal responses the results clearly demonstrate the potential advantages offered by this approach

regression models for mixed discrete and continuous responses with potentially missing values in this paper a likelihood based method for analyzing mixed discrete and continuous regression models is proposed we focus on marginal regression models that is models in which the marginal expectation of the response vector is related to covariates by known link functions the proposed model is based on an extension of the general location model of olkin and tate 1961 annals of mathematical statistics 32 448 465 and can accommodate missing responses when there are no missing data our particular choice of parameterization yields maximum likelihood estimates of the marginal mean parameters that are robust to misspecification of the association between the responses this robustness property does not in general hold for the case of incomplete data there are a number of potential benefits of a multivariate approach over separate analyses of the distinct responses first a multivariate analysis can exploit the correlation structure of the response vector to address intrinsically multivariate questions second multivariate test statistics allow for control over the inflation of the type i error that results when separate analyses of the distinct responses are performed without accounting for multiple comparisons third it is generally possible to obtain more precise parameter estimates by accounting for the association between the responses finally separate analyses of the distinct responses may be difficult to interpret when there is nonresponse because different sets of individuals contribute to each analysis furthermore separate analyses can introduce bias when the missing responses are missing at random mar a multivariate analysis can circumvent both of these problems the proposed methods are applied to two biomedical datasets

on fourier coefficients of modular forms of half integral weight at squarefree integers we show that the dirichlet series associated to the fourier coefficients of a half integral weight hecke eigenform at squarefree integers extends analytically to a holomorphic function in the half plane re s tfrac 1 2 this exhibits a high fluctuation of the coefficients at squarefree integers

on two families of transformations to additivity for binary response data summary two families of power transformations for probabilities are introduced to model symmetric and asymmetric departures from the logistic model the aim is to provide in a comprehensive way a representation of alternative scales in terms of which to carry out the analysis of binary response data maximum likelihood methods are employed to estimate an appropriate transformation using glim tests for detection of symmetric and asymmetric departures are proposed

on substantive research hypotheses conditional independence graphs and graphical chain models summary graphs consisting of points and lines or arrows as connections between selected pairs of points are used to formulate hypotheses about relations between variables points stand for variables connections represent associations when a missing connection is interpreted as a conditional independence the graph characterizes a conditional independence structure as well statistical models called graphical chain models correspond to special types of graphs which are interpreted in this fashion examples are used to illustrate how conditional independences are reflected in summary statistics derived from the models and how the graphs help to identify analogies and equivalences between different models graphical chain models are shown to provide a unifying concept for many statistical techniques that in the past have proven to be useful in analyses of data they also provide tools for new types of analysis

extreme values of zeta and l functions we introduce a resonance method to produce large values of the riemann zeta function on the critical line and large and small central values of l functions

on the convergence of collocation methods for boundary integral equations on polygons the integral equations encountered in boundary element methods are frequently solved numerically using collocation with spline trial functions convergence proofs and error estimates for these approximation methods have been only available in the following cases fredholm integral equations of the second kind 4 7 one dimensional pseudodifferential equations and singular integral equations with piecewise smooth coefficients on smooth curves 2 3 17 26 29 and some special results on the classical neumann integral equation of potential theory for polygonal plane domains 5 8 9 here we give convergence proofs for collocation with piecewise linear trial functions for neumann s integral equation and symm s integral equation on plane curves with corners we derive asymptotic error estimates in sobolev norms and analyze the effect of graded meshes 0 introduction in this paper we give convergence proofs and asymptotic error estimates in sobolev norms for collocation with piecewise linear spline trial func tions applied to two basic integral equations of potential theory on plane polygons namely the integral equation of the second kind with the double layer potential neumann s integral equation and the integral equation of the first kind with the simple layer potential symm s integral equation we use an idea of arnold and wendland 2 namely considering dirac delta functions the test functions in the collocation method as second derivatives of piecewise linear functions therefore similar results as presented here should be possible for splines of higher odd order corresponding results for even order splines are not yet available thus for one of the simplest methods of numerically solving dirichlet s problem on a plane domain with corners the midpoint collocation with piecewise constant trial functions for the first kind integral equation with the simple layer potential convergence is still an open problem the method of fourier series that yields the convergence proof in the case of a smooth boundary 27 cannot be applied in the presence of corners we apply the method of local mellin transformation that has previously been used to derive error estimates for galerkin methods for a wide class of operators including those occurring in boundary element methods in acoustics electromag

an approximation to maximum likelihood estimates in reduced models summary an approximation to the maximum likelihood estimates of the parameters in a model can be obtained from the corresponding estimates and information matrices in an extended model i e a model with additional parameters the approximation is close provided that the data are consistent with the first model applications are described to log linear models for discrete data to models for multivariate normal distributions with special covariance matrices and to mixed discrete continuous models

graphical and recursive models for contingency tables summary we discuss two classes of models for contingency tables graphical and recursive models both of which arise from restrictions that are expressible as conditional independencies of variable pairs the first of these is a subclass of hierarchical log linear models each of its models can be represented by an undirected graph in the second class each model corresponds to a particular kind of a directed graph instead and can be characterized by a nontrivial factorization of the joint distribution in terms of response variables we derive decomposable or multiplicative models as the intersecting class this result has useful consequences for exploratory types of analysis as well as for the model interpretation we can give an aid for detecting well fitting decomposable models in a transformation of the observed contingency table and each decomposable model may be interpreted with the help of an undirected or directed graph

a fast trigonometric collocation method for some elliptic pseudodifferential equations abstract in this paper we propose and analyze a fast trigonometric collocation method for a class of periodic elliptic pseudodifferential equations whose pseudodifferential operators can always be represented as the sum of a principal part and a smoothing operator we show that the whole matrix representation for the principal part in our discrete linear system can be generated by only computing o n entries rather than computing all entries of the matrix where 2 n or 2 n 1 is the size of the matrix the dense matrix for the smoothing operator can be compressed into a sparse matrix with only o n log n nonzero entries we also prove that our proposed method preserves the optimal convergent order the same as without compression some numerical experiments for solving three cases of boundary integral equations are presented to demonstrate its approximate accuracy and computational efficiency verifying the theoretical estimates

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12