overcoming scale usage heterogeneity questions that use a discrete ratings scale are commonplace in survey research examples in marketing include customer satisfaction measurement and purchase intention survey research practitioners have long commented that respondents vary in their usage of the scale common patterns include using only the middle of the scale or using the upper or lower end these differences in scale usage can impart biases to correlation and regression analyses to capture scale usage differences we developed a new model with individual scale and location effects and a discrete outcome variable we model the joint distribution of all ratings scale responses rather than specific univariate conditional distributions as in the ordinal probit model we apply our model to a customer satisfaction survey and show that the correlation inferences are much different once proper adjustments are made for the discreteness of the data and scale usage we also show that our adjusted or latent ratings scale is more closely related to

a priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient abstract we consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient such as f x u d u d 2 u c x u m x d u d u h x in a bounded domain with a dirichlet boundary condition here r c h l p p n 1 c 0 and the matrix m satisfies 0 1 i m 2 i recently this problem was studied in the coercive case c 0 where uniqueness of solutions can be expected and it was conjectured that the solution set is more complex for noncoercive equations this conjecture was verified in 2015 by arcoya de coster jeanjean and tanaka for equations in divergence form by exploiting the integral formulation of the problem here we show that similar phenomena occur for general even fully nonlinear equations in nondivergence form we use different techniques based on the maximum principle we develop a new method to obtain the crucial uniform a priori bounds which permit to us to use degree theory this method is based on basic regularity estimates such as half harnack inequalities and on a vazquez type strong maximum principle for our kind of equations

backward stochastic differential equations and partial differential equations with quadratic growth we provide existence comparison and stability results for one dimensional backward stochastic differential equations bsdes when the coefficient or generator f t y z is continuous and has a quadratic growth in z and the terminal condition is bounded we also give in this framework the links between the solutions of bsdes set on a diffusion and viscosity or sobolev solutions of the corresponding semilinear partial differential equations

existence of irreducible representations for knot complements with nonconstant equivariant signature mathematics subject classi cation 1991 57m25 57m05 58g03 57r571 introductionthe purpose of this paper is to establish the existence of irreducible su 2 repre sentations of knot complement fundamental groups near abelian representationswhere the equivariant knot signature changes for knots in s

the method of simulated scores for the estimation of ldv models the method of simulated scores mss is presented for estimating limited dependent variables models ldv with flexible correlation structure in the unobservables we propose simulators that are continuous in the unknown parameter vectors and hence standard optimization methods can be used to compute the mss estimators that employ these simulators the first continuous method relies on a recursive conditioning of the multivariate normal density through a cholesky triangularization of its variance covariance matrix the second method combines results about the conditionals of the multivariate normal distribution with gibbs resampling techniques we establish consistency and asymptotic normality of the mss estimators and derive suitable rates at which the number of simulations must rise if biased simulators are used

uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions in this paper we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions

multiplicity results in the non coercive case for an elliptic problem with critical growth in the gradient abstract we consider the boundary value problem p u c x u x u 2 h x u h 0 1 l where r n n 3 is a bounded domain with smooth boundary it is assumed that c h belong to l p for some p n with c 0 as well as l and 1 0 for some 1 r it is known that when 0 problem p has at most one solution in this paper we study under various assumptions the structure of the set of solutions of p assuming that 0 our study unveils the rich structure of this problem we show in particular that what happen for 0 influences the set of solutions in all the half space 0 h 0 1 l most of our results are valid without assuming that h has a sign if we require h to have a sign we observe that the set of solutions differs completely for h 0 and h 0 we also show when h has a sign that solutions not having this sign may exists some uniqueness results of signed solutions are also derived the paper ends with a list of open problems

using penalized likelihood to select parameters in a random coefficients multinomial logit model abstract this paper is about estimating a random coefficients logit model in which the distribution of each coefficient is characterized by finitely many parameters some of which may be zero the paper gives conditions under which with probability approaching 1 as the sample size increases penalized maximum likelihood pml estimation with the adaptive lasso al penalty distinguishes correctly between zero and non zero parameters the paper also gives conditions under which pml reduces the asymptotic mean square estimation error of any continuously differentiable function of the model s parameters the paper describes a method for computing pml estimates and presents the results of monte carlo experiments that illustrate their performance it also presents the results of pml estimation of a random coefficients logit model of choice among brands of butter and margarine in the british groceries market

