the study of existence of equilibria for generalized games without lower semicontinuity in locally topological vector spaces the aim of this paper is to establish general existence results of maximal elements for l majorized mappings which are in turn used to establish the general existence theorems of equilibria for generalized games resp abstract economics without lower semicontinuity for both constraint and preference mappings correspondences and in which strategic resp commodity spaces may not be compact the set of players resp agents may not be countable and underlying spaces are either finite or infinite dimensional locally topological vector spaces our results unify or improve corresponding results in the literature

optimal rates of convergence for deconvolving a density abstract suppose that the sum of two independent random variables x and z is observed where z denotes measurement error and has a known distribution and where the unknown density f of x is to be estimated one application is the estimation of a prior density for a sequence of location parameters a second application arises in the errors in variables problem for nonlinear and generalized linear models when one attempts to model the distribution of the true but unobservable covariates this article shows that if z is normally distributed and f has k bounded derivatives then the fastest attainable convergence rate of any nonparametric estimator of f is only log n k 2 therefore deconvolution with normal errors may not be a practical proposition other error distributions are also treated stefanski carroll 1987a estimators achieve the optimal rates the results given have versions for multiplicative errors where they imply that even optimal rates are exceptionally slow

identifiability and rates of estimation for scale parameters in location mixture models in this paper we consider the problem of identifiability and estimation for the scale parameter 0 in the location mixture model 0 x y where x has a known distribution independent of the y whose distribution is unknown identification of 0 is ensured by constraining y based on the tail behavior of the distribution for x rates for estimation are described for those x which can be written as a square summable series of exponential variables as a special case our analysis shows that the structural parameters in the weibull semiparametric mixture heckman and singer are not estimable at the usual parametric op 1 xn rate the exact relationship between identifying constraints and achievable rates is explained 1 introduction heckman and singer 1984 studied economic theories concerning continuous durations of occupancy of states in order to properly estimate structural parameters in the presence of population heterogeneity they proposed the use of a semiparametric mixture model as a method for modeling duration data the model they proposed was the weibull semiparametric mixture

information in semiparametric mixtures of exponential families in a class of semiparametric mixture models the score function and consequently the effective information for a finite dimensional parameter can be made arbitrarily small depending upon the direction taken in the parameter space this result holds for a broad range of semiparametric mixtures over exponential families and includes examples such as the gamma semiparametric mixture the normal mean mixture the weibull semiparametric mixture and the negative binomial mixture the near zero information rules out the usual parametric n rate for the finite dimensional parameter but even more surprising is that the rate continues to be unattainable even when the mixing distribution is constrained to be countably discrete two key conditions which lead to a loss of information are the smoothness of the underlying density and whether a sufficient statistic is invertible

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

convergence of latent mixing measures in finite and infinite mixture models this paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models using transportation distances i e wasserstein metrics the relationship between wasserstein distances on the space of mixing measures and f divergence functionals such as hellinger and kullback leibler distances on the space of mixture distributions is investigated in detail using various identifiability conditions convergence in wasserstein metrics for discrete measures implies convergence of individual atoms that provide support for the measures thereby providing a natural interpretation of convergence of clusters in clustering applications where mixture models are typically employed convergence rates of posterior distributions for latent mixing measures are established for both finite mixtures of multivariate distributions and infinite mixtures based on the dirichlet process

sieve m inference on irregular parameters abstract this paper presents sieve inferences on possibly irregular i e slower than root n estimable functionals of semi nonparametric models with i i d data we provide a simple consistent variance estimator of the plug in sieve m estimator of a possibly irregular functional and the asymptotic standard normality of the sieve t statistic we show that for hypothesis testing of irregular functionals the sieve likelihood ratio statistic is asymptotically chi square distributed these results are useful in inference on structural parameters that may have singular semiparametric efficiency bounds a simulation study and an empirical application of heckman and singer 1984 duration model are presented

fixed points selections and best approximation for multivalued mappings in r trees abstract in an r tree we consider multivalued mappings with convex values and develop fixed point selection and best approximation theorems for nonexpansive and almost lower semicontinuous mappings

