corrections to bisection algorithm for computing the frequency response gain of sampled data systems infinite dimensional congruent transformation approach this paper derives a bisection algorithm for computing the frequency response gain of sampled data systems with their intersample behavior taken into account the properties of the infinite dimensional congruent transformation i e the schur complement arguments and the sylvester law of inertia play a key role in the derivation specifically it is highlighted that counting up the numbers of the negative eigenvalues of self adjoint operators is quite important for the computation of the frequency response gain this contrasts with the well known arguments on the related issue of the sampled data h sub spl infin problem where the key role is played by the positivity of operators and the loop shifting technique the effectiveness of the derived algorithm is demonstrated through a numerical example

vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

nonlinear stability of traveling waves to a hyperbolic parabolic system modeling chemotaxis we prove nonlinear stability of traveling waves of arbitrary amplitudes to a hyperbolic parabolic system modeling repulsive chemotaxis in contrast to the previous related results where various sm

model averaged double robust estimation researchers estimating causal effects are increasingly challenged with decisions on how to best control for a potentially high dimensional set of confounders typically a single propensity score model is chosen and used to adjust for confounding while the uncertainty surrounding which covariates to include into the propensity score model is often ignored and failure to include even one important confounder will results in bias we propose a practical and generalizable approach that overcomes the limitations described above through the use of model averaging we develop and evaluate this approach in the context of double robust estimation more specifically we introduce the model averaged double robust ma dr estimators which account for model uncertainty in both the propensity score and outcome model through the use of model averaging the ma dr estimators are defined as weighted averages of double robust estimators where each double robust estimator corresponds to a specific choice of the outcome model and the propensity score model the ma dr estimators extend the desirable double robustness property by achieving consistency under the much weaker assumption that either the true propensity score model or the true outcome model be within a specified possibly large class of models using simulation studies we also assessed small sample properties and found that ma dr estimators can reduce mean squared error substantially particularly when the set of potential confounders is large relative to the sample size we apply the methodology to estimate the average causal effect of temozolomide plus radiotherapy versus radiotherapy alone on one year survival in a cohort of 1887 medicare enrollees who were diagnosed with glioblastoma between june 2005 and december 2009

approximation of delay systems by fourier laguerre series abstract error estimates are given for the approximation of stable recorded delay systems in the l 2 and h norms using two recently advocated techniques based on laguerre series in addition some theoretical results on l 0 approximation are derived

a simple resampling method by perturbing the minimand suppose that under a semiparametric setting an estimator of a vector of parameters of interest is obtained by optimising an objective function which has a u process structure the covariance matrix of the estimator is generally a function of the underlying density function which may be difficult to estimate well by conventional methods in this paper we present a simple resampling method by perturbing the objective function repeatedly inferences of the parameters can then be made based on a large collection of the resulting optimisers we illustrate our proposal by three examples with a heteroscedastic regression model

formulas for hankel singular values and vectors for a class of input delay systems this paper gives explicit formulas for hankel singular values and vectors for a class of multiple input delay systems the class consists of systems whose gramian is reducible to an integral operator with semi separable kernel this structure is exploited in deriving a transcendental equation for singular values

hopf bifurcation of viscous shock waves in compressible gas dynamics and mhd extending our previous results for artificial viscosity systems we show under suitable spectral hypotheses that shock wave solutions of compressible navier stokes and magnetohydrodynamics equations undergo hopf bifurcation to nearby time periodic solutions the main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in h s for data in h s we accomplish this by a novel energy estimate carried out in lagrangian coordinates interestingly this estimate is false in eulerian coordinates at the same time we greatly sharpen and simplify the analysis of the previous work

stability of cellular states of the kuramoto sivashinsky equation this paper is concerned with the instability property of a particular type of solution to the kuramoto sivashinsky equation namely the symmetric time independent cellular states attention is focused on the dimension of their unstable manifolds we show that as a control parameter varies the dimension changes in an ordered way governed by certain properties of an associated ordinary differential equation further change in dimension is shown to be related to hopf bifurcations from the cellular states an asymptotic analysis explains the onset of these bifurcations and predicts approximate parameter values at which they occur

