hypothesis test for normal mixture models the em approach normal mixture distributions are arguably the most important mixture models and also the most technically challenging the likelihood function of the normal mixture model is unbounded based on a set of random samples unless an artificial bound is placed on its component variance parameter moreover the model is not strongly identifiable so it is hard to differentiate between over dispersion caused by the presence of a mixture and that caused by a large variance and it has infinite fisher information with respect to mixing proportions there has been extensive research on finite normal mixture models but much of it addresses merely consistency of the point estimation or useful practical procedures and many results require undesirable restrictions on the parameter space we show that an em test for homogeneity is effective at overcoming many challenges in the context of finite normal mixtures we find that the limiting distribution of the em test is a simple function of the 0 5 2 0 5 2 and 2 distributions when the mixing variances are equal but unknown and the 2 when variances are unequal and unknown simulations show that the limiting distributions approximate the finite sample distribution satisfactorily two genetic examples are used to illustrate the application of the em test

a rasch model for partial credit scoring a unidimensional latent trait model for responses scored in two or more ordered categories is developed this partial credit model is a member of the family of latent trait models which share the property of parameter separability and so permit specifically objective comparisons of persons and items the model can be viewed as an extension of andrich s rating scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item the difference between the parameters in this model and the category boundaries in samejima s graded response model is demonstrated an unconditional maximum likelihood procedure for estimating the model parameters is developed

computation of normal forms abstract in this paper a method of computing a normal form for a system of ordinary differential equations is given a program using the symbolic manipulator macsyma is used on several examples

estimating latent correlations between repeated testings a model for longitudinal latent structure analysis is proposed we assume that test scores for a given mental or attitudinal test are observed for the same individuals at two different points in time the purpose of the analysis is to fit a model that combines the values of the latent variable at the two time points in a two dimensional latent density the correlation coefficient between the two values of the latent variable can then be estimated the theory and methods are illustrated by a danish dataset concerning psychic vulnerability

an irt model with a parameter driven process for change an irt model with a parameter driven process for change is proposed quantitative differences between persons are taken into account by a continuous latent variable as in common irt models in addition qualitative interindividual differences and autodependencies are accounted for by assuming within subject variability with respect to the parameters of the irt model in particular the parameters of the irt model are governed by an unobserved or hidden homogeneous markov process the model includes the mixture linear logistic test model mislevy verhelst 1990 the mixture rasch model rost 1990 and the saltus model wilson 1989 as specific instances the model is applied to a longitudinal experiment on discontinuity in conservation acquisition van der maas 1993

the vanishing twist in the restricted three body problem abstract this paper demonstrates the existence of twistless tori and the associated reconnection bifurcations and meandering curves in the planar circular restricted three body problem near the lagrangian equilibrium l 4 a twistless torus is created near the tripling bifurcation of the short period family decreasing the mass ratio leads to twistless bifurcations which are particularly prominent for rotation numbers 3 10 and 2 7 this scenario is studied by numerically integrating the regularised hamiltonian flow and finding rotation numbers of invariant curves in a two dimensional poincare map to corroborate the numerical results the birkhoff normal form at l 4 is calculated to eighth order truncating at this order gives an integrable system and the rotation numbers obtained from the birkhoff normal form agree well with the numerical results a global overview for the mass ratio 4 3 is presented by showing lines of constant energy and constant rotation number in action space

versal normal form at the lagrange equilibrium l4 a new normal form called versal for the linearized hamiltonian vector field of the planar restricted three body problem at the lagrange equilibrium point l4 depending smoothly on the mass ratio for all values close to the critical routh s ratio is described then a canonical transformation also depending smoothly on the mass ratio which brings the linear hamiltonian vector field into this versal normal form is explicitly calculated

the stability of the lagrange triangular point and a theorem of arnold deprit and deprit bartholome 5 computed the normal form for the hamiltonian of the planar restricted three body problem at the lagrange equilateral triangle equilibrium point up to terms of order 4 this normalization was carried out for all mass ratios p smaller than routh s critical mass ratio p1 i 1 j 9 except for two values p2 and pi at p2 and u the ratio of the linearized frequencies is 1 2 and 1 3 respectively and so low order resonance terms appear this normalization was carried out in order to apply the kam theory to prove the stability of this equilibrium point from the coefficients of the normal form they computed a quantity d defined below which when non zero establishes the stability of the equilibrium point they found that d 0 for 0 cp p u p2 p and p the value u does not correspond to resonance but is simply a point where the quantity d4 changes sign in this paper we shall establish the full stability of the lagrange equilibrium point in the planar restricted three body problem even in the case when u p we do this by computing by machine the normal form up to terms of order 6 and then applying a theorem of arnold 3 which establishes the stability of the equilibrium even in degenerate cases russman lo announced a theorem which implies that the equilibrium is isoenergetic stable when p p by isoenergetic stable he means that the system is stabie only on the energy surface defined by the energy at the equilibrium itself unfortunately there is not even an indication of the proof arnold announced his theorem in a doklady note with only a sketch of a proof the sketch indicates that his proof requires completely redoing all