information and asymptotic efficiency in parametric nonparametric models asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems for regular estimates and asymptotic minimax bounds the methods used involve i the notion of a hellinger differentiable root density where part of the differentiation is with respect to the nonparametric part of the model to obtain appropriate scores and ii calculation of the effective score for the real or vector finite dimensional parameter of interest as that component of the score function orthogonal to all nuisance parameter scores perhaps infinite dimensional the resulting asymptotic information for estimation of the parametric component of the model is just 4 times the squared l 2 norm of the effective score a corollary of these results is a simple necessary condition for adaptive estimation adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model examples considered include the one sample location model with and without symmetry mixture models the two sample shift model and cox s proportional hazards model

likelihood ratio test for multi sample mixture model and its application to genetic imprinting genomic imprinting is a known aspect of the etiology of many diseases the imprinting phenomenon depicts differential expression levels of the allele depending on its parental origin when the parental origin is unknown the expression level has a finite normal mixture distribution in such applications a random sample of expression levels consists of three subsamples according to the number of minor alleles an individual possesses of which one is the mixture and the other two are homogeneous this understanding leads to a likelihood ratio test lrt for the presence of imprinting because of the nonregularity of the finite mixture model the classical asymptotic conclusions on likelihood based inference are not applicable we show that the maximum likelihood estimator of the mixing distribution remains consistent more interestingly thanks to the homogeneous subsamples the lrt statistic has an elegant and rather distinct 0 5 21 0 5 22 null limiting distribution simulation studies confirm that the

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

geometrizing rates of convergence ii consider estimating a functional t f of an unknown distribution f in mathbf f from data x 1 cdots x n i i d f let omega varepsilon denote the modulus of continuity of the functional t over mathbf f computed with respect to hellinger distance for well behaved loss functions l t we show that inf t n sup mathbf f e fl t n t f is equivalent to l omega n 1 2 to within constants whenever t is linear and mathbf f is convex the same conclusion holds in three nonlinear cases estimating the rate of decay of a density estimating the mode and robust nonparametric regression we study the difficulty of testing between the composite infinite dimensional hypotheses h 0 t f leq t and h 1 t f geq t delta our results hold in the cases studied because the difficulty of the full infinite dimensional composite testing problem is comparable to the difficulty of the hardest simple two point testing subproblem

a selection theorem for quasi lower semicontinuous mappings in hyperconvex spaces we prove a continuous selection theorem for quasi lower semicontinuous mappings with values that are closed sub admissible subsets of a hyperconvex metric space and apply this result to obtain fixed point theorems in these spaces

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

hyperconvexity and nonexpansive multifunctions it is shown that a ball intersection valued nonexpansive multifunction on a hyperconvex space admits a nonexpansive point valued selection this implies fixed point theorems for such multifunctions and to certain point valued nonexpansive maps the result is used to study best approximation and to show the space of all nonexpansive maps of a bounded hyperconvex space is hyperconvex

remarks on approximate fixed points abstract a mapping f from a topological space x to a metric space m d is said to be r continuous r 0 if for any x x there exists a neighborhood u x of x such that diam f u x r in 1961 victor klee proved that an r continuous self mapping of a compact convex subset of a banach space always has a invariant point for any r recently h x phu extended klee s result to mappings that are almost r continuous in this note we extend phu s result to compact hyperconvex spaces and to geodesically bounded complete r trees we also extend klee s stability result to certain geodesic spaces

approximate selection theorems and their applications abstract we prove three approximate selection theorems and give an improved version of the michael selection theorem as their applications new fixed point theorems and equilibrium theorems for generalized games are established

approximate selections fixed points almost fixed points of multivalued mapping and generalized quasi variational inequalities in h spaces a fixed point theorem for continuous multivalued mapping with closed h convex values is analyzed using the almost fixed point theorem new approximate selection theorems a new almost fixed point theorem for a lower semicontinuous multivalued mapping and a new existence theorem for solutions of a generalized quasi variational inequality in h spaces are discussed

michael s selection theorem under an assumption weaker than lower semicontinuous in h spaces a new continuous selection theorem is proved which unifies and generalizes some known results

