a note on stabilization by dynamic high gain output feedback in this note we study the problem of stabilization by dynamic output feedback for a certain class of linear systems we establish the existence of pairs of subspaces im upsilon a where im is a minimum phase input subspace contained in an almost stabilizability snbspace upsilon a by applying a recent theorem by sehumacber this result leads to the existence of stabilizing high gain compensators for the linear systems under consideration in particular it will be shown that invertible minimum phase systems can always be stabilized by dynamic compensators of dynamic order equal to the excess of poles over zeros minus the number of inputs

on the order of stable compensators abstract by means of two typical examples it is shown that in general the minimal order of a stabilizing stable compensator may not be bounded in terms of the plant order such bounds can only be given in the special case of plants with at most one right half plane zero implications for the simultaneous stabilization problem are elucidated

working correlation structure misspecification estimation and covariate design implications for generalised estimating equations performance the method of generalised estimating equations for regression modelling of clustered outcomes allows for specification of a working matrix that is intended to approximate the true correlation matrix of the observations we investigate the asymptotic relative efficiency of the generalised estimating equation for the mean parameters when the correlation parameters are estimated by various methods the asymptotic relative efficiency depends on three features of the analysis namely i the discrepancy between the working correlation structure and the unobservable true correlation structure ii the method by which the correlation parameters are estimated and iii the design by which we refer to both the structures of the predictor matrices within clusters and distribution of cluster sizes analytical and numerical studies of realistic data analysis scenarios show that choice of working covariance model has a substantial impact on regression estimator efficiency protection against avoidable loss of efficiency associated with covariance misspecification is obtained when a gaussian estimation pseudolikelihood procedure is used with an ar 1 structure

working covariance model selection for generalized estimating equations we investigate methods for data based selection of working covariance models in the analysis of correlated data with generalized estimating equations we study two selection criteria gaussian pseudolikelihood and a geodesic distance based on discrepancy between model sensitive and model robust regression parameter covariance estimators the gaussian pseudolikelihood is found in simulation to be reasonably sensitive for several response distributions and noncanonical mean variance relations for longitudinal data application is also made to a clinical dataset assessment of adequacy of both correlation and variance models for longitudinal data should be routine in applications and we describe open source software supporting this practice

low order integral action controller synthesis abstract a simple controller synthesis method is developed for certain classes of linear time invariant multi input multi output plants the number of poles in each entry of these controllers depends on the number of right half plane plant zeros and is independent of the number of poles of the plant to be stabilized furthermore these controllers have integral action so that they achieve asymptotic tracking of step input references with zero steady state error the designed controller s poles and zeros are all in the stable region with the exception of one pole at the origin for the integral action design requirement the freedom available in the design parameters may be used for additional performance objectives although the only goal here is stabilization and tracking of constant references

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

smooth bootstrap methods for analysis of longitudinal data in analysis of longitudinal data the variance matrix of the parameter estimates is usually estimated by the sandwich method in which the variance for each subject is estimated by its residual products we propose smooth bootstrap methods by perturbing the estimating functions to obtain bootstrapped realizations of the parameter estimates for statistical inference our extensive simulation studies indicate that the variance estimators by our proposed methods can not only correct the bias of the sandwich estimator but also improve the confidence interval coverage we applied the proposed method to a data set from a clinical trial of antibiotics for leprosy

working correlation structure identification in generalized estimating equations selecting an appropriate working correlation structure is pertinent to clustered data analysis using generalized estimating equations gee because an inappropriate choice will lead to inefficient parameter estimation we investigate the well known criterion of qic for selecting a working correlation structure and have found that performance of the qic is deteriorated by a term that is theoretically independent of the correlation structures but has to be estimated with an error this leads us to propose a correlation information criterion cic that substantially improves the qic performance extensive simulation studies indicate that the cic has remarkable improvement in selecting the correct correlation structures we also illustrate our findings using a data set from the madras longitudinal schizophrenia study