symbolic computation of normal forms for nonlinear hamiltonian systems abstract in this paper a method of computing normal forms for time independent nonlinear hamiltonian systems is given a program using the symbolic manipulator macsyma is used to do an example

rates of convergence for the posterior distributions of mixtures of betas and adaptive nonparametric estimation of the density in this paper we investigate the asymptotic properties of nonparametric bayesian mixtures of betas for estimating a smooth density on 0 1 we consider a parametrization of beta distributions in terms of mean and scale parameters and construct a mixture of these betas in the mean parameter while putting a prior on this scaling parameter we prove that such bayesian nonparametric models have good frequentist asymptotic properties we determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a holder class with regularity for all positive leading to a minimax adaptive estimating procedure of the density we also believe that the approximating results obtained on these mixtures of beta densities can be of interest in a frequentist framework

some neglected problems in irt the paper addresses three neglected questions from irt in section 1 the properties of the measurement of ability or trait parameters and item difficulty parameters in the rasch model are discussed it is shown that the solution to this problem is rather complex and depends both on general assumptions about properties of the item response functions and on assumptions about the available item universe section 2 deals with the measurement of individual change or modifiability based on a rasch test a conditional likelihood approach is presented that yields a an ml estimator of modifiability for given item parameters b allows one to test hypotheses about change by means of a clopper pearson confidence interval for the modifiability parameter or c to estimate modifiability jointly with the item parameters uniqueness results for all three methods are also presented in section 3 the mantel haenszel method for detecting dif is discussed under a novel perspective what is the most general framework within which the mantel haenszel method correctly detects dif of a studied item the answer is that this is a 2pl model where however all discrimination parameters are known and the studied item has the same discrimination in both populations since these requirements would hardly be satisfied in practical applications the case of constant discrimination parameters that is the rasch model is the only realistic framework a simple pearsonx2 test for dif of one studied item is proposed as an alternative to the mantel haenszel test moreover this test is generalized to the case of two items simultaneously studied for dif

nonlinear stability of the equilibria in a double bar rotating system we study the nonlinear stability of the equilibria corresponding to the motion of a particle orbiting around a two finite orthogonal straight segment the potential is a logarithmic function and may be considered as an approximation to the one generated by irregular celestial bodies using arnold s theorem for non definite quadratic forms we determine the nonlinear stability of the equilibria for all values of the parameter of the problem moreover the resonant cases are determined and the stability is investigated

on choosing mixture components via non local priors choosing the number of mixture components remains an elusive challenge model selection criteria can be either overly liberal or conservative and return poorly separated components of limited practical use we formalize non local priors nlps for mixtures and show how they lead to well separated components with non negligible weight interpretable as distinct subpopulations we also propose an estimator for posterior model probabilities under local priors and nlps showing that bayes factors are ratios of posterior to prior empty cluster probabilities the estimator is widely applicable and helps to set thresholds to drop unoccupied components in overfitted mixtures we suggest default prior parameters based on multimodality for normal t mixtures and minimal informativeness for categorical outcomes we characterize theoretically the nlp induced sparsity derive tractable expressions and algorithms we fully develop normal binomial and product binomial mixtures but the theory computation and principles hold more generally we observed a serious lack of sensitivity of the bayesian information criterion insufficient parsimony of the akaike information criterion and a local prior and a mixed behaviour of the singular bayesian information criterion we also considered overfitted mixtures their performance was competitive but depended on tuning parameters under our default prior elicitation nlps offered a good compromise between sparsity and power to detect meaningfully separated components

