conditional scores and optimal scores for generalized linear measurement error models abstract this paper studies estimation of the parameters of generalized linear models in canonical form when the explanatory vector is measured with independent normal error for the functional case i e when the explanatory vectors are fixed constants unbiased score functions are obtained by conditioning on certain sufficient statistics this work generalizes results obtained for logistic regression in the case that the explanatory vectors are independent and identically distributed with unknown distribtuion efficient score functions are obtained using the theory developed in begun et al 1983 keywords conditional score function efficient score function functional model generalized linear model measurement error structural model

a note on efficient estimation from case control data summary it is well known that it is valid to treat case control data as if they had been obtained by random sampling when estimating the odds ratio parameters of logistic regression models the result has been generalised to estimation of the regression parameters in multiplicative intercept models and it has further been pointed out that the approach may be applied to non multiplicative intercept models if the parameter space is first enlarged to include a multiplicative intercept term this note is concerned with the efficiency of the approach in epidemiological case control studies it is common to model the influence of predictors on the risk of disease through a finite dimensional regression parameter while treating the marginal distribution of the predictors as an infinite dimensional nuisance parameter see for example breslow day 1980 it is well known that case control data may be treated as if they had been obtained by random sampling when estimating the regression parameters in a logistic regression model in a multiplicative intercept model the log odds ratio of the risk of disease is modelled as a known function of the regression parameters and the predictors plus a variation independent intercept parameter it is also known that the result in breslow day 1980 holds more generally for estimating other than the intercept parameter in any multiplicative intercept model see for example prentice pyke 1979 hsieh manski mcfadden 1985 anderson 1972 wacholder weinberg 1993 robins rotnitzky zhao 1994 or scott wild 1989 with non multiplicative intercept models the data may be treated as if they had been obtained by random sampling if the model for the risk of disease is first replaced in the likelihood by a larger model that has the multiplicative intercept property the larger model is defined implicitly by treating the logit of the risk of disease as if it were equal to the logit of the risk of disease in the original model plus an unspecified intercept term see for example scott wild 1986 robins blevins 1987 breslow storer 1985 or storer wacholder breslow 1983 the results of cosslett 1981 imply that joint maximisation of the likelihood for the marginal distribution of the predictors and the regression parameters results in efficient estimates of the regression parameters in the asymptotic setting where the numbers of cases and controls tend to infinity together scott wild 1986 note that with randomly sampled data in multiplicative intercept models the case control indicators are ancillary for the regression coefficients thus joint maximisation of the semiparametric likelihood for the case control data and maximisation of the likelihood obtained by treating the data as if they were randomly sampled both result in the same estimate of the regression parameters j robins personal communication has pointed out that the argument may be applied to show efficiency of the approach in non multiplicative intercept models as well because the case control likelihood in the larger model and the case control likelihood in the smaller model are equivalent this note presents a direct calculation that demonstrates the efficiency result both for multiplicative and non multiplicative intercept models

conditional cumulant of exposure method in logistic missing covariate regression summary we consider estimation in logistic regression where some covariate variables may be missing at random satten and kupper 1993 journal of the american statistical association88 200 208 proposed estimating odds ratio parameters using methods based on the probability of exposure by approximating a partial likelihood we extend their idea and propose a method that estimates the cumulant generating function of the missing covariate given observed covariates and surrogates in the controls our proposed method first estimates some lower order cumulants of the conditional distribution of the unobserved data and then solves a resulting estimating equation for the logistic regression parameter a simple version of the proposed method is to replace a missing covariate by the summation of its conditional mean and conditional variance given observed data in the controls we note that one important property of the proposed method is that when the validation is only on controls a class of inverse selection probability weighted semiparametric estimators cannot be applied because selection probabilities on cases are zeros the proposed estimator performs well unless the relative risk parameters are large even though it is technically inconsistent small sample simulations are conducted we illustrate the method by an example of real data analysis

on robust estimation in logistic case control studies consider a logistic regression model under case control sampling prentice pyke 1979 showed that the logistic slope estimates with case control sampling may be estimated from a standard prospective logistic regression program and that the resulting standard errors are asymptotically correct since logistic regression estimates are nonrobust we propose and analyze robust estimates of the slope parameters we focus specifically on estimates which downweight observations on one of three factors i leverage ii extreme fitted values and iii likelihood of misclassification

