approximate radix 8 booth multipliers for low power and high performance operation the booth multiplier has been widely used for high performance signed multiplication by encoding and thereby reducing the number of partial products a multiplier using the radix 4 or modified booth algorithm is very efficient due to the ease of partial product generation whereas the radix 8 booth multiplier is slow due to the complexity of generating the odd multiples of the multiplicand in this paper this issue is alleviated by the application of approximate designs an approximate 2 bit adder is deliberately designed for calculating the sum of 1 times and 2 times of a binary number this adder requires a small area a low power and a short critical path delay subsequently the 2 bit adder is employed to implement the less significant section of a recoding adder for generating the triple multiplicand with no carry propagation in the pursuit of a trade off between accuracy and power consumption two signed 16 times 16 bit approximate radix 8 booth multipliers are designed using the approximate recoding adder with and without the truncation of a number of less significant bits in the partial products the proposed approximate multipliers are faster and more power efficient than the accurate booth multiplier the multiplier with 15 bit truncation achieves the best overall performance in terms of hardware and accuracy when compared to other approximate booth multiplier designs finally the approximate multipliers are applied to the design of a low pass fir filter and they show better performance than other approximate booth multipliers

estimation and comparison of changes in the presence of information right censoring by modeling the censoring process abstract in estimating and comparing the rates of change of a continuous variable between two groups the unweighted averages of individual simple least squares estimates from each group are often used under a linear random effects model when all individuals have completed observations at identical time points these statistics are maximum likelihood estimates for the expected rates of change however with censored of missing data these estimates are no longer efficient when compared to generalized least squares estimates when in addition the right censoring process is dependent upon the individual rates of change i e informative right censoring the generalized least squares estimates will be biased likelihood ratio test for informativeness of the censoring process and maximum likelihood estimates for the expected rates of change and the parameters of the right censoring process are developed under a linear random effect models with a probit model for the right censoring process in realistic situations we illustrate that the bias in estimating group rate of change and the reduction of power in comparing group difference could be substantial when strong dependency of the right censoring process on individual rates of change is ignored author

on doubly robust estimation in a semiparametric odds ratio model we consider the doubly robust estimation of the parameters in a semiparametric conditional odds ratio model our estimators are consistent and asymptotically normal in a union model that assumes either of two variation independent baseline functions is correctly modelled but not necessarily both furthermore when either outcome has finite support our estimators are semiparametric efficient in the union model at the intersection submodel where both nuisance functions models are correct for general outcomes we obtain doubly robust estimators that are nearly efficient at the intersection submodel our methods are easy to implement as they do not require the use of the alternating conditional expectations algorithm of chen 2007

hamiltonian and super hamiltonian systems of a hierarchy of soliton equations abstract by considering an isospectral eigenvalue problem a hierarchy of soliton equations are derived two types of extensions are presented by enlarging the associated spectral problem with the aid of generalized trace identity and the super trace identity the hamiltonian and super hamiltonian structures for the integrable extensions are established

new non isospectral integrable couplings of the akns system abstract by means of semi direct sums of lie algebras and the compatibility condition of lax pair an efficient way to construct non isospectral integrable couplings is proposed as its applications new non isospectral integrable couplings of the akns system are obtained the method can be devoted to constructing non isospectral integrable couplings of other nonlinear evolution equations

addition related arithmetic operations via controlled transport of charge this work investigates the single electron tunneling set technology based computation of basic addition related arithmetic functions e g addition and multiplication via a novel computation paradigm which we refer to as electron counting arithmetic that is based on controlling the transport of discrete quantities of electrons within the set circuit first assuming that the number of controllable electrons within the system is unrestricted we prove that the addition of two n bit operands can be computed with a depth 2 network composed out of 3n 1 circuit elements and that the multiplication of two n bit operands can be computed with a depth 3 network composed out of 4n 1 circuit elements second assuming that the number of controllable electrons cannot be higher than a given constant r determined by practical limitations we prove that the addition of two n bit operands can be computed with a depth n r 3 network composed out of 3n 1 n r circuit elements under the same restriction we suggest methods to reduce the addition network depth in the order of logn r and to perform n bit multiplication in an o logn r delay finally we propose set based implementations for a set of basic electron counting building blocks and implement a number of circuits operating under the electron counting paradigm as follows 4 bit digital to analog converter 5 bit analog to digital converter 4 bit adder and 3 bit multiplier all proposed implementations are verified by means of simulation