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

simple estimation of a duration model with unobserved heterogeneity this paper presents a simple estimator of the shape parameter in a weibull duration model with unobserved heterogeneity the estimator is consistent and asymptotically normal under mild conditions and a consistent estimator of the asymptotic variance is available a monte carlo study indicates that the asymptotic distribution of the estimator provides a good approximation to the finite sample distribution the estimation strategy can be extended to a model with regressors and to a log logistic model with unobserved heterogeneity the advantages of the estimator are that it is easy to calculate and that its asymptotic distribution can be derived copyright 1990 by the econometric society

new results on equilibria of abstract fuzzy economies abstract in this paper we consider the model of an abstract fuzzy economy and we obtain new equilibrium theorems for the situations when the involved correspondences satisfy properties which are weaker than upper semicontinuity in our case the constraint or preference correspondences are almost weakly upper semicontinuous or have the e upper semicontinuous selection property the obtained results either generalize or improve corresponding ones present in the literature

best approximation coincidence and fixed point theorems for quasi lower semicontinuous set valued maps in hyperconvex metric spaces abstract suppose x is a compact admissible subset of a hyperconvex metric spaces m and suppose f x m is a quasi lower semicontinuous set valued map whose values are nonempty admissible suppose also g x x is a continuous onto quasi convex set valued map with compact admissible values then there exists an x 0 x such that d g x 0 f x 0 inf x x d x f x 0 as applications we give some coincidence and fixed point results for weakly inward set valued maps our results generalize some well known results in literature

bounds for bayesian order identification with application to mixtures the efficiency of two bayesian order estimators is studied by using nonparametric techniques we prove new underestimation and overestimation bounds the results apply to various models including mixture models in this case the errors are shown to be o e an and o log n b p n a b 0 respectively 1 introduction order identification deals with the estimation and test of a structural parameter which indexes the complexity of a model in other words the most economical representation of a random phenomenon is sought this problem is encountered in many situations including mixture models 13 19 with an unknown number of components cluster analysis 9 when the number of clusters is unknown autoregressive models 1 when the process memory is not known this paper is devoted to the study of two bayesian estimators of the order of a model frequentist properties of efficiency are particularly investigated we obtain new efficiency bounds under mild assumptions providing a theoretical answer to the questions raised for instance in 7 see their section 4

the singularity of the information matrix of the mixed proportional hazard model this paper presents new identification conditions for the mixed proportional hazard model in particular the baseline hazard is assumed to be bounded away from 0 and near t 0 these conditions ensure that the information matrix is nonsingular the paper also presents an estimator for the mixed proportional hazard model that converges at rate n 1 2

a further generalization of yannelis prabhakar s continuous selection theorem and its applications abstract in this article we first prove one improved version of the yannelis prabhakar continuous selection theorem and next as its applications a fixed point theorem in noncompact product spaces a nonempty intersection theorem some existence theorems of solutions for the generalized quasi variational inequalities and some equilibrium existence theorems for the abstract economies are given

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

asymptotic inference for mixture models by using data dependent priors for certain mixture models improper priors are undesirable because they yield improper posteriors however proper priors may be undesirable because they require subjective input we propose the use of specially chosen data dependent priors we show that in some cases data dependent priors are the only priors that produce intervals with second order correct frequentist coverage the resulting posterior also has another interpretation it is the product of a fixed prior and a pseudolikelihood

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

stabilization of parabolic nonlinear systems with finite dimensional feedback or dynamical controllers application to the navier stokes system let a mathcal d a to mathcal x be the generator of an analytic semigroup and b mathcal v to mathcal d a a relatively bounded control operator in this paper we consider the stabilization of the system y ay bu where u is the linear combination of a family v 1 ldots v k our main result shows that if a b satisfies a unique continuation property and if k is greater than or equal to the maximum of the geometric multiplicities of the unstable modes of a then the system is generically stabilizable with respect to the family v 1 ldots v k with the same functional framework we also prove the stabilizability of a class of nonlinear systems when using feedback or dynamical controllers we apply these results to stabilize the navier stokes equations in two and three dimensions by using boundary controls

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

kernel estimation of hazard functions when observations have dependent and common covariates abstract we propose a hazard model where dependence between events is achieved by assuming dependence between covariates this model allows for correlated variables specific to observations as well as macro variables which all observations share this setup better fits many economic and financial applications where events are not independent nonparametric estimation of the hazard function is then studied kernel estimators proposed in nielsen and linton 1995 and linton et al 2003 are shown to have similar asymptotic properties compared with the i i d case mixing conditions ensure the asymptotic results follow these results depend on adjustments to bandwidth conditions simulations are conducted which verify the impact of dependence on estimators bandwidth selection accounting for dependence is shown to improve performance in an empirical application trade intensity in high frequency financial data is estimated