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

optimal choice of time scaling factor for linear system approximations using laguerre models clowes 1965 illustrated how to select the optimal time scaling factor for approximations of rational transfer functions using a series expansion of laguerre functions the present paper extends this result to approximations for general l sub 2 stable linear systems as a direct application of this extended result we are able to obtain approximations for irrational transfer functions which minimize the frequency domain error in the l sub 2 sense also we give empirical solutions for the optimal time scaling factor for approximations of first order plus delay systems which can then be applied to a larger class of overdamped systems

o 2 hopf bifurcation for a model of cellular shock instability abstract we study by center manifold and normal form reduction an o 2 hopf bifurcation arising in a simplified model of bifurcating viscous shock waves in a channel suppressing longitudinal dependence and modeling onset of instability via competing stable unstable diffusions for this canonical system a cousin of the kuramoto sivashinsky model we are able to carry out a complete bifurcation analysis

approximation of infinite dimensional systems a fourier series based method for approximation of stable infinite dimensional linear time invariant system models is discussed the basic idea is to compute the fourier series coefficients of the associated transfer function t sub d z and then take a high order partial sum two results on h sup infinity convergence and associated error bounds of the partial sum approximation are established it is shown that the fourier coefficients can be replaced by the discrete fourier transform coefficients while maintaining h sup infinity convergence thus a fast fourier transform algorithm can be used to compute the high order approximation this high order finite dimensional approximation can then be reduced using balanced truncation or optimal hankel approximation leading to the final finite dimensional approximation to the original infinite dimensional model this model has been tested on several transfer functions of the time delay type with promising results

discrete fourier transform and h sub infinity approximation it is shown that uniform rational approximation of nonrational transfer functions can always be obtained by means of the discrete fourier transform dft as long as such approximants exist based on this fact it is permissible to apply the fast fourier transform fft algorithm in carrying out rational approximations without being apprehensive of convergence the dft is used to obtain traditional approximations for transfer functions of infinite dimensional systems justification is provided for using the dft in such approximations it is established that whenever a stable transfer function can be approximated uniformly on the right half plane by a rational function its approximants can always be recognized by means of a dft

on the adaptive elastic net with a diverging number of parameters we consider the problem of model selection and estimation in situations where the number of parameters diverges with the sample size when the dimension is high an ideal method should have the oracle property fan and li 2001 fan and peng 2004 which ensures the optimal large sample performance furthermore the high dimensionality often induces the collinearity problem which should be properly handled by the ideal method many existing variable selection methods fail to achieve both goals simultaneously in this paper we propose the adaptive elastic net that combines the strengths of the quadratic regularization and the adaptively weighted lasso shrinkage under weak regularity conditions we establish the oracle property of the adaptive elastic net we show by simulations that the adaptive elastic net deals with the collinearity problem better than the other oracle like methods thus enjoying much improved finite sample performance

stratification and weighting via the propensity score in estimation of causal treatment effects a comparative study estimation of treatment effects with causal interpretation from observational data is complicated because exposure to treatment may be confounded with subject characteristics the propensity score the probability of treatment exposure conditional on covariates is the basis for two approaches to adjusting for confounding methods based on stratification of observations by quantiles of estimated propensity scores and methods based on weighting observations by the inverse of estimated propensity scores we review popular versions of these approaches and related methods offering improved precision describe theoretical properties and highlight their implications for practice and present extensive comparisons of performance that provide guidance for practical use

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

a new approximation method for time delay systems we develop a rational approximant b sub n sup spl alpha of e sup ds and make a detailed analysis on the approximant b sub n sup spl alpha g of e sup ds g where g spl isin rl sup 2 we prove l sup p 2 spl infin error convergence and provide corresponding error bounds explicitly comparative study with other approximants is performed in order to make clear the properties of b sub n sup spl alpha making full use of its recursive structure b sub n sup spl alpha is shown to be a suitable approximant for a class of h sup spl infin problems involving time delay

controller design for distributed systems based on hankel norm approximations a procedure for obtaining a finite dimensional reduced order model for a class of distributed systems is developed it is based on a combination of a modal approximation and an optimal hankel norm approximation and the approximation error can be accurately predicted the reduced order model can be used to design a precompensator for the original system as its performance and robust stability characteristics can be deduced from the reduced order model and finitely many modal parameters the procedure is applied to a parabolic system

semiparametric double balancing score estimation for incomplete data with ignorable missingness when estimating the marginal mean response with missing observations a critical issue is robustness to model misspecification in this article we propose a semiparametric estimation method with extended double robustness that attains the optimal efficiency under less stringent requirement for model specifications than the doubly robust estimators in this semiparametric estimation covariate information is collapsed into a two dimensional score s with one dimension for i the pattern of missingness and the other for ii the pattern of response both estimated from some working parametric models the mean response e y is then estimated by the sample mean of e y s which is estimated via nonparametric regression the semiparametric estimator is consistent if either the core of i or the core of ii is captured by s and attains the optimal efficiency if both are captured by s as the cores can be obtained without correctly specifying the full parametric models for i or ii the proposed es

