norming c algebras by c subalgebras we introduce and investigate the notion of a norming c subalgebra of c algebra we characterize when von neumann algebras norm b h and construct various examples of abelian and hyperfinite norming subalgebras we then apply these results to von neumann algebra cohomology theory and the bounded projection problem for von neumann algebras

comments on algorithm for simultaneous stabilization of single input systems via dynamic feedback wu et al 1990 presented a sufficient condition for the simultaneous stabilizability of single input systems their condition is shown in this note by a counterexample to be incorrect the flaw leading to this error is pointed out and the remedy for their error is provided

order selection in finite mixture models with a nonsmooth penalty order selection is a fundamental and challenging problem in the application of finite mixture models in this article we develop a new penalized likelihood approach the new method modified smoothly clipped absolute deviation mscad deviates from information based methods such as akaike information criterion aic and bayesian information criterion bic by introducing two penalty functions that depend on the mixing proportions and the component parameters it is consistent at estimating both the order of the mixture model and the mixing distribution simulations show that mscad has much better performance than a number of existing methods two real data examples are examined to illustrate the performance of mscad

testing the order of a model using locally conic parametrization population mixtures and stationary arma processes in this paper we address the problem of testing hypotheses using the likelihood ratio test statistic in nonidentifiable models with application to model selection in situations where the parametrization for the larger model leads to nonidentifiability in the smaller model we give two major applications the case where the number of populations has to be tested in a mixture and the case of stationary arma p q processes where the order p q has to be tested we give the asymptotic distribution for the likelihood ratio test statistic when testing the order of the model in the case of order selection for armas the asymptotic distribution is invariant with respect to the parameters generating the process a locally conic parametrization is a key tool in deriving the limiting distributions it allows one to discover the deep similarity between the two problems

detecting features in spatial point processes with clutter via model based clustering abstract we consider the problem of detecting features such as minefields or seismic faults in spatial point processes when there is substantial clutter we use model based clustering based on a mixture model for the process in which features are assumed to generate points according to highly linear multivariate normal densities and the clutter arises according to a spatial poisson process nonlinear features are represented by several densities giving a piecewise linear representation hierarchical model based clustering provides a first estimate of the features and this is then refined using the em algorithm the number of features is estimated from an approximation to its posterior distribution the method gives good results for the minefield and seismic fault problems software to implement it is available on the world wide web

mixture densities maximum likelihood and the em algorithm the problem of estimating the parameters which determine a mixture density has been the subject of a large diverse body of literature spanning nearly ninety years during the last two decades the

mixture of regression models with varying mixing proportions a semiparametric approach in this article we study a class of semiparametric mixtures of regression models in which the regression functions are linear functions of the predictors but the mixing proportions are smoothing functions of a covariate we propose a one step backfitting estimation procedure to achieve the optimal convergence rates for both regression parameters and the nonparametric functions of mixing proportions we derive the asymptotic bias and variance of the one step estimate and further establish its asymptotic normality a modified expectation maximization type em type estimation procedure is investigated we show that the modified em algorithms preserve the asymptotic ascent property numerical simulations are conducted to examine the finite sample performance of the estimation procedures the proposed methodology is further illustrated via an analysis of a real dataset

algorithm for simultaneous stabilization of single input systems via dynamic feedback using a sufficient algebraic stability criterion for polynomials an algorithm is deduced for the design of dynamic feedback which simultaneously stabilizes a set of single input systems the application of this algorithm to the stabilization problem of an aircraft is also presented

a constrained formulation of maximum likelihood estimation for normal mixture distributions in this context the normal densities are sometimes referred to as component densities the log likelihood function l zy corresponding to a random sample xl xnj is defined by l ty ek 1 10g pm xk y the first problem associated with maximum likelihood estimation arises from the unboundedness of l qy on r day 1969 a global maximum likelihood estimate always fails to exist in addition the unboundedness of l y causes failures of optimization algorithms of both the em redner and walker 1984 and quasi newton fowlkes 1979 types in spite of the unboundedness of l 4y statistical theory kiefer 1978 guarantees that a particular local maximizer of l y is strongly consistent and asymptotically efficient several local maximizers can exist for a given sample and the other major maximum likelihood difficulty is in determining when the correct one has been found day 1969 noted that spurious maximizers corresponding to parameter points having some component standard deviations very small relative to others are generated by any small number of sample points grouped sufficiently close together the spurious maximizers like the unboundedness of l zy can create difflculties when using the em or quasi newton algorithms the constrained maximum likelihood formulation presented in the next section avoids the unconstrained problems of singularities and spurious maximizers of the type noted by day section 3 contains the statement and proof of a consistency result for the corresponding constrained estimator

