a simple converse of burnashev s reliability function in a remarkable paper published in 1976 burnashev determined the reliability function of variable length block codes over discrete memoryless channels dmcs with feedback subsequently an alternative achievability proof was obtained by yamamoto and itoh via a particularly simple and instructive scheme their idea is to alternate between a communication and a confirmation phase until the receiver detects the codeword used by the sender to acknowledge that the message is correct we provide a converse that parallels the yamamoto itoh achievability construction besides being simpler than the original the proposed converse suggests that a communication and a confirmation phase are implicit in any scheme for which the probability of error decreases with the largest possible exponent the proposed converse also makes it intuitively clear why the terms that appear in burnashev s exponent are necessary

bayesian active learning with non persistent noise we consider the problem of noisy bayesian active learning where we are given a finite set of functions mathcal h a sample space mathcal x and a label set mathcal l one of the functions in mathcal h assigns labels to samples in mathcal x the goal is to identify the function that generates the labels even though the result of a label query on a sample is corrupted by independent noise more precisely the objective is to declare one of the functions in mathcal h as the true label generating function with high confidence using as a few label queries as possible by selecting the queries adaptively and in a strategic manner previous work in bayesian active learning considers generalized binary search and its variants for the noisy case and analyzes the number of queries required by these sampling strategies in this paper we show that these schemes are in general suboptimal instead we propose and analyze an alternative strategy for sample collection our sampling strategy is motivated by a connection between bayesian active learning and active hypothesis testing and is based on querying the label of a sample which maximizes the extrinsic jensen shannon divergence at each step we provide upper and lower bounds on the performance of this sampling strategy and show that these bounds are better than the previous bounds in the literature

moment reconstruction and moment adjusted imputation when exposure is generated by a complex nonlinear random effects modeling process for the classical homoscedastic measurement error model moment reconstruction freedman et al 2004 2008 and moment adjusted imputation thomas et al 2011 are appealing computationally simple imputation like methods for general model fitting like classical regression calibration the idea is to replace the unobserved variable subject to measurement error with a proxy that can be used in a variety of analyses moment reconstruction and moment adjusted imputation differ from regression calibration in that they attempt to match multiple features of the latent variable and also to match some of the latent variable s relationships with the response and additional covariates in this note we consider a problem where true exposure is generated by a complex nonlinear random effects modeling process and develop analogues of moment reconstruction and moment adjusted imputation for this case this general model includes classical measurement errors berkson measurement errors mixtures of berkson and classical errors and problems that are not measurement error problems but also cases where the data generating process for true exposure is a complex nonlinear random effects modeling process the methods are illustrated using the national institutes of health aarp diet and health study where the latent variable is a dietary pattern score called the healthy eating index 2005 we also show how our general model includes methods used in radiation epidemiology as a special case simulations are used to illustrate the methods

regressions with berkson errors in covariates a nonparametric approach this paper establishes that so called instrumental variables enable the identification and the estimation of a fully nonparametric regression model with berkson type measurement error in the regressors an estimator is proposed and proven to be consistent its practical performance and feasibility are investigated via monte carlo simulations as well as through an epidemiological application investigating the effect of particulate air pollution on respiratory health these examples illustrate that berkson errors can clearly not be neglected in nonlinear regression models and that the proposed method represents an effective remedy

semiparametric regression modeling with mixtures of berkson and classical error with application to fallout from the nevada test site nous construisons des methodes bayesiennes pour une modelisation semi parametrique d une fonction de regression monotone quand les predicteurs sont mesures avec une erreur classique une erreur de berkson ou les deux de telles methodes demandent une distribution pour le predicteur non observe variable latente distribution que nous modelisons aussi de facon semi parametrique de telles combinaisons de methodes semi parametriques pour la reponse a des doses aussi bien que pour la distribution de la variable latente n ont pas ete etudiees dans la litterature sur les erreurs de mesure quelle que soit la forme de l erreur de mesure de plus nos methodes proposent une nouvelle approche des problemes ou l erreur de mesure combine une composante de berkson et une composante classique ces methodes sont generales mais nous les developpons autour d une application particuliere qui est l etude des maladies de la thyroide en relation avec les radiations venant du site de test du nevada nous utilisons ces donnees pour illustrer nos methodes lesquelles suggerent une estimation ponctuelle moyenne a posteriori du risque relatif a des doses pres de deux fois plus elevee que celle obtenue par les analyses precedentes mais suggere aussi une bien plus grande incertitude sur le risque relatif