on the structure of suboptimal h controllers in the sensitivity minimization problem for distributed stable plants in this paper we consider the h sensitivity minimization problem for siso distributed stable systems we elucidate the structure of all the suboptimal h controllers for stable distributed plants with invertible outer parts and rational weights we identify the finite and infinite dimensional parts of the controller the conditions under which one can obtain a finite dimensional suboptimal controller by approximating the infinite dimensional parts of the optimal controller are discussed the case where plant transfer function is continuous on the boundary is also discussed

frequency response of sampled data systems this paper introduces the concept of frequency response for sampled data systems and explores some basic properties as well as its computational procedures it is shown that 1 by making use of the lifting technique the notion of frequency response can be naturally introduced to sampled data systems in spite of their time varying characteristics 2 it represents a frequency domain steady state behaviour and 3 it is also closely related to the original transfer function representation via an integral formula it is shown that the computation of the frequency response can be reduced to a finite dimensional eigenvalue problem and some examples are presented to illustrate the results

positive realness analysis of sampled data systems and its applications in this paper we first study strong positive realness of sampled data systems and introduce a measure called positive realness gap index we show that this index can be computed efficiently with a bisection method and provide state space formulas for its computation the importance of this index lies in that it is useful for robust stability analysis of sampled data systems an iterative procedure for computing an exact robust stability margin is given and illustrated through a numerical example

covariate selection with group lasso and doubly robust estimation of causal effects summary the efficiency of doubly robust estimators of the average causal effect ace of a treatment can be improved by including in the treatment and outcome models only those covariates which are related to both treatment and outcome i e confounders or related only to the outcome however it is often challenging to identify such covariates among the large number that may be measured in a given study in this article we propose glider group lasso and doubly robust estimation a novel variable selection technique for identifying confounders and predictors of outcome using an adaptive group lasso approach that simultaneously performs coefficient selection regularization and estimation across the treatment and outcome models the selected variables and corresponding coefficient estimates are used in a standard doubly robust ace estimator we provide asymptotic results showing that for a broad class of data generating mechanisms glider yields a consistent estimator of the ace when either the outcome or treatment model is correctly specified a comprehensive simulation study shows that glider is more efficient than doubly robust methods using standard variable selection techniques and has substantial computational advantages over a recently proposed doubly robust bayesian model averaging method we illustrate our method by estimating the causal treatment effect of bilateral versus single lung transplant on forced expiratory volume in one year after transplant using an observational registry

on the robustness of the adaptive lasso to model misspecification penalization methods have been shown to yield both consistent variable selection and oracle parameter estimation under correct model specification in this article we study such methods under model misspecification where the assumed form of the regression function is incorrect including generalized linear models for uncensored outcomes and the proportional hazards model for censored responses estimation with the adaptive least absolute shrinkage and selection operator lasso penalty is proven to achieve sparse estimation of regression coefficients under misspecification the resulting estimators are selection consistent asymptotically normal and oracle where the selection is based on the limiting values of the parameter estimators obtained using the misspecified model without penalization we further derive conditions under which the penalized estimators from the misspecified model may yield selection consistency under the true model the robustness is explored numerically via simulation and an application to the wisconsin epidemiological study of diabetic retinopathy

nyquist stability criterion and positive realness of sampled data systems abstract in this paper we first derive a nyquist stability criterion of sampled data systems in such a form that reflects the treatment of sampled data systems with their intersample behavior taken into account we next analyze the positive realness of sampled data systems with d 11 matrix i e the continuous time direct feedthrough matrix from the disturbance input w to the controlled output z being 0 and show that such sampled data systems cannot be made positive real unless the relative degree of the system from w to z is 1 we further show that adding some appropriate d 11 leads to positive real sampled data systems and give a method to find such d 11 that is as small as possible this should be useful for the stability analysis of sampled data systems with nonlinearities all these analyses as well as the small gain analysis are compared from the viewpoint of the gain margin analysis

tuning parameter selection for the adaptive lasso using eric the adaptive lasso is a commonly applied penalty for variable selection in regression modeling like all penalties though its performance depends critically on the choice of the tuning parameter one method for choosing the tuning parameter is via information criteria such as those based on aic and bic however these criteria were developed for use with unpenalized maximum likelihood estimators and it is not clear that they take into account the effects of penalization in this article we propose the extended regularized information criterion eric for choosing the tuning parameter in adaptive lasso regression eric extends the bic to account for the effect of applying the adaptive lasso on the bias variance tradeoff this leads to a criterion whose penalty for model complexity is itself a function of the tuning parameter we show the tuning parameter chosen by eric is selection consistent when the number of variables grows with sample size and that this consistency holds in a wider range of contex

realisation and approximation of linear infinite dimensional systems with error bounds the class of linear infinite dimensional systems with finite dimensional inputs and outputs whose impulse response h satisfies h in l 1 cap l 2 0 infty mathbb c p times m and induces a nuclear hankel operator is said to be of nuclear type for this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies furthermore explicit l infty bounds on the transfer function errors l 1 and l 2 bounds on the impulse response errors and hilbert schmidt and nuclear bounds on the hankel operator errors are obtained these truncations also generate an approximating sequence to the optimal hankel norm approximations to the original system and various error bounds of these approximants are deduced

transformations in density estimation abstract for the density estimation problem the global window width kernel density estimator does not perform well when the underlying density has features that require different amounts of smoothing at different locations in this article we propose to transform the data with the intention that a global window width is more appropriate for the density of the transformed data the density estimate of the original data is the back transform by change of variables of the global window width estimate of the transformed data s density we explore choosing the transformation from suitable parametric families data based selection rules for the choice of transformations and the window width are discussed application to real and simulated data demonstrates the usefulness of our proposals

brief paper separator type robust stability theorem of sampled data systems allowing noncausal lptv scaling this paper derives a necessary and sufficient condition for robust stability of sampled data systems which is stated by using the notion of separators that are dealt with in an operator theoretic framework such operator theoretic treatment of separators provides a new perspective which we call noncausal linear periodically time varying scaling and leads to reducing conservativeness in robust stability analysis a numerical example is given to demonstrate the results

estimating average treatment effects with a double index propensity score we consider estimating average treatment effects ate of a binary treatment in observational data when data driven variable selection is needed to select relevant covariates from a moderately large number of available covariates x to leverage covariates among x predictive of the outcome for efficiency gain while using regularization to fit a parametric propensity score ps model we consider a dimension reduction of x based on fitting both working ps and outcome models using adaptive lasso a novel ps estimator the double index propensity score dips is proposed in which the treatment status is smoothed over the linear predictors for x from both the initial working models the ate is estimated by using the dips in a normalized inverse probability weighting estimator which is found to maintain double robustness and also local semiparametric efficiency with a fixed number of covariates p under misspecification of working models the smoothing step leads to gains in efficiency and robustness over traditional doubly robust estimators these results are extended to the case where p diverges with sample size and working models are sparse simulations show the benefits of the approach in finite samples we illustrate the method by estimating the ate of statins on colorectal cancer risk in an electronic medical record study and the effect of smoking on c reactive protein in the framingham offspring study

robust stabilization of delay systems by approximation of coprime factors given a delay system with transfer function g s h2 s h1 s where h1 s 01npi s e is and h2 s 02nqi s e is with 0 0 1 n1 0 0 n2 the pi being polynomials of degree i and i 0for i 0 and the qi polynomials of degree di 0 for each i the robust stabilization of a class of perturbed coprime factors of this system is considered asymptotic estimates are obtained based on recent results on the approximation and stabilization of normalized coprime factors an explicit formula is given for the normalized coprime factor stability margin for the case of a multivariable transfer function of the form g s e str s with r rational

approximation of transfer functions of infinite dimensional dynamical systems by rational interpolants with prescribed poles rational interpolants with prescribed poles are used to approximate holomorphic functions on the closure of their region of analyticity under natural assumptions of their properties on the boundary the transfer functions of some infinite dimensional dynamical systems of interest in applications satisfy the restrictions we impose this is the case for discrete time fractional filters time delay systems and heat transfer control systems we give two general results by which in particular the transfer functions that arise in such dynamical systems may be approximated estimates for the rate of convergence are given we also include some numerical examples which compare the performance of the method we propose with others commonly used in systems theory

equivariant hopf bifurcation with general pressure laws abstract the equivariant hopf bifurcation dynamics of a class of system of partial differential equations is studied the connections between the current dynamics and fundamental concepts in hyperbolic conservation laws are explained the unique approximation property of center manifold reduction function is used in the current work to determine certain parameter in the normal form the current work generalizes the study of the second author j yao 2014 and discovers a class of examples of o 2 hopf bifurcation with two parameters arising from systems of partial differential equations