selection of mixed copula model via penalized likelihood a fundamental issue of applying a copula method in applications is how to choose an appropriate copula function for a given problem in this article we address this issue by proposing a new copula selection approach via penalized likelihood plus a shrinkage operator the proposed method selects an appropriate copula function and estimates the related parameters simultaneously we establish the asymptotic properties of the proposed penalized likelihood estimator including the rate of convergence and asymptotic normality and abnormality particularly when the true coefficient parameters may be on the boundary of the parameter space and the dependence parameters are in an unidentified subset of the parameter space we show that the limiting distribution for boundary parameter estimator is half normal and the penalized likelihood estimator for unidentified parameter converges to an arbitrary value finally monte carlo simulation studies are carried out to illustrate the finite sample performance of the pro

mixtures of exponential distributions arbitrary nonparametric mixtures of exponential and weibull fixed shape distributions are considered as possible models for a lifetime distribution a characterization of such distributions is given by the well known characterization of laplace transforms the maximum likelihood estimate of the mixing distribution is investigated and found to be supported on a finite number of points it is shown to be unique and weakly convergent to the true mixing measure with probability one a practical algorithm for computing the maximum likelihood estimate is described its performance is briefly discussed and some illustrative examples given

testing the order of a model this paper deals with order identification for nested models in the i i d framework we study the asymptotic efficiency of two generalized likelihood ratio tests of the order they are based on two estimators which are proved to be strongly consistent a version of stein s lemma yields an optimal underestimation error exponent the lemma also implies that the overestimation error exponent is necessarily trivial our tests admit non trivial underestimation error exponents the optimal underestimation error exponent is achieved in some situations the overestimation error can decay exponentially fast with respect to a positive power of the number of observations these results are proved under mild assumptions by relating the underestimation resp overestimation error to large resp moderate deviations of the log likelihood process in particular it is not necessary that the classical cramer condition be satisfied namely the log densities are not required to admit every exponential moment three benchmark examples with specific difficulties location mixture of normal distributions abrupt changes and various regressions are detailed so as to illustrate the generality of our results

duration dependence and nonparametric heterogeneity a monte carlo study we examine the behaviour of the nonparametric maximum likelihood estimator npmle for a discrete duration model with unobserved heterogeneity and unknown duration dependence we find that a nonparametric specification of either the duration dependence or unobserved heterogeneity when the other feature of the hazard is known to be absent leads to estimators that are well behaved even in modestly sized samples in contrast there is a large and systematic bias in the parameters of these components when both are specified nonparametrically as well as a complementary bias in the coefficients on observed heterogeneity furthermore these biases diminish very gradually as sample size increases we find that a minor modification of the quasilikelihood that penalizes specifications with many points of support leads to a dramatic improvement

testing for a finite mixture model with two components summary we consider a finite mixture model with k components and a kernel distribution from a general one parameter family the problem of testing the hypothesis k 2 versus k 3 is studied there has been no general statistical testing procedure for this problem we propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function it is shown that estimators of the support points are consistent the asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal binomial and poisson kernels suggest that the test proposed performs well simulation studies are also conducted for a bootstrap method with normal kernels an example involving foetal movement data from a medical study illustrates the testing procedure