active sequential hypothesis testing consider a decision maker who is responsible to dynamically collect observations so as to enhance his information about an underlying phenomena of interest in a speedy manner while accounting for the penalty of wrong declaration due to the sequential nature of the problem the decision maker relies on his current information state to adaptively select the most informative sensing action among the available ones in this paper using results in dynamic programming lower bounds for the optimal total cost are established the lower bounds characterize the fundamental limits on the maximum achievable information acquisition rate and the optimal reliability moreover upper bounds are obtained via an analysis of two heuristic policies for dynamic selection of actions it is shown that the first proposed heuristic achieves asymptotic optimality where the notion of asymptotic optimality due to chernoff implies that the relative difference between the total cost achieved by the proposed policy and the optimal total cost approaches zero as the penalty of wrong declaration hence the number of collected samples increases the second heuristic is shown to achieve asymptotic optimality only in a limited setting such as the problem of a noisy dynamic search however by considering the dependency on the number of hypotheses under a technical condition this second heuristic is shown to achieve a nonzero information acquisition rate establishing a lower bound for the maximum achievable rate and error exponent in the case of a noisy dynamic search with size independent noise the obtained nonzero rate and error exponent are shown to be maximum

mathieu subspaces of associative algebras abstract motivated by the mathieu conjecture mathieu 1997 m the image conjecture zhao 2010 z3 and the well known jacobian conjecture keller 1939 k see also bass et al 1982 bcw and van den essen 2000 e1 the notion of mathieu subspaces as a natural generalization of the notion of ideals has been introduced recently in zhao 2010 z4 for associative algebras in this paper we first study algebraic elements in the radicals of mathieu subspaces of associative algebras over fields and prove some properties and characterizations of mathieu subspaces with algebraic radicals we then give some characterizations or classifications for strongly simple algebras the algebras with no non trivial mathieu subspaces over arbitrary commutative rings and for quasi stable algebras the algebras all of whose subspaces that do not contain the identity element of the algebra are mathieu subspaces over arbitrary fields furthermore co dimension one mathieu subspaces and the minimal non trivial mathieu subspaces of the matrix algebras over fields are also completely determined

a new method for dealing with measurement error in explanatory variables of regression models we introduce a new method moment reconstruction of correcting for measurement error in covariates in regression models the central idea is similar to regression calibration in that the values of the covariates that are measured with error are replaced by adjusted values in regression calibration the adjusted value is the expectation of the true value conditional on the measured value in moment reconstruction the adjusted value is the variance preserving empirical bayes estimate of the true value conditional on the outcome variable the adjusted values thereby have the same first two moments and the same covariance with the outcome variable as the unobserved true covariate values we show that moment reconstruction is equivalent to regression calibration in the case of linear regression but leads to different results for logistic regression for case control studies with logistic regression and covariates that are normally distributed within cases and controls we show that the resulting estimates of the regression coefficients are consistent in simulations we demonstrate that for logistic regression moment reconstruction carries less bias than regression calibration and for case control studies is superior in mean square error to the standard regression calibration approach finally we give an example of the use of moment reconstruction in linear discriminant analysis and a nonstandard problem where we wish to adjust a classification tree for measurement error in the explanatory variables

covariate measurement errors and parameter estimation in a failure time regression model summary estimation in cox s failure time regression model is considered when the regression vector is subject to measurement error a hazard function model is induced for the failure rate given the measured covariate and a partial likelihood function is derived for the relative risk parameters this partial likelihood function may involve the baseline hazard function as well as the regression parameter but useful inference techniques arise for testing whether the regression parameter equals zero and for more general inferences in important special cases explicit consideration is given to testing equality of survival curves when group membership is subject to misclassification and to relative risk estimation with normally distributed covariates approaches to regression estimation using the overall likelihood function and a marginal likelihood function based on failure time ranks are also indicated illustration of the possible effect of covariate errors on relative risk estimation is provided

efficient regression calibration for logistic regression in main study internal validation study designs with an imperfect reference instrument an extension to the version of the regression calibration estimator proposed by rosner et al for logistic and other generalized linear regression models is given for main study internal validation study designs this estimator combines the information about the parameter of interest contained in the internal validation study with rosner et al s regression calibration estimate using a generalized inverse variance weighted average it is shown that the validation study selection model can be ignored as long as this model is jointly independent of the outcome and the incompletely observed covariates conditional at most upon the surrogates and other completely observed covariates in an extensive simulation study designed to follow a complex multivariate setting in nutritional epidemiology it is shown that with validation study sizes of 340 or more this estimator appears to be asymptotically optimal in the sense that it is nearly unbiased and nearly as efficient as a properly specified maximum likelihood estimator a modification to the regression calibration variance estimator which replaces the standard uncorrected logistic regression coefficient variance with the sandwich estimator to account for the possible misspecification of the logistic regression fit to the surrogate covariates in the main study was also studied in this same simulation experiment in this study the alternative variance formula yielded results virtually identical to the original formula a version of the proposed estimator is also derived for the case where the reference instrument available only in the validation study is imperfect but unbiased at the individual level and contains error that is uncorrelated with other covariates and with error in the surrogate instrument replicate measures are obtained in a subset of study participants in this case it is shown that the validation study selection model can be ignored when sampling into the validation study depends at most only upon perfectly measured covariates two data sets a study of fever in relation to occupational exposure to antineoplastics among hospital pharmacists and a study of breast cancer incidence in relation to dietary intakes of alcohol and vitamin a adjusted for total energy intake from the nurses health study were analysed using these new methods in these data because the validation studies contained less than 200 observations and the events of interest were relatively rare as is typical the potential improvements offered by this new estimator were not apparent

some aspects of measurement error in explanatory variables for continuous and binary regression models a simple form of measurement error model for explanatory variables is studied incorporating classical and berkson cases as particular forms and allowing for either additive or multiplicative errors the work is motivated by epidemiological problems and therefore consideration is given not only to continuous response variables but also to logistic regression models the possibility that different individuals in a study have errors of different types is also considered the relatively simple estimation procedures proposed for use with cohort data and case control data are checked by simulation under the assumption of various error structures the results show that even in situations where conventional analysis yields slope estimates that are on average attenuated by a factor of approximately 50 per cent estimates obtained using the proposed amended likelihood functions are within 5 per cent of their true values the work was carried out to provide a method for the analysis of lung cancer risk following residential radon exposure but it should be applicable to a wide variety of situations

estimation and inference for logistic regression with covariate misclassification and measurement error in main study validation study designs abstract in epidemiological studies continuous covariates often are measured with error and categorical covariates often are misclassified using the logistic regression model to represent the relationship between the binary outcome and the perfectly measured and classified covariates the model for the observed main study data is derived this derivation relies on the assumption that the error in the continuous covariates is multivariate normally distributed and uses a chain of logistic regression models to describe the misclassification processes these model assumptions are empirically verified in the validation study where the misclassified and mismeasured covariates are validated using perfectly measured and classified data the full data likelihood including contributions from both the main study and the validation study is maximized to obtain the maximum likelihood estimates for the parameters of the underlying logistic regression model and of the measurement error model and reclassification mo

a simple solution of some composition conjectures for abel equations abstract trigonometric abel differential equations appear in the study of the number of limit cycles and the center focus problem for certain families of planar polynomial systems the composition centers are a class of centers for trigonometric abel equations which have been widely studied during last years we characterize this type of centers as the ones given by couples of trigonometric polynomials for which all the generalized moments vanish they also coincide with the strongly and the highly persistent centers our result gives a simple and self contained proof of the so called composition conjecture for trigonometric abel differential equations we also prove a similar version of this result for abel equations with polynomial coefficients

abel like differential equations with a unique limit cycle abstract for the family of scalar abel like equations x a t x n b b t x m where a t l 1 k a l sin i l t cos j l t b t sin i b t cos i b t a l b r n m k i l j l i b j b z n m 2 and k 1 we characterize the existence of non trivial limit cycles periodic solutions that are isolated in the set of periodic solutions different from the trivial x t 0 in terms of n m k i l j l i b j b

logistic regression with exposure biomarkers and flexible measurement error regression calibration refined regression calibration and conditional scores estimation procedures are extended to a measurement model that is motivated by nutritional and physical activity epidemiology biomarker data available on a small subset of a study cohort for reasons of cost are assumed to adhere to a classical measurement error model while corresponding self report nutrient consumption or activity related energy expenditure data are available for the entire cohort the self report assessment measurement model includes a person specific random effect the mean and variance of which may depend on individual characteristics such as body mass index or ethnicity logistic regression is used to relate the disease odds ratio to the actual but unmeasured dietary or physical activity exposure simulation studies are presented to evaluate and contrast the three estimation procedures and to provide insight into preferred biomarker subsample size under selected cohort study configurations

center conditions at infinity for abel differential equations an abel differential equation y p x y 2 q x y 3 is said to have a center at a set a a 1 a r of complex numbers if y a 1 y a 2 y a r for any solution y x with the initial value y a 1 small enough the polynomials p q are said to satisfy the polynomial composition condition on a if there exist polynomials p q and w such that p p and q q are representable as p x p w x q x q w x and w a 1 w a 2 w a r we show that for wide ranges of degrees of p and q restricted only by certain assumptions on the common divisors of these degrees the composition condition provides a very accurate approximation of the center one up to a finite number of configurations not accounted for to our best knowledge this is the first general i e not restricted to small degrees of p and q or to a very special form of these polynomials result in the center problem for abel equations as an important intermediate result we show that at infinity according to an appropriate projectivization of the parameter space the center conditions are given by a system of the moment equations of the form as a1 p k q 0 s 2 r k 0 1

feedback in the non asymptotic regime without feedback the backoff from capacity due to non asymptotic blocklength can be quite substantial for blocklengths and error probabilities of interest in many practical applications in this paper novel achievability bounds are used to demonstrate that in the non asymptotic regime the maximal achievable rate improves dramatically thanks to variable length coding and feedback for example for the binary symmetric channel with capacity 1 2 the blocklength required to achieve 90 of the capacity is smaller than 200 compared to at least 3100 for the best fixed blocklength code even with noiseless feedback virtually all the advantages of noiseless feedback are shown to be achievable even if the feedback link is used only to send a single signal informing the encoder to terminate the transmission stop feedback it is demonstrated that the non asymptotic behavior of the fundamental limit depends crucially on the particular model chosen for the end of packet control signal fixed blocklength codes and related questions concerning communicating with a guaranteed delay are discussed in which situation feedback is demonstrated to be almost useless even non asymptotically

algebraic particular integrals integrability and the problem of the center in this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields we first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center these curves control the changes in the systems as parameters vary the bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered namely the real four dimensional projective space next we consider the known four algebraic conditions for the center for quadratic vector fields one of them says that the system is hamiltonian a condition which has a clear geometric meaning we determine the geometric meaning of the remaining other three algebraic conditions i ii iii

uniform upper bounds for the cyclicity of the zero solution of the abel differential equation abstract given two polynomials p q we consider the following question how large can the index of the first non zero moment m k a b p k q be assuming the sequence is not identically zero the answer k to this question is known as the moment bautin index and we provide the first general upper bound k 2 deg q 3 deg p 1 2 the proof is based on qualitative analysis of linear odes applied to cauchy type integrals of certain algebraic functions the moment bautin index plays an important role in the study of bifurcations of periodic solution in the polynomial abel equation y p y 2 e q y 3 for p q polynomials and e 1 in particular our result implies that for p satisfying a well known generic condition the number of periodic solutions near the zero solution does not exceed 5 deg q 3 deg 2 p this is the first such bound depending solely on the degrees of the abel equation

the geometry of generalized binary search this paper investigates the problem of determining a binary valued function through a sequence of strategically selected queries the focus is an algorithm called generalized binary search gbs gbs is a well known greedy algorithm for determining a binary valued function through a sequence of strategically selected queries at each step a query is selected that most evenly splits the hypotheses under consideration into two disjoint subsets a natural generalization of the idea underlying classic binary search this paper develops novel incoherence and geometric conditions under which gbs achieves the information theoretically optimal query complexity i e given a collection of n hypotheses gbs terminates with the correct function after no more than a constant times logn queries furthermore a noise tolerant version of gbs is developed that also achieves the optimal query complexity these results are applied to learning halfspaces a problem arising routinely in image processing and machine learning

how many queries are needed to learn we investigate the query complexity of exact learning in the membership and proper equivalence query model we give a complete characterization of concept classes that are learnable with a polynomial number of polynomial sized queries in this model we give applications of this characterization including results on learning a natural subclass of dnf formulas and on learning with membership queries alone query complexity has previously been used to prove lower bounds on the time complexity of exact learning we show a new relationship between query complexity and time complexity in exact learning if any honest class is exactly and properly learnable with polynomial query complexity but not learnable in polynomial time then p np in particular we show that an honest class is exactly polynomial query learnable if and only if it is learnable using an oracle for p 4

nonparametric methods for solving the berkson errors in variables problem summary it is common in errors in variables problems in regression to assume that the errors are incurred after the experiment in that the observed value of the explanatory variable is an independent perturbation of its true value however if the errors are incurred before the experiment then the true value of the explanatory variable equals a perturbation of its observed value this is the context of the berkson model which is increasingly attracting attention in parametric and semiparametric settings we introduce and discuss nonparametric techniques for analysing data that are generated by the berkson model our approach permits both random and regularly spaced values of the target doses in the absence of data on dosage error it is necessary to propose a distribution for the latter but we show numerically that our method is robust against that assumption the case of dosage error data is also discussed a practical method for smoothing parameter choice is suggested our techniques for errors in variables regression are shown to achieve theoretically optimal convergence rates

a semiparametric transformation approach to estimating usual daily intake distributions abstract the distribution of usual intakes of dietary components is important to individuals formulating food policy and to persons designing nutrition education programs the usual intake of a dietary component for a person is the long run average of daily intakes of that component for that person because it is impossible to directly observe usual intake for an individual it is necessary to develop an estimator of the distribution of usual intakes based on a sample of individuals with a small number of daily observations on a subsample of the individuals daily intake data for individuals are nonnegative and often very skewed also there is large day to day variation relative to the individual to individual variation and the within individual variance is correlated with the individual means we suggest a methodology for estimating usual intake distributions that allows for varying degrees of departure from normality and recognizes the measurement error associated with one day dietary intakes the est

an explicit expression for the first return map in the center problem abstract the classical center focus problem posed by h poincare in 1880 s asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point a center in this paper we present a method allowing for the first time to obtain an explicit expression for the first return map in the center problem

extrinsic jensen shannon divergence applications to variable length coding this paper considers the problem of variable length coding over a discrete memoryless channel with noiseless feedback this paper provides a stochastic control view of the problem whose solution is analyzed via a newly proposed symmetrized divergence termed extrinsic jensen shannon ejs divergence it is shown that strictly positive lower bounds on ejs divergence provide nonasymptotic upper bounds on the expected code length this paper presents strictly positive lower bounds on ejs divergence and hence nonasymptotic upper bounds on the expected code length for the following two coding schemes 1 variable length posterior matching and 2 maxejs coding scheme that is based on a greedy maximization of the ejs divergence as an asymptotic corollary of the main results this paper also provides a rate reliability test variable length coding schemes that satisfy the condition s of the test for parameters r and e are guaranteed to achieve a rate r and an error exponent e the results are specialized for posterior matching and maxejs to obtain deterministic one phase coding schemes achieving capacity and optimal error exponent for the special case of symmetric binary input channels simpler deterministic schemes of optimal performance are proposed and analyzed