approximation of delay systems a case study abstract the approximation of delay systems is studied a numerically well behaved method for computing hankel optima rational approximations for delay systems is presented based on certain properties of hankel matrices and the theory of laguerre polynomials the cf method of trefethen and certain pade approximations of delay systems are also considered the importance of a certain wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the l norm and the hankel norm is shown completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions a case study and numerical comparison is presented for the approximation of the parametric family of first order stable delay systems g s exp s ts 1 numerical experience indicates that l optimized cf approximations based on short truncated maclaurin series give in general somewhat smaller l approximation errors than hankel optim

dealing with limited overlap in estimation of average treatment effects summary estimation of average treatment effects under unconfounded or ignorable treatment assignment is often hampered by lack of overlap in the covariate distributions between treatment groups this lack of overlap can lead to imprecise estimates and can make commonly used estimators sensitive to the choice of specification in such cases researchers have often used ad hoc methods for trimming the sample we develop a systematic approach to addressing lack of overlap we characterize optimal subsamples for which the average treatment effect can be estimated most precisely under some conditions the optimal selection rules depend solely on the propensity score for a wide range of distributions a good approximation to the optimal rule is provided by the simple rule of thumb to discard all units with estimated propensity scores outside the range 0 1 0 9

l approximation and nuclearity of delay systems abstract the l convergence of partial fraction decompositions of single input single output delay systems is studied conditions for the nuclearity of such systems are derived and examples given

l approximation of a parabolic system regarded as a delay system abstract we present an approach to the problem of finding an l approximant of the infinite dimensional system describing the diffusion of heat in a wall we show that this system can be regarded as a delay system with laplace variable s we are able to get results established in zwart et al 1988 about partial fraction expansion for delay systems achieved by some adjustment to the specificities of our particular case the determination of an l approximant is realised in two steps using the optimal hankel norm approximation

approximation of unstable infinite dimensional systems using coprime factors abstract the effectiveness of comprime factor techniques in l 2 and l model reduction of unstable linear systems is analysed asymptotic estimates are given of the achievable error in the stable and unstable parts of the approximate system measured in a number of different norms some involving the associated hankel operators the chordal metric is introduced as a measure of approximation and is shown to yield the graph topology finally it is deduced that asymptotically optimal l 2 and l convergence rates can be obtained for a large class of unstable systems

a necessary and sufficient frequency domain criterion for the passivity of siso sampled data systems we present a frequency domain solution to the sampled data passivity problem our analysis is exact in the sense that we take into account the intersample behavior of the system we use frequency response fr operators to first obtain necessary conditions on the sampling rate t and the relative degree of the open loop transfer function g 11 for achieving a passive continuous time closed loop system then assuming passivity of g 11 and closed loop stability we derive a necessary and sufficient condition for discrete time controllers that render a passive closed loop system we apply the obtained results to the problem of stability of haptic systems

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

covariate selection for the nonparametric estimation of an average treatment effect observational studies in which the effect of a nonrandomized treatment on an outcome of interest is estimated are common in domains such as labour economics and epidemiology such studies often rel

the standard h sub spl infin problem in systems with a single input delay the standard h sub spl infin problem is solved for lti systems with a single pure input lag the solution is based on state space analysis mixing a finite dimensional and an abstract evolution model utilizing the relatively simple structure of these distributed systems the associated operator riccati equations are reduced to a combination of two algebraic riccati equations and one differential riccati equation over the delay interval the results easily extend to finite time and time varying problems where the algebraic riccati equations are substituted by differential riccati equations over the process time duration

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

the prognostic analogue of the propensity score summary the propensity score collapses the covariates of an observational study into a single measure summarizing their joint association with treatment conditions prognostic scores summarize covariates association with potential responses as with propensity scores stratification on prognostic scores brings to uncontrolled studies a concrete and desirable form of balance a balance that is more familiar as an objective of experimental control like propensity scores prognostic scores can reduce the dimension of the covariate yet causal inferences conditional on them are as valid as are inferences conditional only on the unreduced covariate as a method of adjustment unto itself prognostic scoring has limitations not shared with propensity scoring but it holds promise as a complement to the propensity score particularly in certain designs for which unassisted propensity adjustment is difficult or infeasible

partial fraction expansions for delay systems abstract partial fraction decompositions for single input single output delay systems are studied the results have application in rational l approximations which play an important role in robust compensator design for delay systems