measuring growth in a longitudinal large scale assessment with a general latent variable model the aim of the research presented here is the use of extensions of longitudinal item response theory irt models in the analysis and comparison of group specific growth in large scale assessments of educational outcomes a general discrete latent variable model was used to specify and compare two types of multidimensional item response theory mirt models for longitudinal data a a model that handles repeated measurements as multiple correlated variables over time and b a model that assumes one common variable over time and additional variables that quantify the change using extensions of these mirt models we approach the issue of modeling and comparing group specific growth in observed and unobserved subpopulations the analyses presented in this paper aim at answering the question whether academic growth is homogeneous across types of schools defined by academic demands and curricular differences in order to facilitate answering this research question a a model with a single two dimensional ability distribution was compared to b a model assuming multiple populations with potentially different two dimensional ability distributions based on type of school and to c a model that assumes that the observations are sampled from a discrete mixture of unobserved populations allowing for differences across schools with respect to mixing proportions for this purpose we specified a hierarchical mixture distribution variant of the two mirt models the latter model c is a growth mixture mirt model that allows for variation of the mixing proportions across clusters in a hierarchically organized sample we applied the proposed models to the pisa i plus data for assessing learning and change across multiple subpopulations the results of this study support the hypothesis of differential growth

a multidimensional latent trait model for measuring learning and change a latent trait model is presented for the repeated measurement of ability based on a multidimensional conceptualization of the change process a simplex structure is postulated to link item performance under a given measurement condition or occasion to initial ability and to one or more modifiabilities that represent individual differences in change since item discriminations are constrained to be equal within a measurement condition the model belongs to the family of multidimensional rasch models maximum likelihood estimators of the item parameters and abilities are derived and an example provided that shows good recovery of both item and ability parameters properties of the model are explored particularly for several classical issues in measuring change

testing the number of components in normal mixture regression models testing the number of components in finite normal mixture models is a long standing challenge because of its nonregularity this article studies likelihood based testing of the number of components in normal mixture regression models with heteroscedastic components we construct a likelihood based test of the null hypothesis of m0 components against the alternative hypothesis of m0 1 components for any m0 the null asymptotic distribution of the proposed modified em test statistic is the maximum of m0 random variables that can be easily simulated the simulations show that the proposed test has very good finite sample size and power properties supplementary materials for this article are available online

inference on the order of a normal mixture finite normal mixture models are used in a wide range of applications hypothesis testing on the order of the normal mixture is an important yet unsolved problem existing procedures often lack a rigorous theoretical foundation many are also hard to implement numerically in this article we develop a new method to fill the void in this important area an effective expectation maximization em test is invented for testing the null hypothesis of arbitrary order m 0 under a finite normal mixture model for any positive integer m 0 2 the limiting distribution of the proposed test statistic is we also use a novel computer experiment to provide empirical formulas for the tuning parameter selection the finite sample performance of the test is examined through simulation studies real data examples are provided the procedure has been implemented in r code the p values for testing the null order of m 0 2 or m 0 3 can be calculated with a single command this article has supplementary materials avail

on the optimal rates of convergence for nonparametric deconvolution problems deconvolution problems arise in a variety of situations in statistics an interesting problem is to estimate the density f of a random variable x based on n i i d observations from y x varepsilon where varepsilon is a measurement error with a known distribution in this paper the effect of errors in variables of nonparametric deconvolution is examined insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions the smoother the harder in fact there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth it is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators

high dimensional exploratory item factor analysis by a metropolis hastings robbins monro algorithm a metropolis hastings robbins monro mh rm algorithm for high dimensional maximum marginal likelihood exploratory item factor analysis is proposed the sequence of estimates from the mh rm algorithm converges with probability one to the maximum likelihood solution details on the computer implementation of this algorithm are provided the accuracy of the proposed algorithm is demonstrated with simulations as an illustration the proposed algorithm is applied to explore the factor structure underlying a new quality of life scale for children it is shown that when the dimensionality is high mh rm has advantages over existing methods such as numerical quadrature based em algorithm extensions of the algorithm to other modeling frameworks are discussed

rates of convergence for the gaussian mixture sieve gaussian mixtures provide a convenient method of densityestimation that lies somewhere between parametric models and kernel densityestimators when the number of components of the mixture is allowed to increase as sample size increases the model is called a mixture sieve we establish a bound on the rate of convergence in hellinger distance for densityestimation using the gaussian mixture sieve assuming that the true density is itself a mixture of gaussians the underlying mixing measure of the true densityis not necessarilyassumed to have finite support computing the rate involves some delicate calculations since the size of the sieve as measured bybracketing entropy and the saturation rate cannot be found using standard methods when the mixing measure has compact support using kn n 2 3 log n 1 3 components in the mixture yields a rate of order log n 1 6 n1 6 for every 0 the rates depend heavilyon the tail behavior of the true density the sensitivity to the tail behavior is diminished byusing a robust sieve which includes a long tailed component in the

on singular hamiltonians the existence of quasi periodic solutions and nonlinear stability on considere l existence de solutions quasiperiodiques pour une classe de hamiltoniens singuliers et la stabilite non lineaire des singularites

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented