schur recursions error formulas and convergence of rational estimators for stationary stochastic sequences an exact and approximate realization theory for estimation and model filters of second order stationary stochastic sequences is presented the properties of j lossless matrices as a unifying framework are used necessary and sufficient conditions for the exact realization of an estimation filter and a model filter as a submatrix of a j lossless system are deduced an extension of the so called schur algorithm yields an approximate j lossless realization based on partial past information about the process the geometric properties of such partial realizations and their convergence are studied finally connections with the nevanlinna pick problem are made and how the techniques presented constitute a generalization of many aspects of the levinson szego theory of partial realizations is shown as a consequence generalized recursive formulas for reproducing kernels and christoffel darboux formulas are obtained in this paper the scalar case is considered the matrix case will be considered in a separate publication

formative processes with applications to the decision problem in set theory i powerset and singleton operators this paper introduces formative processes composed by transitive partitions given a family f of sets a formative process ending in the venn partition e of f is shown to exist sufficient criteria are also singled out for a transitive partition to model via a function from set variables to unions of sets in the partition all set literals modeled by e on the basis of such criteria a procedure is designed that mimics a given formative process by another where sets have finite rank bounded by c e with c a specific computable function as a by product one of the core results on decidability in computable set theory is rediscovered namely the one that regards the satisfiability of unquantified set theoretic formulae involving boolean operators the singleton former and the powerset operator the method described which is able to exhibit a set solution when the answer is affirmative can be extended to solve the satisfiability problem for broader fragments of set theory

a christoffel darboux formula and a favard s theorem for orthogonal laurent polynomials on the unit circle let f k z k 0 be the family of orthonormal laurent polynomials on the unit circle which spans d in the ordering induced by p n e n 1 2 from the three term recurrence relation satisfied by f k z k 0 we deduce a christoffel darboux formula particular examples are considered and a favard type theorem is proved a connection with the ordering induced by p n e n 2 is also established

illustration of bayesian inference in normal data models using gibbs sampling abstract the use of the gibbs sampler as a method for calculating bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models including variance components unordered and ordered means hierarchical growth curves and missing data in a crossover trial in all cases the approach is straightforward to specify distributionally and to implement computationally with output readily adapted for required inference summaries

a general approach to analyzing epidemiologic data that contain misclassification errors misclassification is a common source of bias and reduced efficiency in the analysis of discrete data several methods have been proposed to adjust for misclassification using information on error rates i gathered by resampling the study population ii gathered by sampling a separate population or iii assumed a priori we present unified methods for incorporating these types of information into analyses based on log linear models and maximum likelihood estimation general variance expressions are developed examples from epidemiologic studies are used to demonstrate the proposed methodology

quadrature formulas on the unit circle based on rational functions abstract quadrature formulas on the unit circle were introduced by jones in 1989 on the other hand bultheel also considered such quadratures by giving results concerning error and convergence in other recent papers a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well known szego polynomials in this paper these quadratures are again analyzed and results about convergence given furthermore an application to the poisson integral is also made

convergence of orthogonal rational functions outside of the unit circle rational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied we study the asymptotic behaviors for the orthogonal rational functions outside of the unit circle

strong stieltjes distributions and orthogonal laurent polynomials with applications to quadratures and pade approximation starting from a strong stieltjes distribution general sequences of orthogonal laurent polynomials are introduced and sonic of their most relevant algebraic properties are studied from this perspective the connection between certain quadrature formulas associated with the distribution and two point pade approximants to the stieltjes transform of is revisited finally illustrative numerical examples are discussed

rates of convergence of multipoint rational approximants and quadrature formulas on the unit circle abstract in this paper multipoint rational approximants to the riesz herglotz transform of a borel measure supported on are considered we give estimates for the rate of convergence of these approximants these estimates are obtained from the asymptotic behaviour of a sequence of rational functions orthogonal with respect to and having prescribed poles 1 k where k 1 is a sequence of complex numbers compactly contained in the open unit disk as an application we also give the rate of convergence of rational szego quadrature formulas for integrals with respect to

conditional independence models for epidemiological studies with covariate measurement error we construct a unifying representation of the structure of measurement error problems with particular reference to situations commonly encountered in epidemiological studies and outline how estimation of the parameters of interest can be carried out in a bayesian framework using gibbs sampling we show how this approach can be implemented for designs involving continuous measurement errors assessed through a validation substudy and discuss our results on simulated data

bayesian analysis of misclassified binary data from a matched case control study with a validation sub study bayesian methods are proposed for analysing matched case control studies in which a binary exposure variable is sometimes measured with error but whose correct values have been validated for a random sample of the matched case control sets three models are considered model 1 makes few assumptions other than randomness and independence between matched sets while models 2 and 3 are logistic models with model 3 making additional distributional assumptions about the variation between matched sets with models 1 and 2 the data are examined in two stages the first stage analyses data from the validation sample and is easy to perform the second stage analyses the main body of data and requires mcmc methods all relevant information is transferred between the stages by using the posterior distributions from the first stage as the prior distributions for the second stage with model 3 a hierarchical structure is used to model the relationship between the exposure probabilities of the matched sets which gives the potential to extract more information from the data all the methods that are proposed are generalized to studies in which there is more than one control for each case the bayesian methods and a maximum likelihood method are applied to a data set for which the exposure of every patient was measured using both an imperfect measure that is subject to misclassification and a much better measure whose classifications may be treated as correct to test methods the latter information was suppressed for all but a random sample of matched sets

quadrature on the half line and two point pade approximants to stieltjes functions ii convergence abstract let be a distribution function on a b 0 a c k b a d x exist for all the integers k the main course of the paper is to give convergence results both for sequences of two point pade approximants to the cauchy transform of the distribution stieltjes function and for quadrature formulas based on exactly integrating laurent polynomials the case when d x k x dx k x being a possibly complex function is also considered and estimates of the rate of convergence are given

orthogonal rational functions on the real half line with poles in 0 the main objective is to generalize previous results obtained for orthogonal laurent polynomials and their application in the context of stieltjes moment problems to the multipoint case the measure of orthogonality is supposed to have support on 0 while the orthogonal rational functions will have poles that are assumed to be in the neighborhood of 0 and in this way orthogonal laurent polynomials will be a special case obtained when all the poles are at 0 and we shall introduce the restrictions on the measure and the locations of the poles gradually and derive recurrence relations christoffel darboux relations and the solution of the rational stieltjes moment problem under appropriate conditions

three partition refinement algorithms we present improved partition refinement algorithms for three problems lexicographic sorting relational coarsest partition and double lexical ordering our double lexical ordering algorithm uses a new efficient method for unmerging two sorted sets

semiparametric estimation in logistic measurement error models summary we describe semiparametric estimation and inference in a logistic regression model with measurement error in the predictors the particular measurement error model consists of a primary data set in which only the response y and a fallible surrogate w of the true predictor x are observed plus a smaller validation data set for which y x w are observed except for the underlying assumption of a logistic model in the true predictor no parametric distributional assumption is made about the true predictor or its surrogate we develop a semiparametric parameter estimate of the logistic regression parameter which is asymptotically normally distributed and computationally feasible the estimate relies on kernel regression techniques for scalar predictors by a detailed analysis of the mean squared error of the parameter estimate we obtain a representation for an optimal bandwidth

on the convergence of certain gauss type quadrature formulas for unbounded intervals we consider the convergence of gauss type quadrature formulas for the integral f 0 f x x dx where w is a weight function on the half line 0 the n point gauss type quadrature formulas are constructed such that they are exact in the set of laurent polynomials p q 1 k p q 1 p a k x k where p p n is a sequence of integers satisfying 0 p n 2n and q q n 2n p n it is proved that under certain carleman type conditions for the weight and when p n or q n goes to oo then convergence holds for all functions f for which fw is integrable on 0 some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical gauss quadrature formulas for the half line

quadrature on the half line and two point pade approximants to stieltjes functions subtitle part iii the unbounded case abstract let be a general absolutely continuous measure possibly complex supported on 0 let f z denote its cauchy transform in this paper we prove under suitable conditions the convergence of two point pade type approximants to f and of the associated quadrature formulas j 1 n a j x j to the intergral 0 f x d x these quadrature formulas can be of gaussian or of interpolatory type estimates for the rate of convergence are also included

covariate measurement error in generalized linear models summary the em algorithm is used to obtain estimators of regression coefficients for generalized linear models with canonical link when normally distributed covariates are masked by normally distributed measurement errors by casting the true covariates as missing data the em procedure suggests an iterative scheme in which each cycle consists of an e step requiring the computation of approximate first and second conditional moments of the true covariates given the observed data followed by an m step in which regression parameters are updated by iteratively reweighted least squares based on these approximations the proposed procedure is compared numerically with the exact maximum likelihood solution obtained by using gaussian quadrature instead of the approximations in the e step of the em algorithm and with alternative estimators for simple logistic regression with measurement error the results for the proposed procedure are encouraging a procedure is proposed in this paper for estimating regression coefficients in generalized linear models when one or more of the covariates is measured with error this work is related to the paper by carroll et al 1984 in which maximum likelihood estimates are considered for the structural logistic regression model these authors provide estimates for the computationally simpler probit regression model and suggest that the structural logistic regression can be done in principle this paper follows up on that suggestion but differs in detail from similar work by armstrong 1985 other procedures with related aims have been suggested by stefanski carroll 1985 wolter fuller 1982 and prentice 1982 for a quick introduction to the proposed method it is convenient to display the naive estimator which ignores measurement error in its standard computational form if yi is an observed response variable and xi is the observed covariate then for the model which equates the canonical parameter of the distribution of yi to xtf3 the maximum likelihood estimator of f3 can be obtained by iteratively reweighted least squares the estimate of f3 after s 1 cycles is given by

adjusting for measurement error to assess health effects of variability in biomarkers longitudinal studies of health effects often relate individuals biomarker levels to disease progression repeated measurements also provide an opportunity to assess within individual biomarker variability and it is reasonable to postulate that this measure might provide additional information about a particular outcome variable given the existing precedent for application of adjustment methods to account for measurement error in subject specific average levels of a covariate this concept motivates the application of such methods to incorporate variability as well in this paper we investigate the nature of the relationship between the decline of cd4 cell count induced by infection with human immunodeficiency virus and cd4 level and variability prior to infection we first describe the distribution of repeated cd4 measurements prior to infection using a model that accounts both for random average levels and random subject specific variance components based on this model we define true unobservable random variables that correspond to prior level and stability we perform a linear regression analysis using these latent variables as covariates by means of a full maximum likelihood approach we compare the resulting parameter estimates with those based on regressions employing sample based estimates of pre infection levels and variances and empirical bayes estimates of these quantities although the final inferences are similar to those based on the unadjusted analysis we find that the magnitude of association with prior level decreases while that with prior stability increases stratified analyses indicate that smoking status affects the relationship between prior cd4 level and initial cd4 decline we point out advantages associated with the maximum likelihood approach in this particular application

strong and weak convergence of orthogonal polynomials le cercle unite fonctionnelles continues sur des mesures resultats principaux sur le cercle unite corollaires de convergence faible les parties absolument continues et les parties singulieres d une mesure integrabilite uniforme et inegalite de turan series orthogonales avec trous la ligne reelle convergence forte convergence faible convergence forte pour des moyennes arithmetiques

covariate measurement error in logistic regression abstract in a logistic regression model when covariates are subject to measurement error the naive estimator obtained by regressing on the observed covariates is asymptotically biased this document introduces a bias adjusted estimator and two estimators appropriate for normally distributed measurement errors a functional maximum likelihood estimator and an estimator which exploits the consequences of sufficiency the four proposals are studied asymptotically under conditions which are appropriate when the measurement error is small a small monte carlo study illustrates the superiority of the measurement error estimators in certain situations additional keywords mathematical models

interpolatory quadrature formulas on the unit circle for chebyshev weight functions in this paper interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands a comparison is made with szego quadrature formulas finally an application to the interval 1 1 is also carried out

on orthogonal polynomials with respect to varying measures on the unit circle let io d i be a system of orthonormal polynomials on the unit circle with respect to d l and yi m dli be a system of orthonormal polynomials on the unit circle with respect to the varying measures d u l iw z 2 z eio where wn z is a sequence of polynomials deg w n whose zeros wn i n lie in izi 1 the asymptotic behavior of the ratio of the two systems on and outside the unit circle is obtained

rational functions associated with double infinite sequences of complex numbers abstract let k be a given double infinite sequence of complex numbers by defining a linear functional on the space of the laurent polynomials certain rational functions are first constructed and some algebraic properties studied the hermitian case i e k k k z is separately considered and it is shown how the theory of polynomials orthogonal on the unit circle can be used in order to prove geometric convergence for sequences such as these rational functions

moment problems and orthogonal functions abstract the caratheodory coefficient problem for an infinite sequence c n can be formulated as follows find a caratheodory function f z i e an analytic function mapping the unit disk d z z into the right half plane p w re w 0 whose taylor coefficients at a fixed points in d for convenience the origin are the c n this problem is equivalent to the trigonometric moment problem find a measure on such that e i n d c n for n 0 1 2 these problems are closely related to the theory of szego polynomials i e orthogonal polynomials on the unit circle t z z 1 let n be an arbitrary given sequence of not necessarily distinct points in d and let w n be a given sequence of numbers the nevanlinna pick problem for this situation is to find a caratheodory function f z with the interpolation property f n w n for all n when points n are repeated the interpolation property involves the appropriate number of taylor coefficients also this problem is connected with the problem of finding a measure generating moments the nevanlinna pick problem is related to a theory of rational functions which are orthogonal on t with poles at the points 1 n this relationship is analogous to and generalizes the relationship between the caratheodory coefficient problem or trigonometric moment problem and polynomials orthogonal on t the situation when points n are given on the boundary t or d is also briefly discussed

convergence analysis of the extended krylov subspace method for the lyapunov equation the extended krylov subspace method has recently arisen as a competitive method for solving large scale lyapunov equations using the theoretical framework of orthogonal rational functions in this paper we provide a general a priori error estimate when the known term has rank one special cases such as symmetric coefficient matrix are also treated numerical experiments confirm the proved theoretical assertions

szego functions and multipoint pade approximation abstract rational functions analogous to szego polynomials are introduced and their application in multipoint pade approximation is discussed

generalized szego theory in frequency analysis abstract a sequence of positive borel measures with infinite support on the unit circle is given converging in the weak star topology to a measure with support consisting of a finite number n 0 of mass points the sequences of monic rational functions orthogonal with respect to inner products determined by the measures are studied with the focus being on the asymptotic behavior of the zeros the main result is twofold for a fixed n n 0 n 0 of the zeros interesting zeros tend to the mass points of the limiting measure while the remaining n n 0 zeros uninteresting zeros may not converge at all however by considering subsequences limits may be obtained and the remaining zeros and their limits will be located in a closed disk z r

orthogonal laurent polynomials corresponding to certain strong stieltjes distributions with applications to numerical quadratures in this paper we shall be mainly concerned with sequences of orthogonal laurent polynomials associated with a class of strong stieltjes distributions introduced by a s ranga algebraic properties of certain quadratures formulae exactly integrating laurent polynomials along with an application to estimate weighted integrals on 1 1 with nearby singularities are given finally numerical examples involving interpolatory rules whose nodes are zeros of orthogonal laurent polynomials are also presented

