prolongation structure of the krichever novikov equation we completely describe wahlquist estabrook prolongation structures coverings dependent on u u x u xx u xxx for the krichever novikov equation u t u xxx 3u xx 2 2u x p u u x au x in the case when the polynomial p u 4u 3 g 2u g 3 has distinct roots we prove that there is a universal prolongation algebra isomorphic to the direct sum of a commutative 2 dimensional algebra and a certain subalgebra of the tensor product of mathfrak sl 2 mathbb c with the algebra of regular functions on an affine elliptic curve this is achieved by identifying this prolongation algebra with the one for the anisotropic landau lifshitz equation using these results we find for the krichever novikov equation a new zero curvature representation which is polynomial in the spectral parameter in contrast to the known elliptic ones

on multivariate ridge regression summary a multivariate linear regression model with q responses as a linear function of p independent variables is considered with a p x q parameter matrix b the least squares or normal theory maximum likelihood estimate of b is deficient in that it takes no account of the across regression correlations and ignores the stein effect a remedy was offered by brown zidek 1980 in the form of a multivariate ridge estimator a richer class of estimators is obtained here by casting the model in a linear hierarchical framework obtaining the brown zidek multivariate ridge estimates efron morris s estimates of several normal mean vectors and fearn s bayesian estimates of growth curves as special cases the unknown covariance case results in an identifiability problem which can be overcome by a bayesian approach using conjugate priors for the unidentified covariance matrices

assessing normality in random effects models when one uses the unbalanced mixed linear model mathbf y i mathbf x i mathbf alpha mathbf z i mathbf beta i varepsilon i i 1 cdots n to analyze data from longitudinal experiments with continuous outcomes it is customary to assume varepsilon i sim operatorname ind mathscr n mathbf 0 sigma 2 mathbf i i independent of mathbf beta i sim operatorname iid mathscr n mathbf 0 delta where sigma 2 and the elements of an arbitrary mathbf delta are unknown variance and covariance components in this paper we describe a method for checking model adequacy and in particular the distributional assumption on the random effects mathbf beta i we generalize the weighted normal plot to accommodate dependent nonidentically distributed observations subject to multiple random effects for each individual unit under study one can detect various departures from the normality assumption by comparing the expected and empirical cumulative distribution functions of standardized linear combinations of estimated residuals for each of the individual units through application of distributional results for a certain class of estimators to our context we adjust the estimated covariance of the empirical cumulative distribution function to account for estimation of unknown parameters several examples of our method demonstrate its usefulness in the analysis of longitudinal data

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

estimation and prediction in a multivariate random effects generalized linear model abstract estimation of an individual s regression coefficients in a multivariate random effects linear model is considered the problem of prediction of future responses of an individual based on the random effects model is also considered and the prediction mean squared errors of various predictors are derived and compared for the practical situation where all parameters of the model are unknown and estimates must be used

large scale estimation of variance and covariance components this paper concerns matrix computations within algorithms for variance and covariance component estimation hemmerle and hartley technometrics 15 1973 pp 819 831 showed how to compute the objective function and its derivatives for maximum likelihood estimation of variance components using matrices with dimensions of the order of the number of coefficients rather than that of the number of observations their approach was extended by corbeil and searle technometrics 18 1976 pp 31 38 for restricted maximum likelihood estimation a similar reduction in dimension is possible using expectation maximization em algorithms in most cases variance components are assumed to be strictly positive we advocate the use of a modification that is numerically stable even if variance component estimates are small in magnitude for problems in which the number of coefficients is large fellner proc statistical computing section american statistical association 1984 pp 150 154 comm statist simulat