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

a caveat concerning independence estimating equations with multivariate binary data clustered binary data occur commonly in both the biomedical and health sciences in this paper we consider logistic regression models for multivariate binary responses where the association between the responses is largely regarded as a nuisance characteristic of the data in particular we consider the estimator based on independence estimating equations iee which assumes that the responses are independent this estimator has been shown to be nearly efficient when compared with maximum likelihood ml and generalized estimating equations gee in a variety of settings the purpose of this paper is to highlight a circumstance where assuming independence can lead to quite substantial losses of efficiency in particular when the covariate design includes within cluster covariates assuming independence can lead to a considerable loss of efficiency in estimating the regression parameters associated with those covariates

akaike s information criterion in generalized estimating equations correlated response data are common in biomedical studies regression analysis based on the generalized estimating equations gee is an increasingly important method for such data however there seem to be few model selection criteria available in gee the well known akaike information criterion aic cannot be directly applied since aic is based on maximum likelihood estimation while gee is nonlikelihood based we propose a modification to aic where the likelihood is replaced by the quasi likelihood and a proper adjustment is made for the penalty term its performance is investigated through simulation studies for illustration the method is applied to a real data set

information ratio test for model misspecification in quasi likelihood inference in this article we focus on the circumstances in quasi likelihood inference that the estimation accuracy of mean structure parameters is guaranteed by correct specification of the first moment but the estimation efficiency could be diminished due to misspecification of the second moment we propose an information ratio ir statistic to test for model misspecification of the variance covariance structure through a comparison between two forms of information matrix the negative sensitivity matrix and the variability matrix we establish asymptotic distributions of the proposed ir test statistics we also suggest an approximation to the asymptotic distribution of the ir statistic via a perturbation resampling method moreover we propose a selection criterion based on the ir test to select the best fitting variance covariance structure from a class of candidates through simulation studies it is shown that the ir statistic provides a powerful statistical tool to detect different scenarios of misspecific

a modified pseudolikelihood approach for analysis of longitudinal data we consider the analysis of longitudinal data when the covariance function is modeled by additional parameters to the mean parameters in general inconsistent estimators of the covariance variance correlation parameters will be produced when the working correlation matrix is misspecified which may result in great loss of efficiency of the mean parameter estimators albeit the consistency is preserved we consider using different working correlation models for the variance and the mean parameters in particular we find that an independence working model should be used for estimating the variance parameters to ensure their consistency in case the correlation structure is misspecified the designated working correlation matrices should be used for estimating the mean and the correlation parameters to attain high efficiency for estimating the mean parameters simulation studies indicate that the proposed algorithm performs very well we also applied different estimation procedures to a data set from a clinical trial for illustration

low order stabilizers for linear systems the existence and construction of low order stabilizers for linear systems are considered firstly it is shown that for an all pole plant the stability of the high degree part of the plant transfer function s denominator guarantees the existence of a low order stabilizer secondly if this high degree part is unstable a method is presented to modify it such that the above result is applicable thirdly an algorithm for constructing a low order stabilizer for a general plant is developed where only a few linear algebraic equations need be solved several examples are included for illustration of the results

time integration algorithm based on divergent series resummation for ordinary and partial differential equations borel s technique of divergent series resummation is transformed into a numerical code and used as a time integration scheme it is applied to the resolution of regular and singular problems arising in fluid mechanics its efficiency is compared to those of classical discretization schemes

existence and uniqueness for a class of nonlinear higher order partial differential equations in the complex plane we prove existence and uniqueness results for nonlinear third order partial differential equations of the form

miscellanea on the efficiency of regression estimators in generalised linear models for longitudinal data summary liang zeger 1986 introduced a generalised estimating equations approach based on a working correlation matrix to obtain consistent and efficient estimators of regression parameters in the class of generalised linear models for repeated measures data as demonstrated by crowder 1995 because of the uncertainty of definition of the working correlation matrix the liang zeger approach may in some cases lead to a complete breakdown of the estimation of the regression parameters in this paper we show that even though the liang zeger approach in many situations yields consistent estimators for the regression parameters these estimators are usually inefficient as compared to the regression estimators obtained by using the independence estimating equations approach

