some measures of robustness for unbiased estimators in one parameter natural exponential families with quadratic variance function in this paper two measures to highlight the possible effect of an observation on the umvu estimate are proposed our study is based in expansions in terms of orthogonal polynomials for the umvue when sampling from a nef qvf we obtain the conditional bias and the asymptotic mean sensitivity curve amsc for the umvue we observe that these measures depend on parametric function under consideration at the true and unknown value of the parameter we study in detail their properties and relationships as well as to the hampel s influence function in fact we note that the amsc also verifies for the umvue in the nef qvf some of most relevant properties of influence function also a case deletion influence diagnostic and some simulations are included to illustrate our results

bias reduction in exponential family nonlinear models in firth 1993 biometrika it was shown how the leading term in the asymptotic bias of the maximum likelihood estimator is removed by adjusting the score vector and that in canonical link generalized linear models the method is equivalent to maximizing a penalized likelihood that is easily implemented via iterative adjustment of the data here a more general family of bias reducing adjustments is developed for a broad class of univariate and multivariate generalized nonlinear models the resulting formulae for the adjusted score vector are computationally convenient and in univariate models they directly suggest implementation through an iterative scheme of data adjustment for generalized linear models a necessary and sufficient condition is given for the existence of a penalized likelihood interpretation of the method an illustrative application to the goodman row column association model shows how the computational simplicity and statistical benefits of bias reduction extend beyond generalized linear models

low dimensional real lie algebras all real lie algebras of dimension up to 8 that admit a nontrivial levi decomposition are found

conjugate priors for exponential families having quadratic variance functions abstract consider a natural exponential family parameterized by it is well known that the standard conjugate prior on is characterized by a condition of posterior linearity for the expectation of the model mean parameter often however this family is not parameterized in terms of but rather in terms of a more usual parameter such at the mean the main question we address is under what conditions does a standard conjugate prior on induce a linear posterior expectation on itself we prove that essentially this happens iff the exponential family has quadratic variance function a consequence of this result is that the standard conjugate on coincides with the prior on induced by the standard conjugate on iff the variance function is quadratic the rest of the article covers more specific issues related to conjugate priors for exponential families in particular we analyze the monotonicity of the expected posterior variance for with respect to the sample size and the hyperparameter

conjugate parameterizations for natural exponential families abstract recently consonni and veronese have shown that the form of the standard conjugate distribution for the mean parameter of a univariate natural exponential family f coincides with that of the distribution induced on by the standard conjugate distribution for the canonical parameter if and only if f has a quadratic variance function in this article we present significant extensions of this result identifying conditions under which transformations of the canonical or mean parameter preserve the form of the standard conjugate family generalizations to the multivariate case are also considered and results relating jeffreys s prior to the standard conjugate family are presented the variance function is seen to play an important role throughout

moment based oscillation properties of mixture models consider finite mixture models of the form g x q f f x dq where f is a parametric density and q is a discrete probability measure an important and difficult statistical problem concerns the determination of the number of support points usually known as components of q from a sample of observations from g for an important class of exponential family models we have the following result if p has more than p components and q is an appropriately chosen p component approximation of p then g x p g x q demonstrates a prescribed sign change behavior as does the corresponding difference in the distribution functions these strong structural properties have implications for diagnostic plots for the number of components in a finite mixture

single observation unbiased priors this paper studies a class of default priors which we call single observation unbiased priors soup a prior for a parameter is a soup if the corresponding posterior mean of the parameter based on a single observation is an unbiased estimator of the parameter we prove that under mild regularity conditions a default prior for a convolution parameter is noninformative in the sense of yielding a posterior inference invariant under amalgamation only if it is a soup therefore when amalgamation invariance is desirable as in our motivating example of performing imputation for census undercount soup is the only possible coherent noninformative prior for bayesian predictions that will be utilized under aggregation the use of soup also mutually calibrates bayesian and frequentist inferences for aggregates of convolution parameters across many small areas we describe approaches that identify soups in many common models in particular a constructive duality method that identifies soups in pairs of distribution families we introduce o completeness a necessary and sufficient condition for a prior distribution to be uniquely characterized by the corresponding posterior mean uniqueness of the soup is determined by the o completeness of the dual family o completeness of a natural exponential family is implied by its completeness hence the diaconis ylvisaker characterization of the conjugate prior for natural exponential families via linear posterior expectation is a direct consequence of the completeness of such families for most of the examples we have examined the inverse of the variance function is the soup for the mean parameter of the corresponding family suggesting that hartigan s results on asymptotic unbiasedness can be generalized to some families with discrete parameters we also discuss a possible extension of berger s result on the inadmissibility of unbiased estimators as the nonexistence of soup can be a first step in establishing such inadmissibility