orthogonal rational functions with poles on the unit circle let n be a sequence of not necessarily distinct points on the unit circle t z c z 1 set ln span 1 1 1 1 n l n 0ln where we have used the notation n nk 1 z k let m be a positive linear functional defined on the space l l with m r real for functions that are real on t define r s m r z s 1 z for r s l in particular if m is given as m r r ei d for some measure then r s r ei s ei d let the orthogonal system n be obtained from 1 n by orthogonalization three term recurrence relations quadrature formulas moment theory and interpolation properties connected with the functional m and the system n are discussed

improved regression calibration the likelihood for generalized linear models with covariate measurement error cannot in general be expressed in closed form which makes maximum likelihood estimation taxing a popular alternative is regression calibration which is computationally efficient at the cost of inconsistent estimation we propose an improved regression calibration approach a general pseudo maximum likelihood estimation method based on a conveniently decomposed form of the likelihood it is both consistent and computationally efficient and produces point estimates and estimated standard errors which are practically identical to those obtained by maximum likelihood simulations suggest that improved regression calibration which is easy to implement in standard software works well in a range of situations

a note on certain generalizations of the midpoint rule abstract in this paper we shall be concerned with some interesting generalizations of the so called midpoint rule to weighted integrals introduced by jagerman 1966 and stetter 1968 first the convergence of the stetter rule on the class of the riemann integrable functions is proved and numerical experiments are also carried out secondly the introduction of a correction factor in order to accelerate the asymptotic rate of convergence is discussed

on the convergence of two point pade type approximants for a given function f holomorphic in a domain omega containing the origin and the point at infinity rational interpolants with prescribed poles are constructed and their convergence properties analyzed numerical results are also given

an indeterminate rational moment problem and caratheodory functions let a n n 1 be a sequence of points in the open unit disk in the complex plane and letb 0 1andb n z email protected email protected email protected a k a email protected email protected z n 1 2 a email protected a k 1 when a k 0 we put l span b n n 0 1 2 and we consider the following moment problem given a positive definite hermitian inner product in l find all positive borel measures n on email protected p such that email protected p pf e i q g e i q email protected q forf email protected l we assume that this moment problem is indeterminate under some additional condition on the a n we will describe a one to one correspondence between the collection of all solutions to this moment problem and the collection of all caratheodory functions augmented by the constant

rational quadrature formulae on the unit circle with arbitrary poles interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered the poles are prescribed under the only restriction of not lying on the unit circle a computable upper bound of the error is obtained which is valid for any choice of poles arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle a number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound also a comparison is made with other error bounds appearing in the literature

a measurement error model for binary and ordinal regression exposure assessment poses special problems in air pollution epidemiology this paper proposes a probit regression model for binary and ordinal outcomes that uses exposure validation information to develop estimates for the coefficient of the true exposure when only the inaccurate surrogate measure of exposure is available for the individuals in the health study this method is closely related to recently developed measurement error methods and is based on the assumption that the outcome and the surrogate exposure are conditionally independent given the true exposure a test statistic is proposed for checking this conditional independence assumption when more than one surrogate is available and an interpretation of the coefficient estimate is provided in the event that the assumption is violated the methods are applied to an example involving nitrogen dioxide exposure and wheeze in children

a density problem for orthogonal rational functions abstract let n n 1 be a sequence of points in the open unit disk in the complex plane and let b 0 1 and b n z k 0 n k k k z 1 k z n 1 2 k k 1 when k 0 we put l span b n n 0 1 2 and we consider the following moment problem given a positive definite hermitian inner product on l l find a nondecreasing function on or a positive borel measure on such that f g f e i g e i d for f g l we give a necessary and sufficient condition called n extremality on a solution of the moment problem in order that l is dense in l2

approximations for regression with covariate measurement error abstract the effect of covariate measurement error on maximum likelihood and maximum quasilikelihood estimates mle s or mqle s of regression parameters is considered the naive estimates obtained by ignoring error are modified to account for it the modified estimates can be computed explicitly from the naive ones and have less bias for small error they are particularly useful for berkson error models examples illustrate their application to linear and nonlinear regression errors in continuous covariates and misclassification of discrete covariates the estimates are applied to data relating air pollution to respiratory disease in children analytic methods and simulations are used to compare the estimates to others that have been proposed for dealing with covariate measurement errors the basic idea is that the mle or mqle of the parameter is a function of the size of the error when 0 and there is no error is the usual mle or mqle for small an improvement over is in many cases of

covariance analysis in generalized linear measurement error models we summarize some of the recent work on the errors in variables problem in generalized linear models the focus is on covariance analysis and in particular testing for and estimation of treatment effects there is a considerable difference between the randomized and non randomized models when testing for an effect in randomized studies simple techniques exist for testing for a treatment effect in some instances such as linear and multiplicative regression simple methods exist for estimating the treatment effect in other examples such as logistic regression estimating a treatment effect requires careful attention to measurement error in non randomized studies there is no recourse to understanding and modelling measurement error in particular ignoring measurement error can lead to the wrong conclusions for example the true but unobserved data may indicate a positive effect for treatment while the observed data indicate the opposite some of the possible methods are outlined and compared

optimization of two point pade type approximants abstract in this paper we first give characterization theorems for the best two point pade type approximants 2ptas in the uniform norm secondly we consider sequences of 2ptas in a domain of the complex plane from the viewpoint of the asymptotic degree of convergence and we also give conditions for geometric convergence

quadrature and orthogonal rational functions abstract classical interpolatory or gaussian quadrature formulas are exact on sets of polynomials the szeg quadrature formulas are the analogs for quadrature on the complex unit circle here the formulas are exact on sets of laurent polynomials in this paper we consider generalizations of these ideas where the laurent polynomials are replaced by rational functions that have prescribed poles these quadrature formulas are closely related to certain multipoint rational approximants of cauchy or riesz herglotz transforms of a positive or general complex measure we consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence

solvable set hyperset contexts i some decision procedures for the pure finite case hereditarily finite sets and hypersets are characterized both as algorithmic data structures and by means of a first order axiomatization which despite being rather weak suffices to make the following two problems decidable 1 establishing whether a conjunction r of formulae of the form y1 ym y1 w1 ym wm q with q quantifier free and involving only the relators and propositional connectives and each yi distinct from all wj s is satisfiable 2 establishing whether a formula of the form y q q quantifier free is satisfiable concerning 1 an explicit decision algorithm is provided moreover significantly broad subproblems of 1 are singled out in which a classification that we call the syllogistic decomposition of r of all possible ways of satisfying the input conjunction r can be obtained automatically for one of these subproblems carrying out the decomposition results in a finite family of syntactic substitutions that generate the space of all solutions to r in this sense one has a unification algorithm concerning 2 a technique is provided for reducing it to a subproblem of 1 for which a decomposition method is available the algorithmic complexity of the problems under study is highlighted a generalization of the decidability results to a theory where sets are blended with free herbrand functors is announced 1996 john wiley sons inc

on the convergence of quadrature formulas for complex weight functions abstract we consider quadrature formulas for complex weights and prove their convergence for a large class of functions

pade approximants and quadratures related to certain strong distributions letbe a c inversive strong distribution as de5ned in ranga numer math 68 1994 283 in this paper two point pad e approximants and quadrature formulas related to the distributionare analyzed c 2001 elsevier science b v all rights reserved msc 41a21 30e05 34e05

the effect of measurement error summary measurement error causes the distribution generating data to differ from the distribution that is of substantive interest to understand the effect of measurement error on the information produced by statistical procedures it is thus necessary to understand the distortions induced by measurement error this paper sheds light on this by providing small error variance approximations to the distribution of variates subject to contamination by measurement error these provide the basis for the analysis of the properties of estimators and tests and less formal procedures when measurement error is present and the basis for the development of tests to detect measurement error and of models which incorporate its influence

approximate quasi likelihood estimation in models with surrogate predictors abstract we consider quasi likelihood estimation with estimated parameters in the variance function when some of the predictors are measured with error we review and extend four approaches to estimation in this problem all of them based on small measurement error approximations a taxonomy of the data sets likely to be available in measurement error studies is developed an asymptotic theory based on this taxonomy is obtained and includes measurement error and berkson error models as special cases

favard theorem for reproducing kernels abstract consider for n 0 1 the nested spaces l n of rational functions of degree n at most with given poles 1 i i i 1 n let l 0 l n given a finite positive measure on the unit circle we associate with it an inner product on l by g g d suppose kn z w is the reproducing kernel for l n i e z k n z w w for all l n w then it is known that they satisfy a coupled recurrence relation in this paper we shall prove a favard type theorem which says that if you have a sequence of kernel functions kn z w which are generated by such a recurrence then there will be a measure supported on the unit circle so that kn is the reproducing kernel for ln the measure is unique under certain extra conditions on the points i

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

sequences of orthogonal laurent polynomials bi orthogonality and quadrature formulas on the unit circle in this paper the construction of orthogonal bases in the space of laurent polynomials on the unit circle is considered as an application a connection with the so called bi orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on laurent polynomials are studied

fully parametric and semi parametric regression models for common events with covariate measurement error in main study validation study designs the derivation of the likelihood function for binary data from two types of main study validation study designs where model covariates are measured with error is elaborated rather than limiting consideration to a restricted family of models with convenient mathematical properties we suggest that empirical considerations customized to the data at hand should drive model choices the joint likelihood function for the main study in which the covariates are measured with error and the validation study in which they are not is maximized and estimation and inference proceeds using standard theory although the choice of the measurement error model is driven by empirical considerations the relatively small validation study sizes typically seen may lead to misspecification resulting in bias in estimation and inference about exposure disease relationships by using a nonparametric form for the measurement error model the resulting semi parametric methods suggested by robins rotnitzky and zhao 1994 journal of the american statistical association 89 864 866 and robins hsieh and newey 1995 journal of the royal statistical society series b 57 409 424 are free from bias due to misspecification of the measurement error model trading efficiency for robustness as usual these fully and semi parametric methods are illustrated with a detailed example from a main study validation study of the health effects of occupational exposure to chemotherapeutics among pharmacists valanis et al 1993 american journal of hospital pharmacy 50 455 462 a constant prevalence ratio model for common binary events with gamma covariate measurement error is derived and empirically verified by the available data a careful reanalysis of the data taking measurement error fully into account leads to a threefold increase in the log relative risk and no loss of statistical power the semi parametric estimates are consistent with the parametric results providing reassurance that important bias due to misspecification of the measurement error model is unlikely

strong and weak convergence of orthogonal systems of rational functions on the unit circle abstract pan k strong and weak convergence of orthogonal systems of rational functions on the unit circle journal of computational and applied mathematics 46 1993 427 436 we study weak and strong convergence of rational functions that are orthonormal on the unit circle and have all their poles in a given sequence of points outside the unit disk the erd s measures have been characterized as well

joint modeling of performance and subjective reporting to assess sensitivity to drug induced sleepiness in an nasa ground study two forms of cognitive tests were evaluated in terms of their sensitivity to sleepiness induced by the drug promethazine pmz performance for the two test modes y 1 and y 2 pmz concentration and a self reported sleepiness using the karolinska sleepiness scale kss were monitored for 12 h post dose a problem arises when using kss to establish an association between true sleepiness and performance because kss scores are discrete and also because they tend to concentrate on certain values therefore we define a latent sleepiness measure x as an unobserved continuous random variable describing a subject s actual state of sleepiness under the assumption that drug concentration affects x which then affects y 1 y 2 and kss we use bayesian methods to estimate joint equations that permit unbiased comparison of the performance measures sensitivity to x the equations incorporate subject random effects and include a negativity constraint on subject specific slopes of performance with respect to sleepiness

calculi for synchrony and asynchrony abstract a calculus for distributed computation is studied based upon four combinators a central idea is an abelian group of actions which models the interfaces between components of a distributed computing agent using a notion of bisimulation congruence relations are defined over computing agents and thence an algebraic theory is derived the calculus models both synchronous and asynchronous computation in particular it is shown that the author s calculus of communicating systems 1980 which is an asynchronous model is derivable from the calculus presented here

extensions of szego s theory of rational functions orthogonal on the unit circle rational functions orthogonal on the unit circle with prescribed poles lying outside the unit circle are studied we study the asymptotic behaviors for the orthogonal rational functions when the measure does not satisfy the szego condition

orthogonality interpolation and quadratures on the unit circle and the interval 1 1 given a weight function s x on 1 1 or more generally a positive borel measure the erdos turan theorem assures the convergence in l 2 s norm to a function f of its sequence of interpolating polynomials at the zeros of the orthogonal polynomials or equivalently at the nodes of the gauss christoffel quadrature formulas associated with s in this paper we will extend this result to the nodes of the gauss radau and gauss lobatto quadrature formulas by passing to the unit circle and taking advantage of the results on interpolation by means of laurent polynomials at the zeros of certain para orthogonal polynomials with respect to the weight function w q s cos q sin q on p p as a consequence an application to the construction of certain product integration rules on finite intervals of the real line will be given several numerical experiments are finally carried out

program derivation with verified transformations a case study a program development methodology based on verified program transformations is described and illustrated through derivations of a high level bisimulation algorithm and an improved minimum state dfa algorithm certain doubts that were raised about the correctness of an initial paper and pencil derivation of the dfa minimization algorithm were laid to rest by machine checked formal proofs of the most difficult derivational steps although the protracted labor involved in designing and checking these proofs was almost overwhelming the expense was somewhat offset by a successful reuse of major portions of these proofs in particular the dfa minimization algorithm is obtained by specializing and then extending the last step in the derivation of the high level bisimulation algorithm our experience suggests that a major focus of future research should be aimed towards improving the technology of machine checkable proofs their construction presentation and reuse this paper demonstrates the importance of such a technology to the verification of programs and program transformations we believe that the utility of transformational systems to program development will ultimately rest on a practical program correctness technology 1996 john wiley sons inc

quadratures and orthogonality associated with the cayley transform abstract in this paper we are dealing with the approximate calculation of weighted integrals over the whole real line the method is based in passing to the unit circle by means of the so called cayley transform z i x i x and then making use of a szeg or interpolatory type quadrature formula on the unit circle in order to obtain a gauss like quadrature rule on the real line some properties concerning orthogonality maximal domains of validity of the quadratures and connections with certain orthogonal rational functions are presented finally some numerical experiments are also carried out

adjustment for non differential misclassification error in the generalized linear model it is well known that estimates of association between an outcome variable and a set of categorical covariates some of which are measured with misclassification tend to be biased upon application of the usual methods of estimation that ignore the classification error we propose a method to adjust for misclassification in covariates when one applies the generalized linear model in the case where one can observe some true covariates only through surrogates we combine a latent class analysis with the approach to incorporate multiple surrogates into the model we include discussion on the efficacy of repeated measurements which one can view as a special case of multiple surrogates with identical distribution we provide two examples to demonstrate the applicability of the method and the efficacy of multiple replicates for a covariate subject to misclassification in a regression framework

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

on the convergence of the rational interpolation approximant of carathe odory functions let zn m nm 1 with zn m 1 n 1 2 be an arbitrary sequence of complex numbers we study the convergence of the rational functionpn 1 qn which interpolates the caratheodory function f z 1 2 2 0u u z d t u eit at zn m 1 zn m nm 1 where d is a finite positive measure on the unit circle we also consider the relations between pn 1 qn and the orthogonal polynomials with respect to varying measures

computing orthogonal rational functions with poles near the boundary computing orthogonal rational functions is a far from trivial problem especially for poles close to the boundary of the support of the orthogonality measure in this paper we analyze some of the difficulties involved and present two different approaches for solving this problem

an application of szego quadratures to the computation of the fourier transform abstract szego quadrature formulas represent the analogue on the unit circle of the well known gauss christoffel quadratures for intervals of the real line in the present work szego formulas are revised and an application to the computation of the fourier transform given

on the convergence of rational functions orthogonal on the unit circle rational functions orthogonal on the unit circle with prescribed poles are studied we establish some convergence theorems for the orthogonal rational functions we generalize some of the results for szego polynomials to rational orthogonal functions

measurement error modelling with an approximate instrumental variable summary consider using regression modelling to relate an exposure predictor variable to a disease outcome response variable if the exposure variable is measured with error but this error is ignored in the analysis then misleading inferences can result this problem is well known and has spawned a large literature on methods which adjust for measurement error in predictor variables one theme is that the requisite assumptions about the nature of the measurement error can be stronger than what is actually known in many practical situations in particular the assumptions that are required to yield a model which is formally identified from the observable data can be quite strong the paper deals with one particular strategy for measurement error modelling namely that of seeking an instrumental variable i e a covariate s which is associated with exposure and conditionally independent of the outcome given exposure if these two conditions hold exactly then we call s an exact instrumental variable and an identified model results however the second is not checkable empirically since the actual exposure is unobserved in practice then investigators typically seek a covariate which is plausibly thought to satisfy it we study inferences which acknowledge the approximate nature of this assumption in particular we consider bayesian inference with a prior distribution that posits that s is probably close to conditionally independent of outcome given exposure we refer to this as an approximate instrumental variable assumption although the approximate instrumental variable assumption is more realistic for most applications concern arises that a non identified model may result thus the paper contrasts inferences arising from the approximate instrumental variable assumption with their exact instrumental variable counterparts with particular emphasis on the benefit of basing inferences on a more realistic model versus the cost of basing inferences on a non identified model

foreword to the proceedings of the orthoquad 2014 conference orthoquad 2014 was an international symposium on orthogonality quadrature and related topics held on january 20 24 2014 in puerto de la cruz tenerife spain it was held in memory of professor pablo gonzalez vera 1955 2012

correction of logistic regression relative risk estimates and confidence intervals for systematic within person measurement error errors in the measurement of exposure that are independent of disease status tend to bias relative risk estimates and other measures of effect in epidemiologic studies toward the null value two methods are provided to correct relative risk estimates obtained from logistic regression models for measurement errors in continuous exposures within cohort studies that may be due to either random unbiased within person variation or to systematic errors for individual subjects these methods require a separate validation study to estimate the regression coefficient lambda relating the surrogate measure to true exposure in the linear approximation method the true logistic regression coefficient beta is estimated by beta lambda where beta is the observed logistic regression coefficient based on the surrogate measure in the likelihood approximation method a second order taylor series expansion is used to approximate the logistic function enabling closed form likelihood estimation of beta confidence intervals for the corrected relative risks are provided that include a component representing error in the estimation of lambda based on simulation studies both methods perform well for true odds ratios up to 3 0 for higher odds ratios the likelihood approximation method was superior with respect to both bias and coverage probability an example is provided based on data from a prospective study of dietary fat intake and risk of breast cancer and a validation study of the questionnaire used to assess dietary fat intake

orthogonal laurent polynomials and two point pade approximants associated with dawson s integral starting from the coefficients of the power expansions for the dawson s integral both around the origin and infinity a linear functional acting on the space of laurent polynomials is defined in this paper properties of this functional are studied in connection with certain sequences of orthogonal laurent polynomials and two point pade approximants

correcting for measurement error in binary and continuous variables using replicates measurement error in exposures and confounders leads to bias in regression coefficients it is possible to adjust for this bias if true values or independent replicates are observed on a subsample we extend a method suitable for quantitative variables to the situation where both binary and quantitative variables are present binary variables with independent replicates introduce two extra problems i the error is correlated with the true value and ii the measurement error probabilities are unidentified if only two replicates are available we show that under plausible assumptions adjustment for error in binary confounders does not need to address these problems the regression coefficient for a binary exposure is overadjusted if methods for continuous variables are used correct adjustment is possible either if three replicates are available or if further assumptions can be made otherwise bounds can be put on the correctly adjusted value and these bounds are reasonably close together if the exposure has prevalence near 0 5

modelling complexity applications of gibbs sampling in medicine summary we review applications of gibbs sampling in medicine involving longitudinal spatial covariate measurement and survival models applications in immunology pharmacology transplantation cancer screening industrial epidemiology and genetic epidemiology are discussed

partitioning a graph in o a log2 v abstract we present an o a log 2 v algorithm to partition a graph where v is the set of vertices and a the set of arcs extending hopcroft s fast partitioning algorithm

regression analysis with covariates that have heteroscedastic measurement error we consider the estimation of the regression of an outcome y on a covariate x where x is unobserved but a variable w that measures x with error is observed a calibration sample that measures pairs of values of x and w is also available we consider calibration samples where y is measured internal calibration and not measured external calibration one common approach for measurement error correction is regression calibration rc which substitutes the unknown values of x by predictions from the regression of x on w estimated from the calibration sample an alternative approach is to multiply impute the missing values of x given y and w based on an imputation model and then use multiple imputation mi combining rules for inferences most of current work assumes that the measurement error of w has a constant variance whereas in many situations the variance varies as a function of x we consider extensions of the rc and mi methods that allow for heteroscedastic measurement error and compare them by simulation the mi method is shown to provide better inferences in this setting we also illustrate the proposed methods using a data set from the biocycle study

quadratures associated with pseudo orthogonal rational functions on the real half line with poles in 0 we consider a positive measure on 0 and a sequence of nested spaces 0 1 2 of rational functions with prescribed poles in 0 let f k k 0 with f 0 0 and f k k k 1 k 1 2 be the associated sequence of orthogonal rational functions the zeros of f n can be used as the nodes of a rational gauss quadrature formula that is exact for all functions in n n 1 a space of dimension 2n quasi and pseudo orthogonal functions are functions in n that are orthogonal to some subspace of n 1 both of them are generated from f n and f n 1 and depend on a real parameter t their zeros can be used as the nodes of a rational gauss radau quadrature formula where one node is fixed in advance and the others are chosen to maximize the subspace of n n 1 where the quadrature is exact the parameter t is used to fix a node at a preassigned point the space where the quadratures are exact has dimension 2n 1 in both cases but it is in n 1 n 1 in the quasi orthogonal case and it is in n n 2 in the pseudo orthogonal case

quadrature on the half line and two point pade approximants to stieltjes functions part i algebraic aspects abstract let f z be a stieltjes function with asymptotic expansions l0 and l at z 0 and z respectively let k n denote the two point pade approximant of type m n which matches k of the coefficients of the series l0 and which takes its remaining interpolation conditions from l we discuss in this paper the algebraic aspects of this problem we shall emphasize the relation between quadrature formulas and two point pade approximants and derive expressions for the error in a subsequent paper we shall consider the convergence aspects of these approximants for example the positivity of the error which is obtained here will result in the monotonic convergence of certain sequences of two point pade approximants a property which is well known in the case of classical pade approximants

spatial statistics and bayesian computation on wednesday may 6th 1992 professor b w silverman in the chair summary markov chain monte carlo mcmc algorithms such as the gibbs sampler have provided a bayesian inference machine in image analysis and in other areas of spatial statistics for several years founded on the pioneering ideas of ulf grenander more recently the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a markov random field has opened the way to the use of mcmc in general bayesian computation in this paper we trace the early development of mcmc in bayesian inference review some recent computational progress in statistical physics based on the introduction of auxiliary variables and discuss its current and future relevance in bayesian applications we briefly describe a simple mcmc implementation for the bayesian analysis of agricultural field experiments with which we have some practical experience

a bayesian approach to prospective binary outcome studies with misclassification in a binary risk factor misclassification in a binary exposure variable within an unmatched prospective study may lead to a biased estimate of the disease exposure relationship it usually gives falsely small credible intervals because uncertainty in the recorded exposure is not taken into account when there are several other perfectly measured covariates interrelationships may introduce further potential for bias bayesian methods are proposed for analysing binary outcome studies in which an exposure variable is sometimes misclassified but its correct values have been validated for a random subsample of the subjects this bayesian approach can model relationships between explanatory variables and between exploratory variables and the probabilities of misclassification three logistic regressions are used to relate disease to true exposure misclassified exposure to true exposure and true exposure to other covariates credible intervals may be used to make decisions about whether certain parameters are unnecessary and hence whether the model can be reduced in complexity in the disease exposure model for parameters representing coefficients related to perfectly measured covariates the precision of posterior estimates is only slightly lower than would be found from data with no misclassification for the risk factor which has misclassification the estimates of model coefficients obtained are much less biased than those with misclassification ignored

asymptotic properties of zeros of orthogonal rational functions abstract a sequence x m m 0 1 2 of real numbers gives rise to an absolutely continuous measure n t on the unit circle given by d t n d 12 n 1 m 0 x m t m e im 2 where n is a natural number t 0 1 each measure n t determines a sequence n n t of monic orthogonal rational functions with prescribed poles outside the closed unit disk in particular these rational functions may be polynomials a measure 0 with support consisting of the points 1 n0 on the unit circle is given let n n0 under a suitable condition on weak convergence of the measures and with a proper ordering of the zeros zk n n t of n n t it can be shown that lim t 1 lim t z k n n t k k 1 n 0 and that there exists a positive number n z k n n t n for k n0 1 n for all n and all t 0 1 the theory is applied to the problem of determining unknown frequencies in a trigonometric signal x m ij iajeimwj m 0 1 2