inference for multivariate normal hierarchical models this paper provides a new method and algorithm for making inferences about the parameters of a two level multivariate normal hierarchical model one has observed j p dimensional vector outcomes distributed at level 1 as multivariate normal with unknown mean vectors and with known covariance matrices at level 2 the unknown mean vectors also have normal distributions with common unknown covariance matrix a and with means depending on known covariates and on unknown regression coefficients the algorithm samples independently from the marginal posterior distribution of a by using rejection procedures functions such as posterior means and covariances of the level 1 mean vectors and of the level 2 regression coefficient are estimated by averaging over posterior values calculated conditionally on each value of a drawn this estimation accounts for the uncertainty in a unlike standard restricted maximum likelihood empirical bayes procedures it is based on independent draws from the exact posterior distributions unlike gibbs sampling the procedure is demonstrated for profiling hospitals based on patients responses concerning p 2 types of problems non surgical and surgical the frequency operating characteristics of the rule corresponding to a particular vague multivariate prior distribution are shown via simulation to achieve their nominal values in that setting

estimation in covariance components models abstract estimation techniques for linear covariance components models are developed and illustrated with special emphasis on explaining computational processes the estimation of fixed and random effects when the variances and covariances are known is presented in bayesian terms point estimates of the unknown variances and covariances are computed using the em algorithm for maximum likelihood estimation from incomplete data the techniques are illustrated with data on law schools field mice and professional football teams

empirical bayes estimation of individual growth curve parameters and their relationship to covariates summary the analysis of growth curves has long been important in biostatistics work has focused on two problems the estimation of individual curves based on many data points and the estimation of the mean growth curve for a group of individuals this paper extends a recent approach that seeks to combine data from a group of individuals in order to improve the estimates of individual growth parameters growth is modeled as polynomial in time and the group model is also linear incorporating growth related covariates into the model the estimation used is empirical bayes the estimation formulas are illustrated with a set of data on rat growth originally presented by box 1950 biometrics 6 362 389

prolongation structure of the kdv equation in the bilinear form of hirota the prolongation structure of the kdv equation in the bilinear form of hirota is determined the resulting lie algebra is realised and the backlund transformation obtained from the prolongation structure is derived the results are compared with those found by wahlquist and estabrook 1975 and by hirota 1980

censored observations repeated measures and mixed effects models an approach using the em algorithm and normal errors summary maximum likelihood estimation of a vector regression parameter and variance com ponents is considered for the mixed effects model when observations are right censored a general scheme of estimation is given using the em algorithm and detailed results found for the model with between and within block variation this model is applied to the logarithms of survival times from a repeated measures design in this paper we consider the estimation of the parameters when some of the observa tions are right censored this can occur with repeated measurements on subjects where the response is a waiting time and the variability in log times is modelled by a normal distribution alternative models for this situation are given by clayton cuzick 1985 and their methods involve partially parametric techniques using ranks and are based on models involving extreme value log gamma and logistic distributions we are concerned with use of the em algorithm dempster laird rubin 1977 to estimate the parameters of the mixed model for uncensored data hartley rao 1967 have considered this solution the analysis leads to a method which is straightforward to implement for the repeated measures model dempster et al 1984 give an account of the method with uncensored observations in 2 we consider a general implementation of the em algorithm for the pure random effects model with censored data and then in 3 we give explicit results for the simple random effects model section 4 uses these results for the analysis of repeated measures and 5 discusses some miscellaneous points arising finally in 6 we give an example involving skin graft data where there has been extensive analysis using matched pairs techniques

a general model for two level data with responses missing at random a general model for two level multivariate data with responses possibly missing at random is described the model combines regressions on fixed explanatory variables with structured residual covariance matrices the likelihood function is reduced to a form enabling computational methods for estimating the model to be devised

contextually specific effects and other generalizations of the hierarchical linear model for comparative analysis abstract contextually specific differences among members of a context pose substantive and methodological problems in multilevel analysis an important example is ethnic identity in comparative studies involving different societies ethnic and religious group membership in many societies is a basis not only for differentiation among members but also for the identification and maintenance of deeply rooted integrative ties so important is ethnicity that the existence of different groups within societies sometimes delays or prohibits the taking of censuses moreover even when ethnic information is available in national statistical data sources researchers are sometimes prohibited from reporting the results of analyses that contain ethnic detail despite these constraints information on ethnicity is often available for pluralistic societies where it is available ethnicity poses a challenging analytic problem for comparative analysis if researchers wish to compare different societies how is ethnicity