model selection in high dimensions a quadratic risk based approach we propose a general class of risk measures which can be used for data based evaluation of parametric models the loss function is defined as the generalized quadratic distance between the true density and the model proposed these distances are characterized by a simple quadratic form structure that is adaptable through the choice of a non negative definite kernel and a bandwidth parameter using asymptotic results for the quadratic distances we build a quick to compute approximation for the risk function its derivation is analogous to the akaike information criterion but unlike the akaike information criterion the quadratic risk is a global comparison tool the method does not require resampling which is a great advantage when point estimators are expensive to compute the method is illustrated by using the problem of selecting the number of components in a mixture model where it is shown that by using an appropriate kernel the method is computationally straightforward in arbitrarily high data dimensions in this same context it is shown that the method has some clear advantages over the akaike information criterion and bayesian information criterion

a topology for operator modules overw algebras abstract given a von neumann algebraron a hilbert space h the so calledr topology is introduced into b h which is weaker than the norm and stronger than the ultrastrong operator topology a rightr submodulexof b h is closed in ther topology if and only if for eachb b h the right ideal consisting of alla rsuch thatba x is weak closed inr equivalently xis closed in ther topology if and only if for eachb b h and each orthogonal family of projectionseiinrwith the sum 1 the conditionbei xfor alliimplies thatb x

robust estimation of mixture complexity in many applications it is important to find the mixture with fewest number of components known as the mixture complexity that provides a satisfactory fit to the data this article focuses on developing an estimator of mixture complexity that is consistent when the form of component densities are unknown but are postulated to be members of some parametric family and is simultaneously robust against model misspecification we treat the estimation of mixture complexity as a model selection problem and construct an estimator of mixture complexity as a byproduct of minimizing a hellinger information criterion this estimator is shown to be consistent for any parametric family of mixtures when the model is correctly specified monte carlo simulations for a wide variety of normal mixtures show that our estimator is very competitive with several others in the literature in correctly identifying the true mixture complexity the basic construction being firmly rooted in the minimum hellinger distance approach

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

semiparametric estimation of a two component mixture model suppose that univariate data are drawn from a mixture of two distributions that are equal up to a shift parameter such a model is known to be nonidentifiable from a nonparametric viewpoint however if we assume that the unknown mixed distribution is symmetric we obtain the identifiability of this model which is then defined by four unknown parameters the mixing proportion two location parameters and the cumulative distribution function of the symmetric mixed distribution we propose estimators for these four parameters when no training data is available our estimators are shown to be strongly consistent under mild regularity assumptions and their convergence rates are studied their finite sample properties are illustrated by a monte carlo study and our method is applied to real data

multiple change point analysis fast implementation and strong consistency one of the main challenges in identifying structural changes in stochastic processes is to carry out analysis for time series with dependency structure in a computationally tractable way another challenge is that the number of true change points is usually unknown requiring a suitable model selection criterion to arrive at informative conclusions to address the first challenge we model the data generating process as a segment wise autoregression which is composed of several segments time epochs each of which modeled by an autoregressive model we propose a multiwindow method that is both effective and efficient for discovering the structural changes the proposed approach was motivated by transforming a segment wise autoregression into a multivariate time series that is asymptotically segment wise independent and identically distributed to address the second challenge we derive theoretical guarantees for almost surely selecting the true number of change points of segment wise independent multivariate time series specifically under mild assumptions we show that a bayesian information criterion like criterion gives a strongly consistent selection of the optimal number of change points while an akaike information criterion like criterion cannot finally we demonstrate the theory and strength of the proposed algorithms by experiments on both synthetic and real world data including the eastern u s temperature data and the el nino data the experiment leads to some interesting discoveries about temporal variability of the summer time temperature over the eastern u s and about the most dominant factor of ocean influence on climate which were also discovered by environmental scientists

completely 1 complemented subspaces of schatten spaces we consider the schatten spaces s p in the framework of operator space theory and for any 1 p 2 we characterize the completely 1 complemented subspaces of s p they turn out to be the direct sums of spaces of the form s p h k where h k are hilbert spaces this result is related to some previous work of arazy and friedman giving a description of all 1 complemented subspaces of s p in terms of the cartan factors of types 1 4 we use operator space structures on these cartan factors regarded as subspaces of appropriate noncommutative l p spaces also we show that for any n 2 there is a triple isomorphism on some cartan factor of type 4 and of dimension 2n which is not completely isometric and we investigate l p versions of such isomorphisms