nonlinear super integrable hamiltonian couplings based on a kind of new lie superalgebras we proposed a scheme for constructing nonlinear super integrable couplings super trace identity over the corresponding loop superalgebras is used to furnish the super hamiltonian structures for the resulting nonlinear super integrable couplings we illustrate the application of the scheme by means of an enlarged super ablowitz kaup newell segur akns matrix spectral problem and present a nonlinear super integrable hamiltonian couplings for the super akns hierarchy

on the design of approximate restoring dividers for error tolerant applications this paper proposes several designs of approximate restoring dividers two different levels of approximation cell and array levels are employed three approximate subtractor cells are utilized for integer subtraction as basic step of division these cells tend to mitigate accuracy in subtraction with other metrics such as circuit complexity and power dissipation at array level exact cells are either replaced or truncated in the approximate divider designs a comprehensive evaluation of approximation at both cell and array divider levels is pursued using error analysis and hspice simulation different circuit metrics including complexity and power dissipation are evaluated different applications are investigated by utilizing the proposed approximate arithmetic circuits the simulation results show that with extensive savings for power dissipation and circuit complexity the proposed designs offer better error tolerant capabilities for quotient oriented applications image processing than remainder oriented application modulo operations the proposed approximate restoring divider is significantly better than the approximate non restoring scheme presented in the technical literature

a new matrix lie algebra the multicomponent yang hierarchy and its super integrable coupling system abstract a set of new matrix lie algebra is constructed which is devoted to obtaining a new loop algebra a 2 m it follows that an isospectral problem is established by use of tu scheme a liouville integrable multi component hierarchy of soliton equations is generated which possesses the multi component hamiltonian structures as its reduction cases the multi component yang hierarchy is given finally the multi component super integrable coupling system of yang hierarchy is presented by enlarging matrix spectral problem

two integrable hamiltonian hierarchies in sl 2 r and so 3 r with three potentials by introducing two specific matrix spectral problems associated with sl 2 r and so 3 r matrix lie algebras we generate two integrable hamiltonian hierarchies with three potentials the computation and analysis on their hamiltonian structures by means of the trace identity show that the resulting hierarchies are liouville integrable namely that each hierarchy consists of commuting hamiltonian soliton equations

a new design technique for column compression multipliers in this paper a new design technique for column compression cc multipliers is presented constraints for column compression with full and half adders are analyzed and under these constraints considerable flexibility for implementation of the cc multiplier including the allocation of adders and choosing the length of the final fast adder is exploited using the example of an 8 spl times 8 bit cc multiplier we show that architectures obtained from this new design technique are more area efficient and have shorter interconnections than the classical dadda cc multiplier we finally show that our new technique is also suitable for the design of twos complement multipliers

two 2 1 dimensional expanding dynamical systems associated to the mkp hierarchy an isospectral problem with a parameter is presented for which a modified kdv mkdv hierarchy is easily derived from the tu scheme then two different expanding integrable models are obtained respectively the main purpose for this focuses on deriving a 2 1 dimensional mkdv hierarchy called the mkp hierarchy and the corresponding different 2 1 dimensional expanding models including the linear and nonlinear whose hamiltonian structures are obtained by employing an identity developed by us in the paper as the reduced consequences the linear and nonlinear 2 1 dimensional expanding models of the mkp equation are generated

two 2 1 dimensional hierarchies of evolution equations and their hamiltonian structures two kinds of appropriate isospectral problems are introduced by using a lie algebra with the help of the tah scheme we generate two new 2 1 dimensional hierarchies of evolution equations whose hamiltonian structures are derived from the trace identity proposed by tu guizhang andrushkiw r i and huang x c finally we propose some problems worth thinking about

lie algebras and korteweg de vries equations abstract for each metrizable lie algebra g an infinite commuting hierarchy of bi hamiltonian integrable extensions of the korteweg de vries kdv hierarchy is found the classical kdv hierarchy itself corresponds to the case g 0 the construction is based on the hamiltonian formalism on the dual space to a certain one parameter family of differential lie algebras produced from g

a subalgebra of lie algebra a2 and its associated two types of loop algebras as well as hamiltonian structures of integrable hierarchy in this paper a subalgebra a2 of the lie algebra a2 is constructed which gives a corresponding loop algebra a 2 by properly choosing the gradation of the basis elements it follows that an isospectral problem is established and a new liouville integrable hamiltonian hierarchy is obtained by making use of a matrix transformation a subalgebra a2 of the lie algebra a1 is presented which possesses the same communicative operations of basis elements as those in a2 again we expand the lie algebra a1 into a high dimensional loop algebra g and a type of expanding integrable system of the hierarchy obtained above is worked out furthermore hamiltonian structures of hierarchy are presented by use of the quadratic form identity

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

instrumental variable estimation of nonlinear errors in variables models this paper establishes that instruments enable the identification of nonparametric regression models in the presence of measurement error by providing a closed form solution for the regression function in terms of fourier transforms of conditional expectations of observable variables for parametrically specified regression functions we propose a root n consistent and asymptotically normal estimator that takes the familiar form of a generalized method of moments estimator with a plugged in nonparametric kernel density estimate both the identification and the estimation methodologies rely on fourier analysis and on the theory of generalized functions the finite sample properties of the estimator are investigated through monte carlo simulations

variational identities and applications to hamiltonian structures of soliton equations abstract this is an introductory report concerning our recent research on hamiltonian structures we will discuss variational identities associated with continuous and discrete spectral problems and their applications to hamiltonian structures of soliton equations our illustrative examples are the akns hierarchy and the volterra lattice hierarchy associated with semisimple lie algebras and two hierarchies of their integrable couplings associated with non semisimple lie algebras the resulting hamiltonian structures generate infinitely many commuting symmetries and conservation laws for the four soliton hierarchies the presented variational identities can be applied to hamiltonian structures of other soliton hierarchies

new nonlinear evolution equations related to some superloop algebras constructs hierarchies of nonlinear evolution equations on the superloop algebras related to the super lie algebras b 0 1 and sl 2 1 the hierarchy associated with sl 2 1 contains a generalisation of the super nonlinear schrodinger equation with a cubic non derivative interaction term in complex anticommuting fields in all cases the author gives a recursive procedure for constructing involutive integrals of local polynomials

a discrete variational identity on semi direct sums of lie algebras the discrete variational identity under general bilinear forms on semi direct sums of lie algebras is established the constant involved in the variational identity is determined through the corresponding solution to the stationary discrete zero curvature equation an application of the resulting variational identity to a class of semi direct sums of lie algebras in the volterra lattice case furnishes hamiltonian structures for the associated integrable couplings of the volterra lattice hierarchy

instrumental variable estimator for the nonlinear errors in variables model abstract estimation for the nonlinear errors in variables model is considered it isassumed that additional information is available in the form of observations on instrumental variables an estimation procedure is presented for the parameters of the model asymptotic properties of the estimator are investigated

analysis of semi parametric regression models with non ignorable non response in this article we consider the problem of making inferences about the parameter 0 indexing the conditional mean of an outcome given a vector of regressors when a subset of the variables outcome or covariates are missing for some study subjects and the probability of non response depends upon both observed and unobserved data values that is non response is non ignorable we propose a new class of inverse probability of censoring weighted estimators that are consistent and asymptotically normal can for estimating 0 when the non response probabilities can be parametrically modelled and a can estimator exists the proposed estimators do not require full specification of the likelihood and their computation does not require numerical integration we show that the asymptotic variance of the optimal estimator in our class attains the semi parametric variance bound for the model in some models no can estimator of 0 exists we provide a general algorithm for determining when can estimators of 0 exist our results follow after specializing a general representation described in the article for the efficient score and the influence function of regular asymptotically linear estimators in an arbitrary semi parametric model with non ignorable non response in which the probability of observing complete data is bounded away from zero and the non response probabilities can be parametrically modelled 1997 by john wiley sons ltd

design and analysis of area and power efficient approximate booth multipliers approximate computing is an emerging technique in which power efficient circuits are designed with reduced complexity in exchange for some loss in accuracy such circuits are suitable for applications in which high accuracy is not a strict requirement radix 4 modified booth encoding is a popular multiplication algorithm which reduces the size of the partial product array by half in this paper three approximate booth multiplier models abm m1 abm m2 and abm m3 are proposed in which approximate computing is applied to the radix 4 modified booth algorithm each of the three designs features a unique approximation technique that involves both reducing the logic complexity of the booth partial product generator and modifying the method of partial product accumulation the proposed approximate multipliers are demonstrated to have better performance than existing approximate booth multipliers in terms of accuracy and power compared to the exact booth multiplier abm m1 achieves up to a 23 percent reduction in area and 15 percent reduction in power with a mean relative error distance mred value of 7 9 times 10 4 7 9 10 4 abm m2 has area and power savings of up to 51 and 46 percent respectively with a mred of 2 7 times 10 2 2 7 10 2 abm m3 has area savings of up to 56 percent and power savings of up to 46 percent with a mred of 3 4 times 10 3 3 4 10 3 the proposed designs are compared with the state of the art existing multipliers and are found to outperform them in terms of area and power savings while maintaining high accuracy the performance of the proposed designs are demonstrated using image transformation matrix multiplication and finite impulse response fir filtering applications

probabilistic error analysis of approximate recursive multipliers approximate multipliers are gaining importance in energy efficient computing and require careful error analysis in this paper we present the error probability analysis for recursive approximate multipliers with approximate partial products since these multipliers are constructed from smaller approximate multiplier building blocks we propose to derive the error probability in an arbitrary bit width multiplier from the probabilistic model of the basic building block and the probability distributions of inputs the analysis is based on common features of recursive multipliers identified by carefully studying the behavioral model of state of the art designs by building further upon the analysis probability mass function pmf of error is computed by individually considering all possible error cases and their inter dependencies we further discuss the generalizations for approximate adder trees signed multipliers squarers and constant multipliers the proposed analysis is validated by applying it to several state of the art approximate multipliers and comparing with corresponding simulation results the results show that the proposed analysis serves as an effective tool for predicting evaluating and comparing the accuracy of various multipliers results show that for the majority of the recursive multipliers we get accurate error performance evaluation we also predict the multipliers performance in an image processing application to demonstrate its practical significance

estimating linear statistical relationships this paper on estimating linear statistical relationships includes three lectures on linear functional and structural relationships factor analysis and simultaneous equations models the emphasis is on relating the several models by a general approach and on the similarity of maximum likelihood estimators under normality in the different models in the first two lectures the observable vector is decomposed into a systematic part and a random error the systematic part satisfies the linear relationships estimators are derived for several cases and some of their properties given estimation of the coefficients of a single equation in a simultaneous equations model is shown to be equivalent to estimation of linear functional relationships

semiparametric analysis of discrete response asymptotic properties of the maximum score estimator abstract this paper is concerned with the estimation of the model med y x x from a random sample of observations on sgn y x manski 1975 introduced the maximum score estimator of the normalized parameter vector in the present paper strong consistency is proved it is also proved that the maximum score estimate lies outside any fixed neighborhood of with probability that goes to zero at exponential rate

high dimensional linear discriminant analysis optimality adaptive algorithm and missing data the paper develops optimality theory for linear discriminant analysis in the high dimensional setting a data driven and tuning free classification rule which is based on an adaptive constrained l1 minimization approach is proposed and analysed minimax lower bounds are obtained and this classification rule is shown to be simultaneously rate optimal over a collection of parameter spaces in addition we consider classification with incomplete data under the missingness completely at random model an adaptive classifier with theoretical guarantees is introduced and the optimal rate of convergence for high dimensional linear discriminant analysis under the missingness completely at random model is established the technical analysis for the case of missing data is much more challenging than that for complete data we establish a large deviation result for the generalized sample covariance matrix which serves as a key technical tool and can be of independent interest an application to lung cancer and leukaemia studies is also discussed

design and analysis of approximate compressors for multiplication inexact or approximate computing is an attractive paradigm for digital processing at nanometric scales inexact computing is particularly interesting for computer arithmetic designs this paper deals with the analysis and design of two new approximate 4 2 compressors for utilization in a multiplier these designs rely on different features of compression such that imprecision in computation as measured by the error rate and the so called normalized error distance can meet with respect to circuit based figures of merit of a design number of transistors delay and power consumption four different schemes for utilizing the proposed approximate compressors are proposed and analyzed for a dadda multiplier extensive simulation results are provided and an application of the approximate multipliers to image processing is presented the results show that the proposed designs accomplish significant reductions in power dissipation delay and transistor count compared to an exact design moreover two of the proposed multiplier designs provide excellent capabilities for image multiplication with respect to average normalized error distance and peak signal to noise ratio more than 50 db for the considered image examples

a general instrumental variable framework for regression analysis with outcome missing not at random summary the instrumental variable iv design is a well known approach for unbiased evaluation of causal effects in the presence of unobserved confounding in this article we study the iv approach to account for selection bias in regression analysis with outcome missing not at random in such a setting a valid iv is a variable which i predicts the nonresponse process and ii is independent of the outcome in the underlying population we show that under the additional assumption iii that the iv is independent of the magnitude of selection bias due to nonresponse the population regression in view is nonparametrically identified for point estimation under i iii we propose a simple complete case analysis which modifies the regression of primary interest by carefully incorporating the iv to account for selection bias the approach is developed for the identity log and logit link functions for inferences about the marginal mean of a binary outcome assuming i and ii only we describe novel and approximately sharp bounds which unlike robins manski bounds are smooth in model parameters therefore allowing for a straightforward approach to account for uncertainty due to sampling variability these bounds provide a more honest account of uncertainty and allows one to assess the extent to which a violation of the key identifying condition iii might affect inferences for illustration the methods are used to account for selection bias induced by hiv testing nonparticipation in the evaluation of hiv prevalence in the zambian demographic and health surveys

new metrics for the reliability of approximate and probabilistic adders addition is a fundamental function in arithmetic operation several adder designs have been proposed for implementations in inexact computing these adders show different operational profiles some of them are approximate in nature while others rely on probabilistic features of nanoscale circuits however there has been a lack of appropriate metrics to evaluate the efficacy of various inexact designs in this paper new metrics are proposed for evaluating the reliability as well as the power efficiency of approximate and probabilistic adders reliability is analyzed using the so called sequential probability transition matrices sptms error distance ed is initially defined as the arithmetic distance between an erroneous output and the correct output for a given input the mean error distance med and normalized error distance ned are then proposed as unified figures that consider the averaging effect of multiple inputs and the normalization of multiple bit adders it is shown that the med is an effective metric for measuring the implementation accuracy of a multiple bit adder and that the ned is a nearly invariant metric independent of the size of an adder the med is therefore useful in assessing the effectiveness of an approximate or probabilistic adder implementation while the ned is useful in characterizing the reliability of a specific design since inexact adders are often used for saving power the product of power and ned is further utilized for evaluating the tradeoffs between power consumption and precision although illustrated using adders the proposed metrics are potentially useful in assessing other arithmetic circuit designs for applications of inexact computing

an approach to generate superextensions of integrable systems an algorithm to generate integrable systems is extended to the super case some new examples of superextensions of integrable systems are illustrated we also generalize the trace identity due to tu to the super case and use it to establish hamiltonian structures of superextensions of integrable systems under consideration

a powerful approach to generate new integrable systems in this paper an effective algorithm to generate integrable systems is given as a result many new integrable equations are derived in a systematic way

design of approximate radix 4 booth multipliers for error tolerant computing approximate computing is an attractive design methodology to achieve low power high performance low delay and reduced circuit complexity by relaxing the requirement of accuracy in this paper approximate booth multipliers are designed based on approximate radix 4 modified booth encoding mbe algorithms and a regular partial product array that employs an approximate wallace tree two approximate booth encoders are proposed and analyzed for error tolerant computing the error characteristics are analyzed with respect to the so called approximation factor that is related to the inexact bit width of the booth multipliers simulation results at 45 nm feature size in cmos for delay area and power consumption are also provided the results show that the proposed 16 bit approximate radix 4 booth multipliers with approximate factors of 12 and 14 are more accurate than existing approximate booth multipliers with moderate power consumption the proposed r4abm2 multiplier with an approximation factor of 14 is the most efficient design when considering both power delay product and the error metric nmed case studies for image processing show the validity of the proposed approximate radix 4 booth multipliers

a nonlinear integrable couplings of c kdv soliton hierarchy and its infinite conservation laws abstract a kind of nonlinear integrable couplings of c kdv soliton hierarchy is constructed then we present the infinitely many conservation laws for the nonlinear integrable couplings of the c kdv soliton hierarchy

a lie algebra containing four parameters for the generalized dirac hierarchy a lie algebra containing four parameters is obtained whose commutation operation is concise and the corresponding computing formula of constant c in the variational identity is presented in this paper as application a new liouville integrable hierarchy which can be reduced to dirac hierarchy is derived by designing a special isospectral problem we call it generalized dirac hierarchy

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

bounding mean regressions when a binary regressor is mismeasured in this paper i examine identification and estimation of mean regression models when a binary regressor is mismeasured i prove that bounds for the model parameters are identified and provide simple estimators which are consistent and asymptotically normal when stronger prior information about the probability of misclassification is available the bounds can be made tighter again a simple estimator for these cases is provided all results apply to parametric and nonparametric models the paper concludes with a short empirical example

an integrable hierarchy and darboux transformations bilinear backlund transformations of a reduced equation a new integrable hierarchy of evolution equations is obtained by making use of a lie algebra and tu ma scheme from which a new generalized broer kaup gbk equation is produced then two kinds of darboux transformations the bilinear presentation the bilinear backlund transformation and the new lax pair of the gbk equation are generated respectively by employing the bell polynomials

identification and estimation of nonlinear models with misclassification error using instrumental variables a general solution abstract this paper provides a general solution to the problem of identification and estimation of nonlinear models with misclassification error in a general discrete explanatory variable using instrumental variables the misclassification error is allowed to be correlated with all the explanatory variables in the model it is not enough to identify the model by simply generalizing the identification in the binary case with a claim that the number of restrictions is no less than that of the unknowns such a claim requires solving a complicated nonlinear system of equations this paper introduces a matrix diagonalization technique which allows one to easily find the unique solution of the system the solution shows that the latent model can be expressed as an explicit function of directly observed distribution functions therefore the latent model is nonparametrically identifiable and directly estimable using instrumental variables the results show that certain monotonicity restrictions on the latent model may lead to its identification with virtually no restrictions on the misclassification probabilities an alternative identification condition suggests that the nonparametric identification may rely on the belief that people always have a higher probability of telling the truth than of misreporting the nonparametric identification in this paper directly leads to a n consistent semiparametric estimator the monte carlo simulation and empirical illustration show that the estimator performs well with a finite sample and real data

on liouville integrability of zero curvature equations and the yang hierarchy sufficient conditions for a zero curvature equation ut vx u v 0 being liouville integrable are investigated in the case that the equation is integrable an explicit formula of the poisson bracket h lambda h mu for hamiltonians h is proposed the yang hierarchy is derived and shown to be liouville integrable

nonlinear schrodinger equations and simple lie algebras we associate a system of integrable generalised nonlinear schrodinger nls equations with each hermitian symmetric space these nls equations are considered as reductions of more general systems this time associated with a reductive homogeneous space the nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space the hamiltonian structure is investigated using r matrix techniques and shown to be canonical for all these equations throughout the reduction procedure this hamiltonian structure does not degenerate each of the above systems of equations is gauge equivalent to a generalised ferromagnet reductions of the latter are discussed in terms of the corresponding nls type equations

further evidence on the robustness of the tobit estimator to heteroskedasticity abstract when the error terms in a tobit model are heteroskedastic the mle which assumes homoskedasticity is inconsistent for the special case of a constant term only model we investigate the size of the inconsistency the inconsistency is greater the greater the heteroskedasticity and the greater the degree of censoring i e the greater the number of limit observations however the inconsistency is much smaller than in the corresponding truncated normal model considered by hurd

modeling longitudinal data with nonignorable dropouts using a latent dropout class model summary in longitudinal studies with dropout pattern mixture models form an attractive modeling framework to account for nonignorable missing data however pattern mixture models assume that the components of the mixture distribution are entirely determined by the dropout times that is two subjects with the same dropout time have the same distribution for their response with probability one as that is unlikely to be the case this assumption made lead to classification error in addition if there are certain dropout patterns with very few subjects which often occurs when the number of observation times is relatively large pattern specific parameters may be weakly identified or require identifying restrictions we propose an alternative approach which is a latent class model the dropout time is assumed to be related to the unobserved latent class membership where the number of classes is less than the number of observed patterns a regression model for the response is specified conditional on the latent variable this is a type of shared parameter model where the shared parameter is discrete parameter estimates are obtained using the method of maximum likelihood averaging the estimates of the conditional parameters over the distribution of the latent variable yields estimates of the marginal regression parameters the methodology is illustrated using longitudinal data on depression from a study of hiv in women

constructing a multi function equation hierarchy and its integrable coupling system with maple in this article by considering a multi dimensional lie algebra system a new multi function integrable equation hierarchy is presented with the help of maple then the integrable coupling system of the new multi function integrable equation hierarchy is obtained through enlarging matrix spectral problem