on the comparison principle for unbounded solutions of elliptic equations with first order terms abstract we prove a comparison principle for unbounded weak sub super solutions of the equation u div a x d u h x d u in where a x is a bounded coercive matrix with measurable ingredients 0 and h x has a super linear growth and is convex at infinity we improve earlier results where the convexity of h x was required to hold globally

comparison results and improved quantified inequalities for semilinear elliptic equations in this paper we prove some pointwise comparison results between the solutions of some second order semilinear elliptic equations in a domain omega of mathbb r n and the solutions of some radially symmetric equations in the equimeasurable ball omega the coefficients of the symmetrized equations in omega satisfy similar constraints as the original ones in omega we consider both the case of equations with linear growth in the gradient and the case of equations with at most quadratic growth in the gradient moreover we show some improved quantified comparisons when the original domain is not a ball the method is based on a symmetrization of the second order terms

consistent sets of estimates for regressions with correlated or uncorrelated measurement errors in arbitrary subsets of all variables over the last decade the problem of measurement errors in the independent variables of a regression equation has attracted renewed interest among econometricians in the fifties and sixties the problem was considered to be more or less hopeless due to its inherent underidentification e g theil 1971 apart from instrumental variables the most frequently cited textbook solution was wald s method of grouping wald 1940 recent insight into the properties of the method of grouping can be interpreted as making this method worthless in most practical cases pakes 1982 since about 1970 new approaches to the problem have been explored basically along three lines viz embedding the error ridden equation into a set of multiple equations e g zellner 1970 goldberger 1972 into a set of simultaneous equations e g hsiao 1976 geraci 1976 and using the dynamics of the equation if present e g maravall and aigner 1977 in view of the underidentification of the basic model it is clear that all these methods invoke additional information of some kind if this information takes the form of exact or stochastic knowledge of certain parameters in the model the construction of consistent estimators is fairly straightforward e g fuller 1980 kapteyn and wansbeek 1984 for an overview of the state of the art see aigner et al 1984 an approach somewhat orthogonal to the ones described above has been to take the model as it is and to use prior ideas about the size of the measurement errors to diagnose how serious the probem is examples are blomqvist 1972 hodges and moore 1972 and davies and hutton 1975 leamer 1983 starts from the opposite direction by asking how serious the measurement error problem has to be in order to render the data useless for inference that is to say when measurement error is large enough to make it impossible to put bounds on regression parameters in an empirical example he shows that even very small measurement errors in some explanatory variables would open up the possibility of perfectly collinear explanatory variables and hence make the data useless for statistical inference at least without additional prior information the most systematic analysis of the information loss caused by measurement error is due to klepper and leamer 1984 they start out by invoking a minimal amount of prior information and then ask the question under what conditions it is still possible to make some inferences regarding the vector of unknown regression parameters p in the special case where the measurement errors are assumed uncorrelated and the k 1 estimates of 3 obtained by regressing each of the k 1 variables involved i e the one dependent variable and the k independent variables on the remaining k variables are all in the same orthant one can bound the ml estimates of p in that case the convex hull of the k 1 regressions contains all possible ml estimates and any point in the hull is a possible ml estimate if the k 1 regressions are not all in the same orthant then the set of ml estimates is unbounded in that case klepper and leamer 1984 introduce extra prior information which allows them to bound the set of maximum likelihood estimates the prior information comes in two forms firstly a researcher is supposed to be able to specify a maximum value of r2 if all exogenous variables were measured accurately it is shown that if this maximum is low enough one can again bound the set of ml estimates by a convex hull secondly if

existence and comparison results for quasilinear elliptic equations with critical growth in the gradient abstract we give a condition of existence of bounded solutions for a general quasilinear elliptic problem with quadratic growth in the gradient when the data are small enough moreover we obtain an a e comparison with the semilinear symmetrized problem

hypothesis testing with a restricted parameter space abstract this paper considers hypothesis tests when the parameter space is restricted under the alternative hypothesis multivariate one sided tests are a leading example optimal tests called directed tests are derived using a weighted average power criterion exact results are established first for gaussian linear regression models with known variance asymptotic analogues are then established for dynamic nonlinear models simulation is used to compare the tests discussed in the paper the d w directed test is found to perform best in an overall sense for multivariate one sided alternatives with the likelihood ratio test being a close second the d w and likelihood ratio tests are found to perform best for mixed one and two sided alternatives