perturbation bounds for quantum markov processes and their fixed points we investigate the stability of quantum markov processes with respect to perturbations of their transition maps in the first part we introduce a condition number that measures the sensitivity of fixed points of a quantum channel to perturbations we establish upper and lower bounds on this condition number in terms of subdominant eigenvalues of the transition map in the second part we consider quantum markov processes that converge to a unique stationary state and we analyze the stability of the evolution at finite times in this way we obtain a linear relation between the mixing time of a quantum markov process and the sensitivity of its fixed point with respect to perturbations of the transition map

a nonparametric method for dealing with mismeasured covariate data abstract mismeasurement of covariate data is a frequent problem in statistical data analysis however when true and mismeasured data are obtained for a subsample of the observations it is possible to estimate the parameters relating the outcome to the covariate of interest maximum likelihood methods that rely on parametric models for the mismeasurement have not met with much success realistic models for the mismeasurement process are difficult to construct the form of the likelihood is often intractable and more important such methods are not robust to model misspecification we propose an easily implemented method that is nonparametric with respect to the mismeasurement process and that is applicable when mismeasurement is due to the problem of missing data errors in variables or use of imperfect surrogate covariates specifically denote the outcome variable by y the covariate data subject to mismeasurement by x and the remaining covariates including perhaps surrogates or mismeasured values

a family of calabi yau varieties and potential automorphy we prove potential modularity theorems for l adic representations of any dimension from these results we deduce the sato tate conjecture for all elliptic curves with nonintegral j invariant defined over a totally real field

binary regression models for contaminated data on etudie la nature des observations aberrantes dans le contexte des donnees de regression binaires on propose un modele simple qui permet qu un petit nombre de reponses binaires soit mal enregistre

explicit forms for ergodicity coefficients and spectrum localization explicit forms for orgodicity coefficients which bound the non unit eigenvalues of finite stochastic matrices are known in the cases p 1 when the lp norm is used 3 6 7 the purpose of this report is to generalize these results to arbitrary real matrices in particular square matrices with a real valued eigenvalue by giving a generalized and to some extent simplified synthesis of methods used hitherto alternative procedures are mentioned in the light of the framework established the results are of particular relevance for a primitive nonnegative matrix for which a bound results for the non perron frobenius eigenvalues

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

upper bounds on the maximum modulus of subdominant eigenvalues of nonnegative matrices abstract upper bounds for the maximum modulus of the subdominant roots of square nonnegative matrices are obtained we provide a unified approach that yields or improves upon most of the bounds that have been obtained so far

response operators for markov processes in a finite state space radius of convergence and link to the response theory for axiom a systems using straightforward linear algebra we derive response operators describing the impact of small perturbations to finite state markov processes the results can be used for studying empirically constructed e g from observations or through coarse graining of model simulations finite state approximation of statistical mechanical systems recent results concerning the convergence of the statistical properties of finite state markov approximation of the full asymptotic dynamics on the srb measure in the limit of finer and finer partitions of the phase space are suggestive of some degree of robustness of the obtained results in the case of axiom a system our findings give closed formulas for the linear and nonlinear response theory at all orders of perturbation and provide matrix expressions that can be directly implemented in any coding language plus providing bounds on the radius of convergence of the perturbative theory in particular we relate the convergence of the response theory to the rate of mixing of the unperturbed system one can use the formulas derived for finite state markov processes to recover previous findings obtained on the response of continuous time axiom a dynamical systems to perturbations by considering the generator of time evolution for the measure and for the observables a very basic low tech and computationally cheap analysis of the response of the lorenz 63 model to perturbations provides rather encouraging results regarding the possibility of using the approximate representation given by finite state markov processes to compute the system s response