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

a modified likelihood ratio test for homogeneity in finite mixture models summary testing for homogeneity in finite mixture models has been investigated by many researchers the asymptotic null distribution of the likelihood ratio test lrt is very complex and difficult to use in practice we propose a modified lrt for homogeneity in finite mixture models with a general parametric kernel distribution family the modified lrt has a x2 type of null limiting distribution and is asymptotically most powerful under local alternatives simulations show that it performs better than competing tests they also reveal that the limiting distribution with some adjustment can satisfactorily approximate the quantiles of the test statistic even for moderate sample sizes

the haagerup norm on the tensor product of operator modules it is proved that the analogy of the haagerup norm on the tenser product of submodules of b h over a von neumann algebra t b h is injective if r s b h are von neumann algebras with s injective and t r s then the natural map froms ts s equipped with the haagerup norm to cb r s the space of all completely bounded maps from r to s is shown to be an isometry and from this we deduce the result of chattejee and smith that the natural map from the central haagerup tenser product r c r to cb r r is an isometry for each von neumann algebra r it is also shown that for an elementary operator on a prime c algebra with zero socle or on a continuous von neumann algebra the norm is equal to the completely bounded norm

a mixed non homogeneous hidden markov model for categorical data with application to alcohol consumption hidden markov models hmms are frequently used to analyse longitudinal data where the same set of subjects is repeatedly observed over time in this context several sources of heterogeneity may arise at individual and or time level which affect the hidden process that is the transition probabilities between the hidden states in this paper we propose the use of a finite mixture of non homogeneous hmms nh hmms to face the heterogeneity problem the non homogeneity of the model allows us to take into account observed sources of heterogeneity by means of a proper set of covariates time and or individual dependent explaining the variations in the transition probabilities moreover we handle the unobserved sources of heterogeneity at the individual level due to for example omitted covariates by introducing a random term with a discrete distribution the resulting model is a finite mixture of nh hmm that can be used to classify individuals according to their dynamic behaviour or to estimate a mixed nh hmm without any assumption regarding the distribution of the random term following the non parametric maximum likelihood approach we test the effectiveness of the proposal through a simulation study and an application to real data on alcohol abuse

nonparametric inference in multivariate mixtures summary we consider mixture models in which the components of data vectors from any given subpopulation are statistically independent or independent in blocks we argue that if under this condition of independence we take a nonparametric view of the problem and allow the number of subpopulations to be quite general the distributions and mixing proportions can often be estimated root n consistently indeed we show that if the data are k variate and there are p subpopulations then for each p 2 there is a minimal value of k kp say such that the mixture problem is always nonparametrically identifiable and all distributions and mixture proportions are nonparametrically identifiable when k kp we treat the case p 2 in detail and there we show how to construct explicit distribution density and mixture proportion estimators converging at conventional rates other values of p can be addressed using a similar approach although the methodology becomes rapidly more complex as p increases

optimal rate of convergence for finite mixture models in finite mixture models we establish the best possible rate of convergence for estimating the mixing distribution we find that the key for estimating the mixing distribution is the knowledge of the number of components in the mixture while a sqrt n consistent rate is achievable when the exact number of components is known the best possible rate is only n 1 4 when it is unknown under a strong identifiability condition it is shown that this rate is reached by some minimum distance estimators most commonly used models are found to satisfy the strong identifiability condition

nonparametric estimation of convex models via mixtures we present a general approach to estimating probability measures constrained to lie in a convex set we represent constrained measures as mixtures of simple known extreme measures and so the problem of estimating a constrained measure becomes one of estimating an unconstrained mixing measure convex constraints arise in many modeling situations such as estimation of the mean and estimation under stochastic ordering constraints we describe mixture representation techniques for these and other situations and discuss applications to maximum likelihood and bayesian estimation