hierarchical poisson regression modeling abstract the poisson model and analyses here feature nonexchangeable gamma distributions although exchangeable following a scale transformation for individual parameters with standard deviations proportional to means a relatively uninformative prior distribution for the shrinkage values eliminates the ill behavior of maximum likelihood estimators of the variance components when tested in simulation studies the resulting procedure provides better coverage probabilities and smaller risk than several other published rules and thus works well from bayesian and frequentist perspectives alike the computations provide fast accurate density approximations to individual parameters and to structural regression coefficients the computer program is publicly available through statlib

the finite element approximation of a coupled reaction diffusion problem with non lipschitz nonlinearities a coupled semilinear elliptic problem modelling an irreversible isothermal chemical reaction is introduced and discretised using the usual piecewise linear galerkin finite element approximation an interesting feature of the problem is that a reaction order of less than one gives rise to a dead core region initially one reactant is assumed to be acting as a catalyst and is kept constant it is shown that error bounds previously obtained for a scheme involving numerical integration can be improved upon by considering a quadratic regularisation of the nonlinear term this technique is then applied to the full coupled problem and optimal h 1 and l infty error bounds are proved in the absence of quadrature for a scheme involving numerical integration bounds similar to those obtained for the catalyst problem are shown to hold

symmetry classes of variable coefficient nonlinear schrodinger equations a variable coefficient nonlinear schrodinger vcnls equation involving three arbitrary complex functions of space time in 1 1 dimensions is analysed from the point of view of its symmetries all equations of the type studied having non trivial lie point symmetry groups g are identified the symmetry group is shown to be at most five dimensional and only when the equation is equivalent to the nls equation itself or to a rather special complex ginzburg landau equation lie point transformations are used to obtain solutions of specific vcnls equations that should be of interest in nonlinear optics or other branches of physics

dead cores in nonlinear reaction diffusion systems on considere le systeme elliptique semilineaire u f u g v x u a x v k f u g v x v b x ou est un domaine borne dans r n 0 k 0 a x 0 b x 0 et max x a x max x b x 1

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

multilevel mixed linear model analysis using iterative generalized least squares summary models for the analysis of hierarchically structured data are discussed an iterative generalized least squares estimation procedure is given and shown to be equivalent to maximum likelihood in the normal case there is a discussion of applications to complex surveys longitudinal data and estimation in multivariate models with missing responses an example is given using educational data

classical point symmetries of a porous medium equation the lie group formalism is applied to deduce symmetries of the porous medium equation we study those spatial forms that admit the classical symmetry group the reduction obtained from the optimal system of subalgebras are derived some new exact solutions can be obtained

mean squared error properties of empirical bayes estimators in a multivariate random effects general linear model abstract estimation of an individual s regression coefficients is considered in a multivariate general linear model where it is assumed that the individual s coefficients k are subject to both fixed effects and random effects over different individuals the mean squared error matrix of a natural estimator of k is derived for any individual k in the general situation where all parameters of the model must be estimated and is shown to be smaller than the mean squared error matrix of the individual least squares estimator for every individual extension of this result to more general multiparameter estimation situations is also considered

fast moment based estimation for hierarchical models summary hierarchical models allow for heterogeneous behaviours in a population while simultaneously borrowing estimation strength across all subpopulations unfortunately existing likelihood based methods for fitting hierarchical models have high computational demands and these demands have limited their adoption in large scale prediction and inference problems the paper proposes a moment based procedure for estimating the parameters of a hierarchical model which has its roots in a method originally introduced by cochran in 1937 the method trades statistical efficiency for computational efficiency it gives consistent parameter estimates competitive prediction error performance and substantial computational improvements when applied to a large scale recommender system application and compared with a standard maximum likelihood procedure the method delivers competitive prediction performance while reducing the sequential computation time from hours to minutes