local central limit theorems the high order correlations of rejective sampling and logistic likelihood asymptotics let i 1 i n be independent but not necessarily identically distributed bernoulli random variables and let x n n j 1 i j for v in a bounded region a local central limit theorem expansion of p x n ex n v is developed to any given degree by conditioning this expansion provides information on the high order correlation structure of dependent weighted sampling schemes of a population e a special case of which is simple random sampling where a set d e is sampled with probability proportional to a d x a where x a are positive weights associated with individuals a e these results are used to determine the asymptotic information and demonstrate the consistency and asymptotic normality of the conditional and unconditional logistic likelihood estimator for unmatched case control study designs in which sets of controls of the same size are sampled with equal probability

minimum distance estimation for the logistic regression model summary it is well known that the maximum likelihood fit of the logistic regression parameters can be greatly affected by atypical observations several robust alternatives have been proposed however if we consider the model from the case control viewpoint it is clear that current techniques can exhibit poor behaviour in many common situations a new robust class of estimation procedures is introduced the estimators are constructed via a minimum distance approach after identifying the model with a semiparametric biased sampling model the approach is developed under the case control sampling scheme yet is shown to be applicable under prospective sampling as well a weighted cramer von mises distance is used as an illustrative example of the methodology

measurement error in the explanatory variable of a binary regression regression calibration and integrated conditional likelihood in studies of residential radon and lung cancer in epidemiology one approach to investigating the dependence of disease risk on an explanatory variable in the presence of several confounding variables is by fitting a binary regression using a conditional likelihood thus eliminating the nuisance parameters when the explanatory variable is measured with error the estimated regression coefficient is biased usually towards zero motivated by the need to correct for this bias in analyses that combine data from a number of case control studies of lung cancer risk associated with exposure to residential radon two approaches are investigated both employ the conditional distribution of the true explanatory variable given the measured one the method of regression calibration uses the expected value of the true given measured variable as the covariate the second approach integrates the conditional likelihood numerically by sampling from the distribution of the true given measured explanatory variable the two approaches give very similar point estimates and confidence intervals not only for the motivating example but also for an artificial data set with known properties these results and some further simulations that demonstrate correct coverage for the confidence intervals suggest that for studies of residential radon and lung cancer the regression calibration approach will perform very well so that nothing more sophisticated is needed to correct for measurement error

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

vinogradov s mean value theorem via efficient congruencing ii we apply the efficient congruencing method to estimate vino gradov s integral for moments of order 2s with 1 6 s 6 k 2 1 thereby we show that quasi diagonal behaviour holds when s o k 2 we obtain near optimal estimates for 1 6 s 6 1 k 2 k and optimal estimates for s k 2 1 in this way we come half way to proving the main conjecture in two different directions there are consequences for estimates of weyl type and in several allied applications thus for example the anticipated asymptotic formula in waring s problem is established for sums of s kth powers of natural numbers whenever s 2k 2 2k 8 k 6 estimates stemming from vinogradov s mean value theorem deliver bounds for exponential sums of large degree both in mean and pointwise beyond the competence of alternate approaches the ubiquity of such exponential sums in analytic number theory in the analysis for example of the riemann zeta function in waring s problem and beyond accounts for the high pro file of vinogradov s methods in the associated literature in recent work we established a version of vinogradov s mean value theorem which achieves an essentially optimal upper bound with a number of variables only twice the number conjectured to be best possible see 19 for systems of degree k previous estimates missed such a bound by a factor of order logk our earlier approach provides no upper bounds when the number of variables is smaller precluding the possibility of applications involving the finer features of these mean values our goal in this paper is to remedy this deficiency at the same time strengthening our previous conclusions it transpires that we are able to come within a hair s breadth of proving the main conjecture concerning vino gradov s mean value theorem in half of the basic interval of relevant moments such developments illustrate the flexibility of the new efficient congruencing method introduced in 19 we now introduce some notation when k n and r k define

response and sensitivity using markov chains dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study how this affects their statistical properties a prominent real life example of this class of problems is the investigation of climate response to perturbations in this respect it is crucial to determine what the linear response of a system is as a quantification of sensitivity alongside previous work here we use the transfer operator formalism to study the response and sensitivity of a dynamical system undergoing perturbations by projecting the transfer operator onto a suitable finite dimensional vector space one is able to obtain matrix representations which determine finite markov processes further using perturbation theory for markov matrices it is possible to determine the linear and nonlinear response of the system given a prescribed forcing here we suggest a methodology which puts the scope on the evolution law of densities the liouville fokker planck equation allowing to effectively calculate the sensitivity and response of two representative dynamical systems