shrinkage tuning parameter selection with a diverging number of parameters contemporary statistical research frequently deals with problems involving a diverging number of parameters for those problems various shrinkage methods e g the lasso and smoothly clipped absolute deviation are found to be particularly useful for variable selection nevertheless the desirable performances of those shrinkage methods heavily hinge on an appropriate selection of the tuning parameters with a fixed predictor dimension wang and co worker have demonstrated that the tuning parameters selected by a bayesian information criterion type criterion can identify the true model consistently in this work similar results are further extended to the situation with a diverging number of parameters for both unpenalized and penalized estimators consequently our theoretical results further enlarge not only the scope of applicability of the traditional bayesian information criterion type criteria but also that of those shrinkage estimation methods

the positive semidefiniteness of partitioned matrices abstract the positive semidefiniteness of a partitioned matrix is characterized in terms of its submatrices the result is applied to a variety of problems concerning lowner ordered matrices which need not be partitioned themselves

continuum of solutions for an elliptic problem with critical growth in the gradient abstract we consider the boundary value problem p u h 0 1 l u c x u x u 2 h x where r n n 3 is a bounded domain with smooth boundary it is assumed that c 0 c h belong to l p for some p n 2 and that l we explicitly describe a condition which guarantees the existence of a unique solution of p when 0 and we show that these solutions belong to a continuum the behaviour of the continuum depends in an essential way on the existence of a solution of p 0 it crosses the axis 0 if p 0 has a solution otherwise it bifurcates from infinity at the left of the axis 0 assuming that p 0 has a solution and strengthening our assumptions to x 1 0 and h 0 we show that the continuum bifurcates from infinity on the right of the axis 0 and this implies in particular the existence of two solutions for any 0 sufficiently small

simulated maximum likelihood estimation of dynamic discrete choice statistical models some monte carlo results this article reports monte carlo results on the simulated maximum likelihood estimation of discrete dynamic panel models introduced by james heckman 1981a the simulated maximum likelihood method is numerically stable even for long panels regression models and polya and renewal models can be better estimated than markov models with a moderate number of simulation draws most of these complex models can be adequately estimated for panels with length up to 30 polya and renewal models can be accurately estimated for panels up to 50 periods estimates of markov models can be sensitive to misspecified initial states but polya renewal and habit persistence models may not

nonlinear elliptic equations with natural growth in the gradient and source terms in lorentz spaces abstract in this paper we consider the problem u w 0 1 p div a x u d u b x u d u h x u d u in d where div a x u d u is a leray lions operator which is defined on w 0 1 p with coercivity where the growth with respect to du of h x u d u is controlled by d u p and where b x u d u satisfies a similar growth condition but has the good sign the main feature of the problem is that the source terms belong to the lorentz space l n p when two smallness conditions are satisfied the second one depends on the behavior of b x u d u when u tends to infinity we prove the existence of a solution which further satisfies e p 1 u 1 w 0 1 p for every with 0 for some threshold 0 the key ingredient in the proof of the existence result is an a priori estimate which holds true for every solution to the problem which satisfies the above mentioned exponential regularity condition

proxies versus omitted variables in regression analysis when some variables to be used in a regression analysis contain measurement error the ols estimator will be inconsistent when these variables are left out ols will also be inconsistent in an appropriate metric the first type of inconsistency is smaller than the second type

some clarifications of the tuckals2 algorithm applied to the idioscal problem kroonenberg and de leeuw have suggested fitting the idioscal model by the tuckals2 algorithm for three way components analysis in theory this is problematic because tuckals2 produces two possibly different coordinate matrices that are useless for idioscal unless they are equal kroonenberg has claimed that when idioscal is fitted by tuckals2 the resulting coordinate matrices will be identical in the present paper this claim is proven valid when the data matrices are semidefinite however counterexamples for indefinite matrices are also constructed by examining the global minimum in the case where the data matrices have the same eigenvectors similar counterexamples have been considered by ten berge and kiers in the related context of candecomp parafac to fit the indscal model

robust confidence regions for incomplete models call an economic model incomplete if it does not generate a probabilistic prediction even given knowledge of all parameter values we propose a method of inference about unknown parameters for such models that is robust to heterogeneity and dependence of unknown form the key is a central limit theorem for belief functions robust confidence regions are then constructed in a fashion paralleling the classical approach monte carlo simulations support tractability of the method and demonstrate its enhanced robustness relative to existing methods

likelihood approximation by numerical integration on sparse grids the calculation of likelihood functions of many econometric models requires the evaluation of integrals without analytical solutions approaches for extending gaussian quadrature to multiple dimensions discussed in the literature are either very specific or suffer from exponentially rising computational costs in the number of dimensions we propose an extension that is very general and easily implemented and does not suffer from the curse of dimensionality monte carlo experiments for the mixed logit model indicate the superior performance of the proposed method over simulation techniques

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

reiterated ergodic algebras and applications we redefine the homogenization algebras without requiring the separability assumption we show that this enables one to treat more complicated homogenization problems than those solved by the previous theory in particular we exhibit an example of algebra which contrary to the algebra of almost periodic functions induces no homogenization algebra we prove some general compactness results which are then applied to the resolution of some homogenization problems related to the generalized reynolds type equations and to some nonlinear hyperbolic equations

smooth unbiased multivariate probability simulators for maximum likelihood estimation of limited dependent variable models abstract we apply a new simulation method that solves the multidimensional probability integrals that arise in maximum likelihood estimation of a broad class of limited dependent variable models the simulation method has four key features the simulated choice probabilities are unbiased they are a continuous and differentiable function of the parameters of the model they are bounded between 0 and 1 and their computation takes an effort that is nearly linear in the dimension of the probability integral independent of the magnitudes of the true probabilities we also show that the new simulation method produces probability estimates with substantially smaller variance than those generated by acceptance rejection methods or by stern s 1992 method the simulated probabilities can therefore be used to revive the lerman and manski 1981 procedure of approximating the likelihood function using simulated choice probabilities by overcoming its computational disadvantages

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

extensible lattice sequences for quasi monte carlo quadrature integration lattices are one of the main types of low discrepancy sets used in quasi monte carlo methods however they have the disadvantage of being of fixed size this article describes the construction of an infinite sequence of points the first bm of which forms a lattice for any nonnegative integer m thus if the quadrature error using an initial lattice is too large the lattice can be extended without discarding the original points generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice the spectral test used for finding pseudorandom number generators is one important example of such a discrepancy the performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems

misspecified heteroskedasticity in the panel probit model a small sample comparison of gmm and sml estimators abstract this paper compares generalized method of moments gmm and simulated maximum likelihood sml approaches to the estimation of the panel probit model both techniques circumvent multiple integration of joint density functions without the need to restrict the error term variance covariance matrix of the latent normal regression model particular attention is paid to a three stage gmm estimator based on nonparametric estimation of the optimal instruments for given conditional moment functions monte carlo experiments are carried out which focus on the small sample consequences of misspecification of the error term variance covariance matrix the correctly specified experiment reveals the asymptotic efficiency advantages of sml the gmm estimators outperform sml in the presence of misspecification in terms of multiplicative heteroskedasticity this holds in particular for the three stage gmm estimator allowing for heteroskedasticity over time increases the robustness with respect to misspecification in terms of multiplicative heteroskedasticity an application to the product innovation activities of german manufacturing firms is presented

some further results on hermitian matrix inequalities abstract some new results on the lowner partial ordering between certain sums products and direct products of matrices are derived proofs of these results are based on a necessary and sufficient condition for the nonnegative definiteness of the difference of two hermitian matrices which follows from the criterion for the nonnegative definiteness of a partitioned hermitian matrix due to albert 1969

implications of indeterminate factor error covariances for factor construction prediction and determinacy the assumptions of the model for factor analysis do not exclude a class of indeterminate covariances between factors and error variables grayson 2003 the construction of all factors of the model for factor analysis is generalized to incorporate indeterminate factor error covariances a necessary and sufficient condition is given for indeterminate factor error covariances to be arbitrarily small for mean square convergence of the regression predictor of factor scores and for the existence of a unique determinate factor and error variable the determinate factor and error variable are uncorrelated and satisfy the defining assumptions of factor analysis several examples are given to illustrate the results

backward stochastic partial differential equations with quadratic growth abstract this paper is concerned with the weak solution in analytic sense to the cauchy dirichlet problem of a backward stochastic partial differential equation when the nonhomogeneous term has a quadratic growth in both the gradient of the first unknown and the second unknown existence and uniqueness results are obtained under separate conditions

existence and uniqueness of solutions to a quasilinear parabolic equation with quadratic gradients in financial markets a quasilinear parabolic equation with quadratic gradient terms is analyzed the equation models an optimal portfolio in so called incomplete financial markets consisting of risky assets and non tradable state variables its solution allows to compute an optimal portfolio strategy the quadratic gradient terms are essentially connected to the assumption that the so called relative risk aversion function is not logarithmic the existence of weak global in time solutions in any dimension is shown under natural hypotheses the proof is based on the monotonicity method of frehse furthermore the uniqueness of solutions is shown under a smallness condition on the derivatives of the covariance diffusion matrices using a nonlinear test function technique developed by barles and murat finally the influence of the non tradable state variables on the optimal value function is illustrated by a numerical example in three dimensions

an instanton invariant for 3 manifolds to an oriented closed 3 dimensional manifoldm withh1 m 0 we assign a 8 graded homology groupi m whose euler characteristic is twice casson s invariant the definition uses a construction on the space of instantons onm

alternative sampling methods for estimating multivariate normal probabilities we study the performance of alternative sampling methods for estimating multivariate normal probabilities through the ghk simulator the sampling methods are randomized versions of some quasi monte carlo samples halton niederreiter niederreiter xing sequences and lattice points and some samples based on orthogonal arrays latin hypercube orthogonal array and orthogonal array based latin hypercube samples in general these samples turn out to have a better performance than monte carlo and antithetic monte carlo samples improvements over these are large for low dimensional 4 and 10 cases and still significant for dimensions as large as 50

a method for smoothing simulated moments of discrete probabilities in multinomial probit models there has been much interest over the years in estimating discrete choice models however with the exception of multinomial logit models agents have been restricted to few choices so that multivariate integrals could be feasibly integrated numerically 2 pakes and pollard 1989 and mcfadden 1989 have independently developed the method of simulated moments msm to deal with estimating a wide class of models of which high order discrete choice models are a subset one of the problems a researcher must handle when using msm for a discrete choice model is how to smooth the discrete simulated random variables this is necessary to keep small the number of draws required to simulate derivatives and to simulate variation in the data geweke 1989 and mcfadden 1989 have suggested importance sampling methods and other methods to smooth the simulated variables for general error structures in this paper i present a factor analytic smoothing method that can be applied to probit problems although it is not as general as the methods geweke and mcfadden suggest it is easy to use and has clear intuition furthermore simulated probabilities will be unbiased will be bounded between zero and unity and will have smaller variances than unsmoothed probabilities always and smaller variances than importance sampling estimates for a large class of probabilities the second section of this paper develops notation defines the problem and presents the smoothing method and an algorithm to employ it the last section presents monte carlo comparisons of the smoothing method to the importance sampling method

some qualitative properties of solutions of quasilinear elliptic equations and applications abstract we study quasilinear elliptic equations of leray lions type in w 1 p maximum principles nonexistence and existence of solutions the control of lower upper bound for essential supremum essential infimum of solutions sign changing solutions local and global oscillation of solutions geometry of domain generating singularities of solutions and lower bounds on constants appearing in schauder agmon douglis and nirenberg estimates

uniqueness results for nonlinear elliptic equations with a lower order term abstract we prove the uniqueness of the weak solution of noncoercive nonlinear elliptic problems whose model is u w 0 1 p div a x 1 u 2 p 2 2 u b x 1 u 2 1 2 f in d where is a bounded open subset of r n n 2 p satisfies p 2 n n 1 a is a function belonging to l such that a x 0 f belongs to the dual space w 1 p b belongs to some lebesgue space l r with r r n p and belongs to the interval 0 n p r with n p r and r n p functions which are specified below

existence and non existence of solutions to elliptic equations with a general convection term in this paper we consider the nonlinear elliptic problem u u g u h x in u 0 on where is a smooth bounded domain of n 0 g is an arbitrary c 1 increasing function and h c 1 is non negative we completely analyse the existence and non existence of positive classical solutions in terms of the parameter we show that there exist solutions for every when 0 and the integral 1 g s d s or when 0 and the integral s g s d s conversely when the respective integrals converge and h is non trivial on existence depends on the size of moreover non existence holds for large our proofs mainly rely on comparison arguments and on the construction of suitable supersolutions in annuli our results include some cases where the function g is superquadratic and existence still holds without assuming any smallness condition on

solvability of uniformly elliptic fully nonlinear pde we get existence uniqueness and non uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of the hamilton jacobi bellman isaacs type with unbounded ingredients and quadratic growth in the gradient without hypotheses of convexity or properness some of our results are new even for equations in divergence form

a method of simulated moments for estimation of discrete response models without numerical integration this paper proposes a simple modification of a conventional method of moments estimator for a discrete response model replacing response probabilities that require numerical integration with estimators obtained by monte carlo simulation this method of simulated moments msm does not require precise estimates of these probabilities for consistency and asymptotic normality relying instead on the law of large numbers operating across observations to control simulation error and hence can use simulations of practical size the method is useful for models such as high dimensional multinomial probit mnp where computation has restricted applications

comparison principle for some classes of nonlinear elliptic equations abstract we prove a comparison principle for second order quasilinear elliptic operators in divergence form when a first order term appears we deduce uniqueness results for weak solutions to dirichlet problems when data belong to the natural dual space

backwards sde with random terminal time and applications to semilinear elliptic pde suppose im t is the filtration induced by a wiener process w in r d tau is a finite im t stopping time terminal time xi is an im tau measurable random variable in r k terminal value and f cdot y z is a coefficient process depending on y in r k and z in l r d r k satisfying y tilde y f s y z f s tilde y z leq a y tilde y 2 f need not be lipschitz in y and f s y z f s y tilde z leq b z tilde z for some real a and b plus other mild conditions we identify a hilbert space depending on tau and on the number gamma equiv b 2 2a in which there exists a unique pair of adapted processes y z satisfying the stochastic differential equation dy s 1 s leq tau z s dw s f s y s z s ds with the given terminal condition y tau xi provided a certain integrability condition holds this result is applied to construct a continuous viscosity solution to the dirichlet problem for a class of semilinear elliptic pde s

representations of knot groups in su 2 this paper is a study of the structure of the space r k of representations of classical knot groups into su 2 let r k equal the set of conjugacy classes of irreducible representations in gi we interpret the relations in a presentation of the knot group in terms of the geometry of su 2 using this technique we calculate r k for k equal to the torus knots twist knots and the whitehead link we also determine a formula for the number of binary dihedral representations of an arbitrary knot group we prove using techniques introduced by culler and shalen that if the dimension of r k is greater than 1 then the complement in sj of a tubular neighborhood of k contains closed nonboundary parallel incompressible surfaces we also show how for certain nonprime and doubled knots r k has dimension greater than one in gii we calculate the zariski tangent space tp r k for an arbitrary knot k at a reducible representation p using a technique due to weil we prove that for all but a finite number of the reducible representations dimtp r k 3 these nonexceptional representations possess neighborhoods in r k containing only reducible representations at the exceptional representations which correspond to real roots of the alexander polynomial dim tp r k 3 2k for a positive integer k in those examples analyzed in this paper these exceptional representations can be expressed as limits of arcs of irreducible representations we also give an interpretation of these extra tangent vectors as representations in the group of euclidean isometries of the plane 0 introduction this paper is a study of the space r k of representations of the fundamental group of the complement of a knot k into the lie group su 2 the major results are as follows 1 a calculation of the topological type of r k for k a torus knot a twist knot or the whitehead link 2 a determination of the zariski tangent space to r k at a reducible representation for an arbitrary knot k and 3 a proposition and some examples relating the dimension of r k to incompressible surfaces in the complement of k while the idea of representing 3 manifold groups in su 2 is a fairly recent one the representation of these groups in other groups has a substantial history we will not attempt to chronicle this history in detail but will comment here on received by the editors january 18 1989 and in revised form july 25 1989 1980 mathematics subject classification 1985 revision primary 57m05 57m25

extensions of estimation methods using the em algorithm abstract the em algorithm described by dempster laid and rubin 1977 is reviewed with the purpose of clarifying several misconceptions in the statistical and econometric literature the clarifications lead to several applications of the algorithm to models that have appeared to be less tractable the relationship between the em algorithm and the method of scoring is also explained providing estimators of the score and the information from the em algorithm the em algorithm is extended to missing data problems and an estimation method based on simulations

a convex optimization approach for computing correlated choice probabilities with many alternatives a popular discrete choice model that incorporates correlation information is the multinomial probit mnp model where the random utilities of the alternatives are chosen from a multivariate normal distribution computing the choice probabilities is challenging in the mnp model when the number of alternatives is large mishra et al ieee transactions on automatic control 2012 have proposed a semidefinite optimization approach to compute choice probabilities for the distribution of the random utilities that maximizes expected agent utility given only the mean variance and covariance information their model is referred to as the cross moment cmm model computing the choice probabilities with many alternatives is challenging in the cmm model since one needs to solve large scale semidefinite programs we develop a simpler formulation as a representative agent model by maximizing over the choice probabilities in the unit simplex where the objective function is the sum of the expected utilities and a strongly concave perturbation function by characterizing the perturbation function for the cmm model and its gradient we develop a simple first order gradient method with inexact line search to compute choice probabilities we establish local linear convergence of this algorithm under mild assumptions on the choice probabilities an implication of our results is that inverting the choice probabilities to compute the mean utilities is straightforward given any positive definite covariance matrix numerical experiments show that this method can compute choice probabilities for a large number of alternatives within a reasonable amount of time while explicitly capturing the correlation information comparisons with simulation methods for mnp and semidefinite programming methods for cmm indicate the efficacy of the method

simulation and the asymptotics of optimization estimators on demontre un theoreme de limite centrale general pour des estimateurs definis par minimisation de la longueur d une fonction de critere aleatoire a valeurs vectorielles aucune hypothese de regularite suffisante n est imposee sur la fonction de critere pour que les resultats puissent etre appliques a une classe d estimlateurs de simulation assez large des analyses completes de deux estimateurs de simulation l un introduit par pakes l autre par mcfadden illustrent l application des theoremes generaux

the su 3 casson invariant for integral homology 3 spheres we derive a gauge theoretic invariant of integral homology 3 spheres which counts gauge orbits of irreducible perturbed flat su 3 connections with sign given by spectral flow to compensate for the dependence of this sum on perturbations the invariant includes contributions from the reducible perturbed flat orbits our formula for the correction term generalizes that given by walker in his extension of casson s su 2 invariant to rational homology 3 spheres

bayesian hemodynamic parameter estimation by bolus tracking perfusion weighted imaging a delay insensitive probabilistic method for estimating hemodynamic parameters delays theoretical residue functions and concentration time curves by computed tomography ct and magnetic resonance mr perfusion weighted imaging is presented only a mild stationarity hypothesis is made beyond the standard perfusion model new microvascular parameters with simple hemodynamic interpretation are naturally introduced simulations on standard digital phantoms show that the method outperforms the oscillating singular value decomposition osvd method in terms of goodness of fit linearity statistical and systematic errors on all parameters especially at low signal to noise ratios snrs delay is always estimated sharply with user supplied resolution and is purely arterial by contrast to osvd time to maximum tmax that is very noisy and biased by mean transit time mtt blood volume and snr residue functions and signals estimates do not suffer overfitting anymore one ct acute stroke case confirms simulation results and highlights the ability of the method to reliably estimate mtt when snr is low delays look promising for delineating the arterial occlusion territory and collateral circulation

efficient estimation of probit models with correlated errors abstract maximum likelihood ml estimation of probit models with correlated errors typically requires high dimensional truncated integration prominent examples of such models are multinomial probit models and binomial panel probit models with serially correlated errors in this paper we propose to use a generic procedure known as efficient importance sampling eis for the evaluation of likelihood functions for probit models with correlated errors our proposed eis algorithm covers the standard ghk probability simulator as a special case we perform a set of monte carlo experiments in order to illustrate the relative performance of both procedures for the estimation of a multinomial multiperiod probit model our results indicate substantial numerical efficiency gains for ml estimates based on the ghk eis procedure relative to those obtained by using the ghk procedure

statistical inference in the multinomial multiperiod probit model abstract statistical inference in multinomial multiperiod probit models has been hindered in the past by the high dimensional numerical integrations necessary to form the likelihood functions posterior distributions or moment conditions in these models we describe three alternative estimators implemented using simulation based approaches to inference that circumvent the integration problem posterior means computed using gibbs sampling and data augmentation gibbs simulated maximum likelihood sml estimation using the ghk probability simulator and method of simulated moment msm estimation using ghk we perform a set of monte carlo experiments to compare the sampling distributions of these estimators although all three estimators perform reasonably well some important differences emerge our most important finding is that holding simulation size fixed the relative and absolute performance of the classical methods especially sml gets worse when serial correlation in disturbances is strong in data sets with an ar 1 parameter of 0 50 the rmses for sml and msm based on ghk with 20 draws exceed those of gibbs by 9 and 0 respectively but when the ar 1 parameter is 0 80 the rmses for sml and msm based on 20 draws exceed those of gibbs by 79 and 37 respectively and the number of draws needed to reduce the rmses to within 10 of gibbs are 160 and 80 respectively also the sml estimates of serial correlation parameters exhibit significant downward bias thus while conventional wisdom suggests that 20 draws of ghk is enough to render the bias and noise induced by simulation negligible our results suggest that much larger simulation sizes are needed when serial correlation in disturbances is strong