extended generalized estimating equations for clustered data abstract typically analysis of data consisting of multiple observations on a cluster is complicated by within cluster correlation estimating equations for generalized linear modeling of clustered data have recently received much attention this article proposes an extension to the generalized estimating equation method proposed by liang and zeger which treats within cluster correlations as nuisance parameters using ideas from extended quasi likelihood estimating equations for regression and association parameters are provided simultaneously the resulting estimators are proven to be asymptotically normal and consistent under certain conditions the consistency of regression estimators allows incorrect modeling of the correlation among repeated responses the method is illustrated with an analysis of data from a developmental toxicity study

on the accuracy of efficiency of estimating equation approach summary liang and zeger 1986 biometrika73 13 22 introduced a generalized estimating equation gee approach based on a working correlation matrix to obtain efficient estimators of regression parameters in the class of generalized linear models for repeated measures data as demonstrated by crowder 1995 biometrika82 407 410 because of uncertainty of the definition of the working correlation matrix the liang zeger approach may in some cases lead to a complete breakdown of the estimation of the regression parameters after taking this comment of crowder into account recently sutradhar and das 1999 biometrika86 459 465 examined the loss of efficiency of the regression estimators due to misspecification of the correlation structures but their study was confined to the regression estimation with cluster level covariates as in the original paper of liang and zeger in this paper we study this efficiency loss problem for the generalized regression models with within cluster covariates by utilizing the approach of sutsadhar and das 1999

parametric variance function estimation for nonnormal repeated measurement data zeger and liang 1986 biometrics 42 121 130 proposed a procedure for analyzing nonnormal longitudinal data in the context of the generalized linear model this procedure is extended to model variance heterogeneity allowing the observations to come from distributions with different scale parameters loss of efficiency is evaluated when heterogeneity of scale factor is ignored

brief paper almost optimal lq control using stable periodic controllers this paper presents a scheme for constructing an almost optimal control law for a linear time invariant plant via a bounded input bounded output stable controller the control law is required to be almost optimal with respect to a quadratic performance index the controller is also required to stabilize the system the main result of the paper is a scheme for constructing such an output feedback controller this linear controller is described by a set of state equations which is of twice the plant dimension and allows for jumps in the state at periodic intervals it is shown that for the existence of such a controller it is sufficient that the plant be stabilizable and detectable

the graph metric for unstable plants and robustness estimates for feedback stability in this paper a graph metric is defined that provides a measure of the distance between unstable multivariable plants the graph metric induces a graph topology on unstable plants which is the weakest possible topology in which feedback stability is robust using the graph metric it is possible to derive estimates for the robustness of feedback stability without assuming that the perturbed and unperturbed plants have the same number of rhp poles if the perturbed and unperturbed systems have the same rhp poles then it is possible to obtain necessary and sufficient conditions for robustness with respect to a given class of perturbations as an application of these results the design of stabilizing controllers for unstable singularly perturbed systems is studied finally the relationship of the graph metric to the gap metric introduced by zames and el sakkary is studied in detail in particular it is shown that the robustness results of zames and el sakkary do not enable one to conclude the causality of the perturbed system whereas the present results do

existence uniqueness analyticity and borel summability for boussinesq equations through borel summation methods we analyze the boussinesq equations for coupled fluid velocity and temperature fields 1 ut u p u u ae2 f t u we prove that an equivalent system of integral equations in the borel variable p r dual to 1 t has a unique solution in a class of exponentially bounded functions implying the existence of a classical solution to 1 in a complex t region that includes a real positive time axis segment for analytic initial data and forcing it is shown that the solution is borel summable implying that formal series in powers of t is gevrey 1 asymptotic and within the time interval of existence the solution remains analytic with the same analyticity strip width as the initial data and forcing we also determine conditions on the integral equation solution that improve the estimate for existence time

on minimal order stabilization of single loop plants abstract it is shown by example that a single loop plant of order n relative degree 1 with one right half plane pole and one right half plane zero may not be stabilizable by a compensator of order n 2 a further example shows that a minimum phase plant of order n relative degree r with one right half plane pole may not be stabilizable by a compensator of order r 2

on repeated measures analysis with misspecified covariance structure in recent years various sophisticated methods have been developed for the analysis of repeated measures or longitudinal data the more traditional approach based on a normal likelihood function has been shown to be unsatisfactory in the sense of yielding asymptotically biased estimates when the covariance structure is misspecified more recent methodology based on generalized linear models and quasi likelihood estimation has gained widespread acceptance as generalized estimating equations however this also has theoretical problems in this paper a suggestion is made for improving the asymptotic behaviour of estimators by using the older approach implemented via gaussian estimation the resulting estimating equations include the quasi score function as one component so the methodology proposed can be viewed as a combination of gaussian estimation and generalized estimating equations which has a firmer asymptotic basis than either alone has

effects of variance function misspecification in analysis of longitudinal data the approach of generalized estimating equations gee is based on the framework of generalized linear models but allows for specification of a working matrix for modeling within subject correlations the variance is often assumed to be a known function of the mean this article investigates the impacts of misspecifying the variance function on estimators of the mean parameters for quantitative responses our numerical studies indicate that 1 correct specification of the variance function can improve the estimation efficiency even if the correlation structure is misspecified 2 misspecification of the variance function impacts much more on estimators for within cluster covariates than for cluster level covariates and 3 if the variance function is misspecified correct choice of the correlation structure may not necessarily improve estimation efficiency we illustrate impacts of different variance functions using a real data set from cow growth

robust estimating functions and bias correction for longitudinal data analysis robust methods are useful in making reliable statistical inferences when there are small deviations from the model assumptions the widely used method of the generalized estimating equations can be robustified by replacing the standardized residuals with the m residuals if the pearson residuals are assumed to be unbiased from zero parameter estimators from the robust approach are asymptotically biased when error distributions are not symmetric we propose a distribution free method for correcting this bias our extensive numerical studies show that the proposed method can reduce the bias substantially examples are given for illustration

efficiency of regression estimates for clustered data statistical methods for clustered data such as generalized estimating equations gee and generalized least squares gls require selecting a correlation or convariance structure to specify the dependence between observations within a cluster valid regression estimates can be obtained that do not depend on correct specification of the true correlation but inappropriate specifications can result in a loss of efficiency we derive general expressions for the asymptotic relative efficiency of gee and gls estimators under nested correlation structures efficiency is shown to depend on the covariate distribution the cluster sizes the response variable correlation and the regression parameters the results demonstrate that efficiency is quite sensitive to the between and within cluster variation of the covariates and provide useful characterizations of models for which upper and lower efficiency bounds are attained efficiency losses for simple working correlation matrices such as independence can be large even for small to moderate correlations and cluster sizes

stabilization by reduced order controllers a recent result by a linnemann syst contr lett vol 11 p 27 32 1988 gives conditions under which a continuous time single loop plant of order n can be stabilized by a reduced order controller specifically if the euclidean algorithm is applied to the numerator and denominator polynomials of the transfer function and one of the remainders is a kth order hurwitz polynomial then a stabilizing controller of order n k 1 exists the author provides an alternative proof of this result

a class of single input single output systems stabilizable by reduced order controllers abstract let the euclidian algorithm be applied to the numerator and denominator polynomial of an n th order system transfer function it isshown that there exists an n 1 k nt order stabilizing controller provided one of the remainder polynomials occuring in the algorithm is a k th order hurwitz polunomial the controller is constructed using almost disturbance decoupling techniques the result specifies an algebraic condition under which a reduced order controller can be determined using a reduced order model obtained by minimal partial realization or a certain continued fraction expansion

generalized least squares with misspecified serial correlation structures summary the regression literature contains hundreds of studies on serially correlated disturbances most of these studies assume that the structure of the error covariance matrix q is known or can be estimated consistently from data surprisingly few studies investigate the properties of estimated generalized least squares gls procedures when the structure of q is incorrectly identified and the parameters are inefficiently estimated we compare the finite sample efficiencies of ordinary least squares ols gls and incorrect gls igls estimators we also prove new theorems establishing theoretical efficiency bounds for igls relative to gls and ols results from an exhaustive simulation study are used to evaluate the finite sample performance and to demonstrate the robustness of igls estimates vis a vis ols and gls estimates constructed for models with known and estimated but correctly identified q some of our conclusions for finite samples differ from established asymptotic results

on the use of a working correlation matrix in using generalised linear models for repeated measures summary in a seminal paper liang zeger 1986 extended the use of generalised linear models to repeated measures data they based the analysis on specifications for the means and variances of the observations as usual for generalised linear models but showed how specifications for the correlations between measurements made on the same unit could be avoided by using a working correlation matrix in some cases the parameters involved in this matrix are subject to an uncertainty of definition which can lead to a breakdown of the asymptotic properties of the estimators

rank based regression for analysis of repeated measures we consider rank based regression models for repeated measures to account for possible within subject correlations we decompose the total ranks into between and within subject ranks and obtain two different estimators based on between and within subject ranks a simple perturbation method is then introduced to generate bootstrap replicates of the estimating functions and the parameter estimates this provides a convenient way for combining the corresponding two types of estimating function for more efficient estimation

unbiased estimating equations from working correlation models for irregularly timed repeated measures the method of generalized estimating equations gees has been criticized recently for a failure to protect against misspecification of working correlation models which in some cases leads to loss of efficiency or infeasibility of solutions however the feasibility and efficiency of gee methods can be enhanced considerably by using flexible families of working correlation models we propose two ways of constructing unbiased estimating equations from general correlation models for irregularly timed repeated measures to supplement and enhance gee the supplementary estimating equations are obtained by differentiation of the cholesky decomposition of the working correlation or as score equations for decoupled gaussian pseudolikelihood the estimating equations are solved with computational effort equivalent to that required for a first order gee full details and analytic expressions are developed for a generalized markovian model that was evaluated through simulation large sample sandwich standard errors for working correlation parameter estimates are derived and shown to have good performance the proposed estimating functions are further illustrated in an analysis of repeated measures of pulmonary function in children

power and sample size calculation incorporating misspecifications of the variance function in comparative clinical trials with over dispersed count data over dispersed count data are frequently observed in clinical trials where the primary endpoint is occurrence of clinical events sample sizes of comparative clinical trials with these data are typically calculated under negative binomial models or quasi poisson models with specified variance functions or under the assumption that the specified working variance functions are correctly specified in this article we propose a sample size formula anticipating misspecifications of the working variance function we derived a method based on the asymptotic distribution of a wald test statistic with a sandwich type robust variance estimator under quasi poisson models under misspecifications of the working variance function the asymptotic variance of the estimator of the treatment effect is expressed as a form involving both true and working variance functions our sample size formula includes several existing formulas as special cases when the working variance function is correctly specified as the true variance function we also consider a sensitivity analysis for possible misspecifications of the true variance function when estimating sample sizes using our formula a simulation study demonstrated the adequacy of our formulas in finite sample size settings an application to a clinical trial to evaluate the treatment effect on prevention of copd exacerbation is provided

multivariate regression analyses for categorical data summary it is common to observe a vector of discrete and or continuous responses in scientific problems where the objective is to characterize the dependence of each response on explanatory variables and to account for the association between the outcomes the response vector can comprise repeated observations on one variable as in longitudinal studies or genetic studies of families or can include observations for different variables this paper discusses a class of models for the marginal expectations of each response and for pairwise associations the marginal models are contrasted with log linear models two generalized estimating equation approaches are compared for parameter estimation the first focuses on the regression parameters the second simultaneously estimates the regression and association parameters the robustness and efficiency of each is discussed the methods are illustrated with analyses of two data sets from public health research

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

gaussian estimation for correlated binomial data summary the mean variance structure of correlated binomial data is discussed and whittle s 1961 gaussian estimation is suggested as a useful general method in this context comparisons are made with other monrent methods in current use