testing homogeneity in a mixture distribution via the l2 distance between competing models ascertaining the number of components in a mixture distribution is an interesting and challenging problem for statisticians chen chen and kalbfleisch recently proposed a modified likelihood ratio test mlrt which is distribution free and locally most powerful asymptotically in this article we present a new method for testing whether a finite mixture distribution is homogeneous our method the d test is based on the l2 distance between a fitted homogeneous model and a fitted heterogeneous model for mixture components from standard parametric families the d test statistic has a closed form expression in terms of parameter estimators whereas likelihood ratio type test statistics do not the latter test statistics are nontrivial functions of both the parameter estimators and the full dataset the convergence rates of the d test statistic under a null hypothesis of homogeneity and an alternative hypothesis of heterogeneity are established the d test is shown to be competitive with the mlrt when th

model selection via multifold cross validation a natural extension of the simple leave one out cross validation cv method is to allow the deletion of more than one observations in this article several notions of the multifold cross validation mcv method have been discussed in the context of variable selection under a linear regression model we show that the delete d mcv criterion is asymptotically equivalent to the well known fpe criterion two computationally more feasible methods the r fold cross validation and the repeated learning testing criterion are also studied the performance of these criteria are compared with the simple leave one out cross validation method simulation results are obtained to gain some understanding on the small sample properties of these methods

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

minimum hellinger distance estimation for finite mixture models abstract minimum hellinger distance estimates are considered for finite mixture models when the exact forms of the component densities are unknown in detail but are thought to be close to members of some parametric family minimum hellinger distance estimates are asymptotically efficient if the data come from a member of the parametric family and are robust to certain departures from the parametric family a new algorithm is introduced that is similar to the em algorithm a specialized adaptive density estimate is also introduced standard measures of robustness are discussed some difficulties are noted the robustness and asymptotic efficiency of the estimators are illustrated using simulations

discovering subpopulation structure with latent class mixed models the linear mixed model is a well known method for incorporating heterogeneity for example subject to subject variation into a statistical analysis for continuous responses however heterogeneity cannot always be fully captured by the usual assumptions of normally distributed random effects latent class mixed models offer a way of incorporating additional heterogeneity which can be used to uncover distinct subpopulations to incorporate correlated non normally distributed outcomes and to classify individuals the methodology is motivated with examples in health care studies and a detailed illustration is drawn from the nutritional prevention of cancer trials latent class models are used with longitudinal data on prostate specific antigen psa as well as incidence of prostate cancer the models are extended to accommodate prostate cancer as a survival endpoint this is compared to treating it as a binary endpoint four subpopulations are identified which differ both with regard to their psa trajectories and their incidence rates of prostate cancer

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

robust stability with structured real parameter perturbations this paper considers the problem of robust stabilization of a linear time invariant system subject to variations of a real parameter vector for a given controller the radius of the largest stability hypersphere in this parameter space is calculated this radius is a measure of the stability margin of the closed loop system the results developed are applicable to all systems where the closed loop characteristic polynomial coefficients are linear functions of the parameters of interest in particular this always occurs for single input multioutput or single output multiinput systems where the transfer function coefficients are linear or affine functions of the parameters many problems with transfer function coefficients which are nonlinear functions of physical parameters can be cast into this mathematical framework by suitable weighting and redefinition of functions of physical parameters as new parameters the largest stability hyperellipsoid for the case of weighted perturbations and a stability polytope in parameter space are also determined based on these calculations a design procedure is proposed to robustify a given stabilizing controller this algorithm iteratively enlarges the stability hypersphere or hyperellipsoid in parameter space and can be used to design a controller io stabilize a plant subject to given ranges of parameter excursions these results are illustrated by an example

hidden markov models with applications in cell adhesion experiments estimation of the number of hidden states is challenging in hidden markov models motivated by the analysis of a specific type of cell adhesion experiments a new framework based on a hidden markov model and double penalized order selection is proposed the order selection procedure is shown to be consistent in estimating the number of states a modified expectation maximization algorithm is introduced to efficiently estimate parameters in the model simulations show that the proposed framework outperforms existing methods applications of the proposed methodology to real data demonstrate the accuracy of estimating receptor ligand bond lifetimes and waiting times which are essential in kinetic parameter estimation supplementary materials for this article are available online