the hierarchical logistic regression model for multilevel analysis abstract a hierarchical logistic regression model is proposed for studying data with group structure and a binary response variable the group structure is defined by the presence of micro observations embedded within contexts macro observations and the specification is at both of these levels at the first micro level the usual logistic regression model is defined for each context the same regressors are used in each context but the micro regression coefficients are free to vary over contexts at the second level the micro coefficients are treated as functions of macro regressors an empirical bayes estimation procedure is proposed for estimating the micro and macro coefficients explicit formulas are provided that are computationally feasible for large scale data analyses these include an algorithm for finding the maximum likelihood estimates of the covariance components representing within and between macro equation error variability the methodology is applied to world fertility survey data

maximum likelihood computations with repeated measures application of the em algorithm abstract the purpose of this article is to consider the use of the em algorithm dempster laird and rubin 1977 for both maximum likelihood ml and restricted maximum likelihood reml estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure anderson 1973 several models and methods of analysis have been proposed in recent years for repeated measures data ware 1985 presented an overview because the em algorithm is a general purpose iterative method for computing ml estimates with incomplete data it has often been used in this particular setting dempster et al 1977 andrade and helms 1984 jennrich and schluchter 1985 there are two apparently different approaches to using the em algorithm in this setting in one application each experimental unit is observed under a standard protocol specifying measurements at each of n occasions or under n conditions and incompleteness implies that the number of measurements actua

maximal reductions in the baker hausdorff formula a preliminary expression for the baker hausdorff formula is found up to ninth order i e a series expansion of z in terms of multiple commutators where ez exey with x and y noncommuting up to ninth degree in x y by means of complete sets of linear relations between multiple commutators maximal reduction of the number of different multiple commutators in the series are obtained

constrained maximum likelihood estimation of two level covariance structure model via em type algorithms in this paper the constrained maximum likelihood estimation of a two level covariance structure model with unbalanced designs is considered the two level model is reformulated as a single level model by treating the group level latent random vectors as hypothetical missing data then the popular em algorithm is extended to obtain the constrained maximum likelihood estimates for general nonlinear constraints the multiplier method is used at them step to find the constrained minimum of the conditional expectation an accelerated em gradient procedure is derived to handle linear constraints the empirical performance of the proposed em type algorithms is illustrated by some artifical and real examples

prolongation structures of a higher order nonlinear schrodinger equation a higher order schrodinger equation containing parameters which is used to describe pulse propagation in optical fibres is shown to admit an infinite dimensional prolongation structure for exactly four combinations of the parameters besides the classical nls equation for each of these cases the structure of the resulting prolongation algebra is determined explicitly for the first three cases the prolongation algebra is essentially a sub algebra of a1 1 the fourth case turns out to be sub algebra of the twisted kac moody algebra a2 2 using vector field representations related systems of differential equations for the pseudo potential functions are given for each of the cases the cases found here correspond exactly to the derived nls equations i and ii the hirota equation and the equation recently considered by sasa and satsuma 1991 the result of this paper strongly indicates that the considered higher order nls equation is completely integrable for precisely these four cases

on the convergence and optimization of the baker campbell hausdorff formula abstract in this paper the problem of the convergence of the baker campbell hausdorff series for z log e x e y is revisited we collect some previous results about the convergence domain and present a new estimate which improves all of them we also provide a new expression of the truncated lie presentation of the series up to sixth degree in x and y requiring the minimum number of commutators numerical experiments suggest that a similar accuracy is reached with this approximation at a considerably reduced computational cost

round robin analysis of variance via maximum likelihood abstract a class of two way random effects models is presented for analyzing data that arise in a large variety of round robin designs examples of such designs can be found in numerous games tournaments and social interactions the proposed models provide information not only about individual differences but also about the mutual contingency of the behaviors of interaction dyads the statistical inference of the linear effects is discussed and a converging algorithm based on the em algorithm is proposed for obtaining the maximum likelihood estimates of the covariance components a balanced data set is analyzed using the methodology developed

inference for smooth curves in longitudinal data with application to an aids clinical trial we discuss a longitudinal study where data for many subjects are collected at irregular intervals the study is a randomized trial of hiv infected subjects and the response variable of interest is serum neopterin the mean of the outcome variable taken over patients in each treatment group is assumed to follow a smooth curve piecewise cubic polynomials with a moderate number of knots are used to model the curves a general parametric form is assumed for the covariance structure maximum penalized likelihood estimation is used to smooth the over parameterized curves statistical inference for the mean curves including confidence bands and hypothesis tests is discussed two approaches one using a bayesian interpretation of the penalized likelihood and the other based on the asymptotic distribution of the maximum penalized likelihood estimates are discussed and contrasted the properties of the confidence bands obtained from these two approaches are evaluated by examining their coverage rates in a simulation study

just identified versus overidentified two level hierarchical linear models with missing data summary the development ofmodel based methods f incomplete data has been a seminal contribution to statistical practice under the assumption ofignorable missingness one estimates the joint distribution ofthe complete data f y f yobs y mis fo r from the incomplete or observed data yobs many interesting models involve one to one transformations of for example with yi n fo ri 1 n and an ordinary least squares ols regression model is a one to one transformation of inferences based on such a transformation are equivalent to inferences based on ols using data multiply imputed from f ymis yobs for missing ymis thus identification of from yobs is equivalent to identification ofthe regression model in this article we consider a model f or two level data with continuous outcomes where the observations within each cluster are dependent the parameters ofthe hierarchical linear model hlm ofinterest however lie in a subspace of in general this identification ofthe joint distribution overidentifies the hlm we show how to characterize the joint distribution so that its parameters are a one to one transf ormation ofthe parameters ofthe hlm this leads to efficient estimation ofthe hlm from incomplete data using either the transformation method or the method of multiple imputation the approach allows outcomes and covariates to be missing at either ofthe two levels and the hlm ofinterest can involve the regression ofany subset ofvariables on a disjoint subset ofvariables conceived as covariates

the baker campbell hausdorff formula and nested commutator identities the coefficients of the baker campbell hausdorff expansion are calculated by using various methods comparison of the results yields several remarkable identities satisfied by multiple commutators which in turn allow us to greatly simplify the form of the expansion up to order eight

estimating a finite population mean from unequal probability samples abstract two approaches to estimating a finite population mean from unequal probability samples are contrasted in the prediction approach a model is specified for the population values and is used to predict the nonsampled values in the generalized regression approach the horvitz thompson estimator of the population mean is modified to allow the introduction of covariate information sarndal 1980 generalized regression estimates have the desirable property of asymptotic design consistency which is not always enjoyed by estimates in the prediction class however it is suggested that the prediction approach is the more principled method of estimation if asymptotic design consistency is a desirable property in a given application then only models that yield asymptotically design consistent prediction estimates should be used two classes of models with this property are suggested namely fixed and random effects models that allow a separate intercept for subclasses of the population indexed by the pr

empirical bayes estimation of coefficients in the general linear model from data of deficient rank empirical bayes methods are shown to provide a practical alternative to standard least squares methods in fitting high dimensional models to sparse data an example concerning prediction bias in educational testing is presented as an illustration

approximating pseudopotentials scattering problems and backlund transformations for the korteweg de vries and modified korteweg de vries equations with perturbations approximating pseudopotentials associated with scattering problems and with backlund transformations are derived for the perturbed korteweg de vries and modified korteweg de vries equations recursion formulas for approximate conservation laws are obtained for the perturbed korteweg de vries equation as well as an example the auto backlund transformations are used to construct approximate one and two soliton solutions

an efficient algorithm for computing the baker campbell hausdorff series and some of its applications we provide a new algorithm for generating the baker campbell hausdorff bch series z log exey in an arbitrary generalized hall basis of the free lie algebra l x y generated by x and y it is based on the close relationship of l x y with a lie algebraic structure of labeled rooted trees with this algorithm the computation of the bch series up to degree of 20 111 013 independent elements in l x y takes less than 15min on a personal computer and requires 1 5gbytes of memory we also address the issue of the convergence of the series providing an optimal convergence domain when x and y are real or complex matrices

parameter estimation and inference in the linear mixed model the paper reviews the linear mixed model with a focus on parameter estimation and inference parameter estimation for the different components of the model are reviewed with an emphasis on variance parameter estimation inferential procedures for the fixed effects random effects or a combination of both fixed and random effects are also discussed

segmental hidden markov models with random effects for waveform modeling this paper proposes a general probabilistic framework for shape based modeling and classification of waveform data a segmental hidden markov model hmm is used to characterize waveform shape and shape variation is captured by adding random effects to the segmental model the resulting probabilistic framework provides a basis for learning of waveform models from data as well as parsing and recognition of new waveforms expectation maximization em algorithms are derived and investigated for fitting such models to data in particular the expectation conditional maximization either ecme algorithm is shown to provide significantly faster convergence than a standard em procedure experimental results on two real world data sets demonstrate that the proposed approach leads to improved accuracy in classification and segmentation when compared to alternatives such as euclidean distance matching dynamic time warping and segmental hmms without random effects

approximate bayesian inference in conditionally independent hierarchical models parametric empirical bayes models abstract we consider two stage models of the kind used in parametric empirical bayes peb methodology calling them conditionally independent hierarchical models we suppose that there are k units which may be experimental subjects cities study centers etcetera at the first stage the observation vectors yi for units i 1 k are independently distributed with densities p yi i or more generally p yi i at the second stage the unit specific parameter vectors i are iid with densities p i the peb approach proceeds by regarding the second stage distribution as a prior and noting that if were known inference about could be based on its posterior since is not known the simplest peb methods estimate the parameter by maximum likelihood or some variant and then treat as if it were known to be equal to this estimate although this procedure is sometimes satisfactory a well known defect is that it neglects the uncertainty due to the estimation of in this article w

reml estimation in unbalanced multivariate variance components models using an em algorithm estimation of covariance matrices when data are missing is a difficult problem in the multivariate mixed model setting problems can arise either because the data are unbalanced or because the response vectors are incomplete jennrich and schluchter 1986 biometrics 42 805 820 and laird lange and stram 1987 journal of the american statistical association 82 97 105 have proposed em algorithms for estimating the parameters associated with longitudinal data models in this paper a new em algorithm is proposed for the general multivariate mixed model the algorithm is derivativefree and based on an algorithm for balanced data proposed by calvin and dykstra 1991 annals qf statistics 19 850 869 once the algorithm is described it is used to produce estimates of the variance components matrices associated with a completely random two factor nested model for a bivariate response vector the data for this example come from calvin and sedransk 1991 journal of the american statistical association 86 36 48 who analyze a study of the quality of care received by cancer patients receiving radiation therapy

some bayesian and non bayesian procedures for the analysis of comparative experiments and for small area estimation computational aspects frequentist properties and relationships abstract the estimation of a treatment contrast from experimental data and the estimation of a small area mean are special cases of the prediction of the realization of a linear combination of fixed and random effects in a possibly unbalanced two part mixed linear model in this article a bayesian approach to point and interval prediction is presented and its computational requirements are examined differences between the bayesian approach and the traditional classical approach are discussed in general terms and in addition in terms of two examples taken from the literature 1 the comparison of drug formulations in a bioavailability trial westlake and 2 the estimation of corn crop areas using satellite data battese harter and fuller some deficiencies in the classical approach are pointed out and the bayesian approach is considered from a frequentist perspective it is shown via a monte carlo study that for certain noninformative choices of the prior distribution the frequentist prop

a bayesian approach to nonlinear random effects models nonlinear random effects models are considered from the bayesian point of view the method of analysis follows closely that of lindley and smith 1972 journal of the royal statistical society series b 34 1 42 the numerical method is related to the em algorithm

a fast improvement to the em algorithm on its own terms the em algorithm is a numerical technique for the evaluation of maximum likelihood estimates for parameters describing incomplete data models it is easy to apply in many problems and is stable but slow the algorithm fails to provide a consistent estimator of the standard errors of the maximum likelihood estimates unless the additional analysis required by the louis method is performed newton type or other gradient methods are faster and provide error estimates but tend to be unstable and require the analytical evaluation of likelihoods to derive expressions for the score function and at least approximations to the fisher information matrix the purpose of this paper is to expand on a result by fisher that permits a unification of em methodology and newton methods the evaluation of the individual observation by observation score functions of the incomplete data is a byproduct of the application of the e step of the em algorithm once these become available the fisher information matrix may be consistently estimated and the m step may be replaced by a fast newton type step

bayesian approach for a nonlinear growth model nonlinear least squares methods are currently used for fitting a well known growth model namely the jenss model to the length measurements of a child followed throughout the first six years of life an empirical bayes approach is developed for fitting the model and the prior distribution of the growth model parameters is estimated from a large sample of least squares parameters an expression which is proportional to the posterior distribution is derived so that the posterior mode can be estimated given the observations on a child this posterior mode provides bayes estimates of the jenss curve parameters for the child

empirical bayes analysis for systems of mixed models with linked autocorrelated random effects abstract empirical bayes concepts are implemented in a simultaneous analysis of a system of mixed linear models having linked and serially correlated random effects emphasis is placed on the estimation of the random effects and exploration of the relationships between them application is made to the investigation of several series of laboratory assay data that were observed during overlapping time intervals and were therefore subjected to common systematic errors or daily effects the motivation for this work was the need to investigate methods of adjustment for such daily effects and to estimate the degree to which concurrently run series are impacted in common attention is given to the construction of confidence intervals for daily effects tractable methods are proposed that yield approximately correct coverage for large samples although derived within a bayes empirical bayes framework these intervals are somewhat similar to intervals constructed by the method of kackar and harville implement

a no go theorem for a lie consistent q campbell baker hausdorff expansion lie consistency is defined for commutator expansions and it is shown that while a weakly lie consistent q campbell baker hausdorff expansion for a product of two m type but not p type q exponentials can be formulated a strongly lie consistent expansion only exists for the undeformed case q 1

prolongation structures for supersymmetric equations the well known prolongation technique of wahlquist and estabrook 1975 for nonlinear evolution equations is generalized for supersymmetric equations and applied to the supersymmetric extension of the kdv equation of manin radul using the theory of kac moody lie superalgebras the explicit form of the resulting lie superalgebra is determined it is shown to be isomorphic to rmr cov c 2 sigma where rmr is an eight dimensional radical an auto backlund transformation is derived from the prolongation structure and the relationship with known solution methods of the skdv equation is analysed in addition it is indicated how a super position principle for the skdv equation can be obtained

bayesian models for directed graphs abstract holland and leinhardt 1981 proposed a simple exponential model called the p 1 model for analyzing digraphs that arise in studies of networks a digraph consists of a set of g nodes and a g g adjacency matrix xij where xij 1 if node i relates to node j and xij 0 otherwise the underlying cell probabilities pij pr xij 1 are to be estimated from these dichotomous responses the p 1 model imposes an additive structure on a log odds version of the pij it provides information about the abilities of an individual node to attract and to produce relational ties as well as the tendency of a pair of nodes to reciprocate ties for digraphs of realistic sizes the maximum likelihood estimates mle s of the p 1 exponential parameters are often unsatisfactory particularly when some of the row and column marginal totals of the adjacency matrix are small a bayesian approach using an exchangeable normal prior on the parameters representing the attractiveness and expansiveness characteristic