natural exponential families with quadratic variance functions statistical theory the normal poisson gamma binomial negative binomial and nefghs distributions are the six univariate natural exponential families nef with quadratic variance functions qvf this sequel to morris 1982 treats certain statistical topics that can be handled within this unified nef qvf formulation including unbiased estimation bhattacharyya and cramer rao lower bounds conditional distributions and moments quadratic regression conjugate prior distributions moments of conjugate priors and posterior distributions empirical bayes and g 2 minimax marginal distributions and their moments parametric empirical bayes and characterizations

on point transformations of generalized nonlinear diffusion equations this paper classifies all finite point transformations of a general class between generalized diffusion equations of the form ut x1 m xn 1f u ux x these transformations may be divided into three cases depending on the functional form of f u i f arbitrary ii f un and iii f eu in particular these transformations include all the invariant infinitesimal transformations and in addition they include a number of point transformations which relate different equations of the above form many exact solutions are already known and the transformations which are derived may be used to obtain new solutions from these

infinite dimensional symmetries of two dimensional generalized burgers equations the conditions for a class of generalized burgers equations which a priori involve nine arbitrary functions of one or two variables to allow an infinite dimensional symmetry algebra are determined although this algebra can involve up to two arbitrary functions of time it does not allow a virasoro subalgebra this result reconfirms a long standing fact that variable coefficient generalizations of a nonintegrable equation should be expected to remain as such

constrained bayes estimation with applications abstract bayesian techniques are widely used in these days for simultaneous estimation of several parameters in compound decision problems often however the main objective is to produce an ensemble of parameter estimates whose histogram is in some sense close to the histogram of population parameters this is for example the situation in subgroup analysis where the problem is not only to estimate the different components of a parameter vector but also to identify the parameters that are above and the others that are below a certain specified cutoff point we have proposed in this paper bayes estimates in a very general context that meet this need these estimates are obtained by matching the first two moments of the histogram of the estimates and the posterior expectations of the first two moments of the histogram of the parameters and minimizing subject to these conditions the posterior expectation of the euclidean distance between the estimates and the parameters several applications of the m

group classification of the general second order evolution equation semi simple invariance groups in this paper we consider the problem of group classification of the generic second order evolution equation in one spatial variable we construct all inequivalent evolution equations whose invariance groups are either semi simple or semi direct products of semi simple and solvable lie groups the obtained lists of invariant equations contain both already known equations and the broad classes of new evolution equations possessing nontrivial lie symmetry

new scale invariant nonlinear differential equations for a complex scalar field abstract we describe all complex wave equations of the form u f u u invariant under the extended poincare group as a result we have obtained the five new classes of p 1 3 invariant nonlinear partial differential equations for the complex scalar field

a multirate dsp model for estimation of discrete probability density functions the problem of estimating a probability density function pdf from measurements has been widely studied by many researchers even though much work has been done in the area of pdf estimation most of it was focused on the continuous case we propose a new model based approach for modeling and estimating discrete probability density functions or probability mass functions this approach is based on multirate signal processing theory and it has several advantages over the conventional histogram method we illustrate the pdf estimation procedure and analyze the statistical properties of the pdf estimates based on this model a novel scheme is introduced that can be used for estimating the pdf in the presence of noise furthermore the proposed ideas are extended to the more general case of estimating multivariate pdfs finally we also consider practical issues such as optimizing the coefficients of a digital filter which is an integral part of the model this allows us to apply the proposed model to solve real world problems simulation results are given where appropriate in order to demonstrate the ideas

bias reduction of maximum likelihood estimates summary it is shown how in regular parametric problems the first order term is removed from the asymptotic bias of maximum likelihood estimates by a suitable modification of the score function in exponential families with canonical parameterization the effect is to penalize the likelihood by the jeffreys invariant prior in binomial logistic models poisson log linear models and certain other generalized linear models the jeffreys prior penalty function can be imposed in standard regression software using a scheme of iterative adjustments to the data

merging experts opinions a bayesian hierarchical model with mixture of prior distributions abstract in this paper a general approach is proposed to address a full bayesian analysis for the class of quadratic natural exponential families in the presence of several expert sources of prior information by expressing the opinion of each expert as a conjugate prior distribution a mixture model is used by the decision maker to arrive at a consensus of the sources a hyperprior distribution on the mixing parameters is considered and a procedure based on the expected kullback leibler divergence is proposed to analytically calculate the hyperparameter values next the experts prior beliefs are calibrated with respect to the combined posterior belief over the quantity of interest by using expected kullback leibler divergences which are estimated with a computationally low cost method finally it is remarkable that the proposed approach can be easily applied in practice as it is shown with an application

small area estimation based on natural exponential family quadratic variance function models and survey weights summary we propose pseudo empirical best linear unbiased estimators of small area means based on natural exponential family quadratic variance function models when the basic data consist of survey weighted estimators of these means area specific covariates and certain summary measures involving the weights we also provide explicit approximate mean squared errors of these estimators in the spirit of prasad rao 1990 and these estimators can be readily evaluated a simulation study is undertaken to evaluate the performance of the proposed inferential procedure we estimate also the proportion of poor children in the 5 17 years age group for the different counties in one of the states in the united states

multiscale poisson data smoothing summary the paper introduces a framework for non linear multiscale decompositions of poisson data that have piecewise smooth intensity curves the key concept is conditioning on the sum of the observations that are involved in the computation of a given multiscale coefficient within this framework most classical wavelet thresholding schemes for data with additive homoscedastic noise can be used any family of wavelet transforms orthogonal biorthogonal or second generation can be incorporated in this framework our second contribution is to propose a bayesian shrinkage approach with an original prior for coefficients of this decomposition as such the method combines the advantages of the haar fisz transform with wavelet smoothing and bayesian multiscale likelihood models with additional benefits such as extendability towards arbitrary wavelet families simulations show an important reduction in average squared error of the output compared with the present techniques of anscombe or fisz variance stabilization or multiscale likelihood modelling

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

galilei symmetries of kdv type nonlinear evolution equations abstract we perform galilei symmetry group classification of a class of third order nonlinear evolution equations in one spatial variable which generalize kdv and mkdv equations all inequivalent pdes belonging to the class in question which admit the classical galilei group the extended galilei group and the natural extensions of the extended galilei groups are constructed the list of so obtained invariant equations may be used as motion equations for they all satisfy the galilei relativity principle in addition we also give a complete classification of group invariant solutions for one obtained galilei invariant equation

parametric empirical bayes tolerance intervals the theory of a parametric empirical bayes peb normal distribution tolerance bound is derived classical tolerance intervals are based on the assumption of a fixed unknown standard deviation and mean such intervals assume no prior knowledge about the possible value of the empirical bayes approach assumes that there is prior knowledge of the value of which can be characterized in terms of a translated beta density with shape parameters and range parameters a b the resulting peb k factors are smaller than those of the classical approach so fewer samples are required to achieve equal precision an example illustrates potential cost savings extensive tables are not provided since the peb k factors depend on a b as well as the sample size confidence and coverage probability since the exact computation is intensive i suggest an approximation based on the noncentral t distribution the degrees of freedom and noncentrality parameters are obtained by matching the first two

the natural variability of vital rates and associated statistics the first concern of this work is the development of approximations to the distributions of crude mortality rates age specific mortality rates age standardized rates standardized mortality ratios and the like for the case of a closed population or period study it is found that assuming poisson birthtimes and independent lifetimes implies that the number of deaths and the corresponding midyear population have a bivariate poisson distribution it is suggested that situations in which explanatory variables are present may be modelled via a doubly stochastic poisson distribution for the number of deaths with mean proportional to the population size and an exponential function of a linear combination of the explanatories such a model is fit to mortality data for canadian females classified by age and year a dynamic variant of the model is further fit to the time series of total female deaths alone by year the models with extra poisson variation are found to lead to substantially improved fits excerpt

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

unbalanced repeated measures models with structured covariance matrices the question of how to analyze unbalanced or incomplete repeated measures data is a common problem facing analysts we address this problem through maximum likelihood analysis using a general linear model for expected responses and arbitrary structural models for the within subject covariances models that can be fit include standard univariate and multivariate models with incomplete data random effects models and models with time series and factor analytic error structures we describe newton raphson and fisher scoring algorithms for computing maximum likelihood estimates and generalized em algorithms for computing restricted and unrestricted maximum likelihood estimates an example fitting several models to a set of growth data is included

extension of the stein estimating procedure through the use of estimating functions abstract this article extends the stein method through the use of estimating functions godambe 1960 to address the simultaneous estimation of k population parameters 1 k in a mixed model setting the procedure generalizes the stein method by a allowing us to deal effectively with complications such as inequality of population variances that may arise in non gaussian mixed models b being appropriate for estimating i in populations of varying sizes and in particular populations of small sizes and c applying to situations where it cannot be assumed that the i s have unbiased estimators or even estimators of finite moment the focus of the article is on the quadratic variance function exponential family morris 1983b estimators for the parameters of the mixed model are developed in a regression model setting in which the i s are allowed to vary with a vector of covariates an application to incidence rates for the iceland breast cancer incidence data is presented for illustra

the formation of the dead core in parabolic reaction diffusion problems on donne des conditions pour l existence d une region a concentration de reactant nulle pour des problemes paraboliques non lineaires de reaction diffusion

group classification of nonlinear wave equations we perform complete group classification of the general class of quasilinear wave equations in two variables this class may be seen as a generalization of the nonlinear d alembert liouville sin sinh gordon and tzitzeica equations we derive a number of new genuinely nonlinear invariant models with high symmetry properties in particular we obtain four classes of nonlinear wave equations that admit five dimensional invariance groups

preliminary group classification of a class of fourth order evolution equations we perform preliminary group classification of a class of fourth order evolution equations in one spatial variable following the approach developed by basarab horwath et al acta appl math 69 43 2001 we construct all inequivalent partial differential equations belonging to the class in question which admit semisimple lie groups in addition we describe all fourth order evolution equations from the class under consideration which are invariant under solvable lie groups of dimension n 4 we have constructed all galilei invariant equations belonging to the class of evolution differential equations under study the list of so obtained invariant equations contains both the well known fourth order evolution equations and a variety of new ones possessing rich symmetry and as such may be used to model nonlinear processes in physics chemistry and biology

maximum likelihood algorithms for generalized linear mixed models abstract maximum likelihood algorithms are described for generalized linear mixed models i show how to construct a monte carlo version of the em algorithm propose a monte carlo newton raphson algorithm and evaluate and improve the use of importance sampling ideas calculation of the maximum likelihood estimates is feasible for a wide variety of problems where they were not previously i also use the newton raphson algorithm as a framework to compare maximum likelihood to the joint maximization or penalized quasi likelihood methods and explain why the latter can perform poorly

monotone empirical bayes estimators based on more informative samples abstract an empirical bayes estimator is constructed for the case of nonidentical components coming from an exponential family of distributions that is closed under convolution examples are the normal poisson and binomial distributions the construction is based on robbins approach to empirical bayes estimation and generalizes results of o bryan and stijnen and van houwelingen the method is elaborated for the normal and poisson case and examples are given that show that this approach gives results comparable to those from the nonparametric maximum likelihood approach of laird

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

group classification of heat conductivity equations with a nonlinear source we suggest a systematic procedure for classifying partial differential equations pdes invariant with respect to low dimensional lie algebras this procedure is a proper synthesis of the infinitesimal lie method the technique of equivalence transformations and the theory of classification of abstract low dimensional lie algebras as an application we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms we have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry it is shown that there are 3 7 28 and 12 inequivalent classes of pdes of the type considered that are invariant under the one two three and four dimensional lie algebras correspondingly furthermore we prove that any pde belonging to the class under study and admitting the symmetry group of a dimension higher than four is locally equivalent to a linear equation this classification is compared with existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach furthermore a number of new invariant equations are constructed which have rich symmetry properties and therefore may be used for mathematical modelling of say nonlinear heat transfer processes

on relation between potential and contact symmetries of evolution equations we prove that any evolution equation that can be written in the form of a conservation law reduces to an evolution equation admitting the group of contact symmetries leaving the temporal variable invariant and vice versa one of the consequences of this fact is that any evolution equation admitting the potential symmetries can be transformed into another evolution equation so that the potential symmetries map into contact symmetries of the latter based on this fact is out group approach to classification of evolution equations possessing nonlocal symmetries we present several examples of classifications of second order evolution equations admitting potential symmetries

using empirical bayes methods in biopharmaceutical research a compound sampling model where a unit specific parameter is sampled from a prior distribution and then observed are generated by a sampling distribution depending on the parameter underlies a wide variety of biopharmaceutical data for example in a multi centre clinical trial the true treatment effect varies from centre to centre observed treatment effects deviate from these true effects through sampling variation knowledge of the prior distribution allows use of bayesian analysis to compute the posterior distribution of clinic specific treatment effects frequently summarized by the posterior mean and variance more commonly with the prior not completely specified observed data can be used to estimate the prior and use it to produce the posterior distribution an empirical bayes or variance component analysis in the empirical bayes model the estimated prior mean gives the typical treatment effect and the estimated prior standard deviation indicates the heterogeneity of treatment effects in both the bayes and empirical bayes approaches estimated clinic effects are shrunken towards a common value from estimates based on single clinics this shrinkage produces more efficient estimates in addition the compound model helps structure approaches to ranking and selection provides adjustments for multiplicity allows estimation of the histogram of clinic specific effects and structures incorporation of external information this paper outlines the empirical bayes approach coverage will include development and comparison of approaches based on parametric priors for example a gaussian prior with unknown mean and variance and non parametric priors discussion of the importance of accounting for uncertainty in the estimated prior comparison of the output and interpretation of fixed and random effects approaches to estimating population values estimating histograms and identification of key considerations in the use and interpretation of empirical bayes methods

on form preserving point transformations of partial differential equations new identities are presented relating arbitrary order partial derivatives of and for the general point transformation these identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 1 partial differential equations these results facilitate the search for point symmetries both discrete and continuous lie and assist the search for point transformations which reduce equations to canonical but similar form a simple test for the existence of hodograph type transformations between equations of similar form is given

new results on group classification of nonlinear diffusion convection equations using a new method and additional conditional and partial equivalence transformations we performed group classification in a class of variable coefficient 1 1 dimensional nonlinear diffusion convection equations of the general form f x ut d u ux x k u ux we obtain new interesting cases of such equations with the density f localized in space which have non trivial invariance algebra exact solutions of these equations are constructed we also consider the problem of investigation of the possible local transformations for an arbitrary pair of equations from the class under consideration i e of describing all the possible partial equivalence transformations in this class

a fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects on decrit un algorithme qui utilise des formules explicites pour l inverse et le determinant de la matrice de covariance donnee par la motte 1972 et evite l inversion des grandes matrices

robust empirical bayes analyses of event rates a collection of i similar items generates point event histories for example machines experience failures or operators make mistakes suppose the intervals between events are modeled as iid exponential i or the counts as poisson i t i for the ith item furthermore so as to represent between item variability each individual rate parameter i is presumed drawn from a fixed super population with density g being a vector parameter a parametric empirical bayes peb setup for g specified alternatively as log student t n or gamma we exhibit the results of numerical procedures for estimating superpopulation parameters ll and for describing pooled estimates of the individual rates i obtained via bayes s formula three data sets are analyzed and convenient explicit approximate formulas are furnished for i estimates in the student t case the individual estimates are seen to have a robust quality

symmetry classification of kdv type nonlinear evolution equations group classification of a class of third order nonlinear evolution equations generalizing kdv and mkdv equations is performed it is shown that there are two equations admitting simple lie algebras of dimension three next we prove that there exist only four equations invariant with respect to lie algebras having nontrivial levi factors of dimension four and six our analysis shows that there are no equations invariant under algebras which are semi direct sums of levi factor and radical making use of these results we prove that there are three nine thirty eight fifty two inequivalent kdv type nonlinear evolution equations admitting one two three and four dimensional solvable lie algebras respectively finally we perform a complete group classification of the most general linear third order evolution equation