a factorization problem for normal completely bounded mappings abstract given an operator space x and a von neumann algebra a we consider a contractive mapping q a eh x eh a ncb x a formally defined by q aj xjk bk xjk ajbk from the extended haagerup tensor product a eh x eh a into the space of w continuous completely bounded maps from x into a we characterize elements of the range space im q by a factorization property involving decomposable operators and investigate various properties of that space in the case when x b is the predual of a von neumann algebra b im q is included in the space dec b a of decomposable operators from b into a regarding q as having values in that space we show that q is a quotient map onto its range then we prove that dec b a is a normal dual operator a bimodule and that im q dec b a is a strong operator a bimodule

bayesian model selection in finite mixtures by marginal density decompositions we consider the problem of estimating the number of components d and the unknown mixing distribution in a finite mixture model in which d is bounded by some fixed finite number n our approach relies on the use of a prior over the space of mixing distributions with at most n components by decomposing the resulting marginal density under this prior we discover a weighted bayes factor method for consistently estimating d that can be implemented by an iid generalized weighted chinese restaurant gwcr monte carlo algorithm we also discuss a gibbs sampling method the blocked gibbs sampler for estimating d and also the mixing distribution we show that our resulting posterior is consistent and achieves the frequentist optimal op n 1 4 rate of estimation we compare the performance of the new gwcr model selection procedure with that of the akaike information criterion and the bayes information criterion implemented through an em algorithm applications of our methods to five real datasets and simulation

consistent estimation of mixture complexity the consistent estimation of mixture complexity is of fundamental importance in many applications of finite mixture models an enormous body of literature exists regarding the application computational issues and theoretical aspects of mixture models when the number of components is known but estimating the unknown number of components remains an area of intense research effort this article presents a semiparametric methodology yielding almost sure convergence of the estimated number of components to the true but unknown number of components the scope of application is vast as mixture models are routinely employed across the entire diverse application range of statistics including nearly all of the social and experimental sciences

the predual of the space of decomposable maps from a c algebra into a von neumann algebra abstract for a c algebra a and a von neumann algebra r we describe the predual of space d a r of decomposable maps from a into r equipped with decomposable norm this predual is found to be the matrix regular operator space structure on a r with a certain universal property its matrix norms are the largest and its positive cones on each matrix level are the smallest among all possible matrix regular operator space structures on a r under the two natural restrictions 1 x y x y for x m k a y m l r and 2 v w is positive if v m k a and w m l r

almost nonparametric inference for repeated measures in mixture models summary we consider ways to estimate the mixing proportions in a finite mixture distribution or to estimate the number of components of the mixture distribution without making parametric assumptions about the component distributions we require a vector of observations on each subject this vector is mapped into a vector of os and 1 s and summed the resulting distribution of sums can be modelled as a mixture of binomials we then work with the binomial mixture the efficiency and robustness of this method are compared with the strategy of assuming multivariate normal mixtures when typically the true underlying mixture distribution is different it is shown that in many cases the approach based on simple binomial mixtures is superior let pi x and p2 x be probability distributions with ai and a2 positive real numbers that sum to 1 then p x al pi x a2 p2 x defines a two component mixture distribution with com ponents pl x and p2 x and mixture weights al and a2 mixture distributions arise when observations come from different populations each with a different distribution and the population sampled is unobserved interesting examples include those in the social sciences in which important variables are latent and unobserved as an example consider the problem of trying to infer the solution strategies which young children employ in solving a cognitive task when different solution strategies can be employed a group of same age children can be considered a sample from a mixture distribution in which the components correspond to the various strategies see thomas and horton 1997 recently there has been an expanding interest in mixtures and there is now a large literature examples and theory may be found in many sources including lindsay 1995 mclachlan and basford 1988 titterington et al 1985 and everitt and hand 1981 computational issues in estimating the model parameters rely mainly on the em algorithm see mclachlan and krishnan 1997 for a detailed account two main issues remain unresolved in applications of mixtures one is the specification of the number of components although there may sometimes be substantive theory to guide this specification in exploratory settings

subspaces of c algebras abstract basic properties of matricially normed spaces are considered and a simple matrix norm characterization of the subspaces of c algebras is given the latter result is used to study the haagerup tensor products and quotients of such subspaces