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

inferences about exposure disease associations using probability of exposure information unconditional and conditional likelihood methods are proposed for modeling odds and odds ratios relating a binary disease variable and an exposure variable that is only known probabilistically e g via measurement of a surrogate exposure variable while adjusting appropriately for a vector of covariables assuming nondifferential errors a probabilistic structure is developed that permits analysis of the marginal relation between disease and the surrogate while maintaining important properties of the marginal relation between disease and true exposure in particular analyses conditioned on marginal totals in surrogate disease tables remove the same nuisance parameters as are removed in a conditional analysis of true exposure disease tables in addition a relation between the probability of exposure poe in the disease and disease free groups is derived permitting the use of information about exposure from populations with structure different from the study population examples are presented illustr

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

weighted semiparametric estimation in regression analysis with missing covariate data abstract this article investigates estimation of the regression coefficients in an assumed mean function when covariates on some subjects are missing we examine the performance of a horvitz and thompson 1952 type weighted estimator by using different estimates of the selection probabilities which may be treated as nuisance parameters or a nuisance function in particular we investigate the properties of the estimate of the regression parameters when the selection probabilities are estimated by kernel smoothers we present large sample theory for the new estimator and conduct simulation studies comparing the proposed estimator to the maximum likelihood estimator and multiple imputation under various model assumptions and different missingness mechanisms in addition we provide two real examples that motivate this investigation

measurement error instrumental variables and corrections for attenuation with applications to meta analyses macmahon et al present a meta analysis of the effect of blood pressure on coronary heart disease as well as new methods for estimation in measurement error models for the case when a replicate or second measurement is made of the fallible predictor the correction for attenuation used by these authors is compared to others already existing in the literature as well as to a new instrumental variable method the assumptions justifying the various methods are examined and their efficiencies are studied via simulation compared to the methods we discuss the method of macmahon et al may have bias in some circumstances because it does not take into account i possible correlations among the predictors within a study ii possible bias in the second measurement or iii possibly differing marginal distributions of the predictors or measurement errors across studies a unifying asymptotic theory using estimating equations is also presented

analysis of longitudinal data in case control studies summary case control studies for longitudinal data are considered among repeated binary measurements of disease status in each subject the exposure levels of risk factors for all diseased cases are identified and the exposure levels for only a small fraction of disease free cases to be regarded as controls are identified case control studies for longitudinal data bring about economies in cost and time when the disease is rare and when assessing the exposure level of risk factors is difficult we propose a way of using an ordinary logistic model to analyse case control longitudinal data we prove that the proposed estimator is consistent and asymptotically normally distributed provided that the choice of control observations is independent of the covariates for those subjects we also discuss the validity of the generalised estimating equation method for case control longitudinal data simulation results are provided and a real example is presented

efficacy of repeated measures in regression models with measurement error ignoring measurement error may cause bias in the estimation of regression parameters when the true covariates are unobservable multiple imprecise measurements can be used in the analysis to correct for the associated bias we suggest a simple estimating procedure that gives consistent estimates of regression parameters by using the repeated measurements with error the relative pitman efficiency of our estimator based on models with and without measurement error has been found to be a simple function of the number of replicates and the ratio of intra to inter variance of the true covariate the procedure thus provides a guide for deciding the number of repeated measurements in the design stage an example from a survey study is presented