h infty feedback boundary stabilization of the two dimensional navier stokes equations we study the robust or h infty exponential stabilization of the linearized navier stokes equations around an unstable stationary solution in a two dimensional domain omega the disturbance is an unknown perturbation in the boundary condition of the fluid flow we determine a feedback boundary control law robust with respect to boundary perturbations by solving a max min linear quadratic control problem next we show that this feedback law locally stabilizes the navier stokes system similar problems have been studied in the literature in the case of distributed controls and disturbances to the authors knowledge it is the first time that the robust stabilization of the navier stokes equations is studied for boundary controls and boundary disturbances

on the existence of equilibrium for abstract economies abstract in this paper several existence theorems for abstract economies have been established they are natural modifications strict generalizations when strategy spaces are metrizable of the main results of yannelis and prabhakar 23 tulcea 22 and ding kim and tan 10 for infinite dimensional casas for the finite dimensional case we show that the topological conditions on correspondences in shafer and sonnenschein s result 20 can be much weaker along the proofs several propositions are established which are of independent interest we also show that many existence results in fixed point theory maximization theory and generalized quasi variational inequalities can be encompassed

convex and nonconvex formulations for mixed regression with two components minimax optimal rates we consider the mixed regression problem with two components under adversarial and stochastic noise we give a convex optimization formulation that provably recovers the true solution as well as a nonconvex formulation that works under more general settings and remains tractable upper bounds are provided on the recovery errors for both arbitrary noise and stochastic noise models we also give matching minimax lower bounds up to log factors showing that our algorithm is information theoretical optimal in a precise sense our results represent the first tractable algorithm guaranteeing successful recovery with tight bounds on recovery errors and sample complexity moreover we pinpoint the statistical cost of mixtures our minimax optimal results indicate that the mixture poses a fundamentally more difficult problem in the low snr regime where the learning rate changes

abstract settings for tangential boundary stabilization of navier stokes equations by high and low gain feedback controllers abstract the present paper seeks to continue the analysis in barbu et al tangential boundary stabilization of navier stokes equations memoir ams to appear on tangential boundary stabilization of navier stokes equations d 2 3 as deduced from well posedness and stability properties of the corresponding linearized equations it intends to complement v barbu i lasiecka r triggiani tangential boundary stabilization of navier stokes equations memoir ams to appear on two levels i by casting the riccati based results of barbu et al tangential boundary stabilization of navier stokes equations memoir ams to appear for d 2 3 in an abstract setting thus extracting the key relevant features so that the resulting framework may be applicable also to other stabilizing boundary feedback operators as well as to other parabolic like equations of fluid dynamics ii by including in the case d 2 this time also the low level gain counterpart of the results in barbu et al tangential boundary stabilization of navier stokes equations memoir ams to appear with both riccati based and spectral based tangential feedback controllers this way new local boundary stabilization results of navier stokes equations are obtained over v barbu i lasiecka r triggiani tangential boundary stabilization of navier stokes equations memoir ams to appear

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

uniform rates of estimation in the semiparametric weibull mixture model this paper presents a uniform estimator for a finite dimensional parameter in the semiparametric weibull mixture model the rates achieved by the estimator hold uniformly over shrinking sequences of models much more general than traditional sequences that are required to satisfy a hellinger differentiable property we show that these rates are optimal in a class of identified models constrained by a moment condition on the nonparametric mixing distribution

hypothesis testing in mixture regression models summary we establish asymptotic theory for both the maximum likelihood and the maximum modified likelihood estimators in mixture regression models moreover under specific and reasonable conditions we show that the optimal convergence rate of n 1 4 for estimating the mixing distribution is achievable for both the maximum likelihood and the maximum modified likelihood estimators we also derive the asymptotic distributions of two log likelihood ratio test statistics for testing homogeneity and we propose a resampling procedure for approximating the p value simulation studies are conducted to investigate the empirical performance of the two test statistics finally two real data sets are analysed to illustrate the application of our theoretical results

lyapunov function and local feedback boundary stabilization of the navier stokes equations we study the local exponential stabilization near a given steady state flow of solutions of the navier stokes equations in a bounded domain the control is performed through a dirichlet boundary condition we apply a linear feedback controller provided by a well posed infinite dimensional riccati equation we give a characterization of the domain of the closed loop operator which is obtained from the closed loop linearized navier stokes system we give a class of initial data for which a lyapunov function is obtained for all s in 0 1 2 the stabilization of the two dimensional navier stokes equations is proved for initial data in mathbf h s omega cap v n 0 omega where v n 0 omega is the space in which the stokes operator is defined we also obtain a three dimensional stabilization result but only for a very specific set of initial data