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

prospective analysis of logistic case control studies abstract in a classical case control study prentice and pyke proposed to ignore the study design and instead base estimation and inference on a random sampling i e prospective formulation we generalize this prospective formulation of case control studies to include multiplicative models stratification missing data measurement error robustness and other examples the resulting estimators which ignore the case control study aspect and instead are based on a random sampling formulation are typically consistent for nonintercept parameters and are asymptotically normally distributed we derive the resulting asymptotic covariance matrix of the parameter estimates the covariance matrix obtained by ignoring the case control sampling scheme and using prospective formulas instead is shown to be at worst asymptotically conservative and asymptotically correct in a variety of problems a simple sufficient condition guaranteeing the latter is obtained

on the density of integer points on generalised markoff hurwitz and dwork hypersurfaces we use bounds of mixed character sums modulo a square free integer q of a special structure to estimate the density of integer points on the hypersurface begin aligned f 1 x 1 cdots f n x n a x 1 k 1 ldots x n k n end aligned for some polynomials f i in mathbb z x and nonzero integers a and k i i 1 ldots n in the case of begin aligned f 1 x cdots f n x x 2 quad text and quad k 1 cdots k n 1 end aligned the above hypersurface is known as the markoff hurwitz hypersurface while for begin aligned f 1 x cdots f n x x n quad text and quad k 1 cdots k n 1 end aligned it is known as the dwork hypersurface our results are substantially stronger than those known for general hypersurfaces

on the density of rational and integral points on algebraic varieties let x n be a projective geometrically integral variety over of dimension r and degree d 4 suppose that there are only finitely many r 1 planes over on x the main result of this paper is a proof of the fact that the number n x b of rational points on xwhich have height at most b satisfies for any 0 the implied constant depends at most on d n and

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

a class of logistic type discriminant functions summary in two group discriminant analysis the neyman pearson lemma establishes that the roc receiver operating characteristic curve for an arbitrary linear function is everywhere below the roc curve for the true likelihood ratio the weighted area between these two curves can be used as a risk function for finding good discriminant functions the weight function corresponds to the objective of the analysis for example to minimise the expected cost of misclassification or to maximise the area under the roc the resulting discriminant functions can be estimated by iteratively reweighted logistic regression we investigate some asymptotic properties in the near logistic setting where we assume the covariates have been chosen such that a linear function gives a reasonable but not necessarily exact

conditionally unbiased bounded influence estimation in general regression models with applications to generalized linear models abstract in this article robust estimation in generalized linear models for the dependence of a response y on an explanatory variable x is studied a subclass of the class of m estimators is defined by imposing the restriction that the score function must be conditionally unbiased given x within this class of conditionally fisher consistent estimators optimal bounded influence estimators of regression parameters are identified and their asymptotic properties are studied the estimators studied in this article and the efficient bounded influence estimators studied by stefanski carroll and ruppert 1986 depend on an auxiliary centering constant and nuisance matrix the centering constant can be given explicitly for the conditionally fisher consistent estimators and thus they are easier to compute than the estimators studied by stefanski et al 1986 in addition estimation of the nuisance matrix has no effect on the asymptotic distribution of the conditionally fisher consistent estimators the sam

modeling subject specific phase dependent effects and variations in longitudinal responses via a geometric brownian motion process we address statistical issues regarding modeling a collection of longitudinal response trajectories characterized by the presence of subject specific phase dependent effects and variation to accommodate these two time varying individual characteristics we employ a geometric stochastic differential equation for modeling based on a brownian motion process and develop a two step paradigm for statistical analysis this paradigm reverses the order of statistical inference in random effects model we first extract individual information about phase dependent treatment effects and volatility parameters for all subjects then we derive the association relationship between the parameters characterizing the individual longitudinal trajectories and the corresponding covariates by means of multiple regression analysis the stochastic differential equation model and the two step paradigm together provide significant advantages both in modeling flexibility and in computational efficiency the modeling flexibility is due to the easy adaptation of temporal change points for subject specific phase transition in treatment effects whereas the computational efficiency benefits in part from the independent increment property of brownian motion that avoids high dimensional integration we demonstrate our modeling approach and statistical analysis on a real data set of longitudinal measurements of disease activity scores from a rheumatoid arthritis study copyright 2011 john wiley sons ltd

comparison of perturbation bounds for the stationary distribution of a markov chain abstract the purpose of this paper is to review and compare the existing perturbation bounds for the stationary distribution of a finite irreducible homogeneous markov chain

case control studies with errors in covariates we devise methods for estimating the parameters of a prospective logistic model with dichotomous response d and arbitrary covariates x from case control data when these covariates are measured with error we suppose that some fraction of the cases and controls provide only the error prone covariate measurements w the incomplete or reduced data whereas some of the cases and controls provide measurements on x and w the complete data we assume a measurement error density with a finite set of parameters namely fw xd w x d and nondifferential error is treated as a special case of this model fw x w x our algorithm estimates both the logistic parameters and from a pseudolikelihood because empirical distribution functions are used in place of needed distributions in the pseudolikelihoods the required asymptotic theory is more elaborate than for pseudolikelihoods based on substitution for a finite number of nuisance parameters we also examine computationally simpler methods under t

prospective analysis of case control data under general multiplicative intercept risk models summary we show that under a general multiplicative intercept model for risk case control data can be analyzed by maximum likelihood as if they had arisen prospectively up to an unidentifiable multiplicative constant which depends on the relative sampling fractions this generalizes earlier work of anderson 1972 1979 by showing that not only the point estimates but also the standard errors based on the observed information matrix are correct when the prospective likelihood is maximized likelihood ratio testing is also valid under this broad class of risk models data on disease status from a much larger cohort are shown to add no information to the estimation of the covariate related parameters

variation of the unit root along the dwork family of calabi yau varieties we study the variation of the unit roots of members of the dwork families of calabi yau varieties over a finite field by the method of dwork katz and also from the point of view of formal group laws a p adic analytic formula for the unit roots away from the hasse locus is obtained

optimal estimation for response dependent retrospective sampling abstract in more conventional analytic surveys we sample the response variates y through a sampling design that is dependent on the covariate x the x values are assumed known for all the units in the population however contrary to these situations there are areas of statistical application when the values of the response variable are known for all the individuals but not the values of covariate for example in epidemiology and reliability here we sample the x values the sampling design used depends on the response variate y the problem that we study is the same as usual namely inference regarding dependence of the response y on the covariate x some work in this direction has already been done in this article we use estimating function theory to establish optimum estimation for the parameter of interest this optimality holds conditionally when the response variable is fixed as well as unconditionally we demonstrate that here for response dependent sampling stratification plays the same role a

logistic regression for two stage case control data summary samples of diseased cases and nondiseased controls are drawn at random from the population at risk after classification according to the exposure of interest subsamples of cases and controls are selected for purposes of covariable ascertainment a modification of the usual logistic regression analysis yields consistent estimates of covariable adjusted relative risks and their standard errors by balancing the numbers of exposed and nonexposed for whom covariable inforniation is ascertained within case and control samples some efficiency may be gained over the usual single stage design particularly when the exposure is rare and the relative risks associated with the covariables are large the procedure may be useful also when covariable information is missing for a large part of the sample

on robustness in the logistic regression model summary we investigate robustness in the logistic regression model copas has studied two forms of robust estimator a robust resistant estimate of pregibon and an estimate based on a misclassification model he concluded that robust resistant estimates are much more biased in small samples than the usual logistic estimate is and recommends a bias corrected version of the misclassification estimate we show that there are other versions of robust resistant estimates which have bias often approximately the same as and sometimes even less than the logistic estimate these estimates belong to the mallows class in addition the corrected misclassification estimate is inconsistent at the logistic model we develop a simple consistent modification the modified estimate is a member of the mallows class but unlike most robust estimates it has an interpretable tuning constant the results are illustrated on data sets featuring different kinds of outliers

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

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

a mean score method for missing and auxiliary covariate data in regression models summary we consider regression analysis when incomplete or auxiliary covariate data are available for all study subjects and in addition for a subset called the validation sample true covariate data of interest have been ascertained the term auxiliary data refers to data not in the regression model but thought to be informative about the true missing covariate data of interest we discuss a method which is nonparametric with respect to the association between available and missing data allows missingness to depend on available response and covariate values and is applicable to both cohort and case control study designs the method previously proposed by flanders greenland 1991 and by zhao lipsitz 1992 is generalised and asymptotic theory is derived our expression for the asymptotic variance of the estimator provides intuition regarding performance of the method optimal sampling strategies for the validation set are also suggested by the asymptotic results