tuning parameter selectors for the smoothly clipped absolute deviation method summary the penalized least squares approach with smoothly clipped absolute deviation penalty has been consistently demonstrated to be an attractive regression shrinkage and selection method it not only automatically and consistently selects the important variables but also produces estimators which are as efficient as the oracle estimator however these attractive features depend on appropriate choice of the tuning parameter we show that the commonly used generalized crossvalidation cannot select the tuning parameter satisfactorily with a nonignorable overfitting effect in the resulting model in addition we propose a bic tuning parameter selector which is shown to be able to identify the true model consistently simulation studies are presented to support theoretical findings and an empirical example is given to illustrate its use in the female labor supply data

monte carlo evaluation of multivariate normal probabilities abstract this paper extends research on the simulation of multivariate normal probabilities of high order dimension by developing a new family of simulators such simulators are useful for models with limited dependent variables including multinomial probit in panel studies spatial analysis and time series analysis the simulators are derived from a cholesky decomposition of the covariance matrix combined with a suitable choice of an importance sampling distribution the paper studies among others the impact of antithetical sampling the insights gained in this paper are of use in bayesian analysis as well in the evaluation of posterior densities

a computationally practical simulation estimator for panel data in this paper i develop a practical extension of mcfadden s method of simulated moments estimator for limited dependent variable models to the panel data case the method is based on a factorization of the msm first order condition into transition probabilities along with the development of a new highly accurate method for similating these transition probabilities a series of monte carlo tests show that this msm estimator performs quite well relative to quadrature based ml estimators even when large numbers of quadrature points are employed the estimator also performs well relative to simulated ml even when a highly accurate method is used to simulate the choice probabilities in terms of computational speed complex panel data models involving random effects and arma errors may be estimated via msm in times similar to those necessary for estimation of simple random effects models via ml quadrature

on the equivalence between strict positive realness and strict passivity of linear systems this technical note presents a proof of the equivalence between the strict passivity of linear time invariant lti controllable and observable systems and the strict positive realness of their transfer function matrices where the direct feedthrough d is possibly a non zero matrix although both properties guarantee asymptotic stability the former is a time domain condition and the latter is a frequency domain concept a numerical example illustrates our main results

simulation of multivariate normal rectangle probabilities and their derivatives theoretical and computational results abstract an extensive literature in econometrics and in numerical analysis has considered the problem of evaluating the multiple integral p b a b n v dv e v 1 v b where v is a m dimensional normal vector with mean covariance matrix and density n v and 1 v b is an indicator for the event b v a a leading case of such an integral is the negative orthant probability where b v v the problem is computationally difficult except in very special cases the multinomial probit mnp model used in econometrics and biometrics has cell probabilities that are negative orthant probabilities with and depending on unknown parameters and in general on covariates estimation of this model requires for each trial parameter vector and each observation in a sample evaluation of p b and of its derivatives with respect to and this paper surveys monte carlo techniques that have been developed for approximations of p b and its linear and logarithmic derivatives that limit computation while possessing properties that facilitate their use in iterative calculations for statistical inference the crude frequency simulator cfs normal importance sampling nis a kernel smoothed frequency simulator kfs stern s decomposition simulator sds the geweke hajivassiliou keane simulator ghk a parabolic cylinder function simulator pcf deak s chi squared simulator dcs an acceptance rejection simulator ars the gibbs sampler simulator gss a sequentially unbiased simulator sus and an approximately unbiased simulator aus we also discuss gauss and fortran implementations of these algorithms and present our computational experience with them we find that ghk is overall the most reliable method

remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions abstract in this note we present some uniqueness and comparison results for a class of problem of the form 0 1 l u h x u u h x u h 0 1 l where r n n 2 is a bounded domain l is a general elliptic second order linear operator with bounded coefficients and h is allowed to have a critical growth in the gradient in some cases our assumptions prove to be sharp

nonnegative and positive definiteness of matrices modified by two matrices of rank one abstract necessary and sufficient conditions are given for the nonnegative and positive definiteness of matrices of the form a a 1 a 1 a 2 a 2 and a a 1 a 1 a 2 a 2 where a is a hermitian matrix and a 1 a 2 are complex vectors they are derived using some auxiliary results which seem to be of independent interest as well some particular cases of these conditions are also discussed especially in the context of related results known in the literature

bayesian analysis of order statistics models for ranking data in this paper a class of probability models for ranking data the order statistics models is investigated we extend the usual normal order statistics model into one where the underlying random variables follow a multivariate normal distribution bayesian approach and the gibbs sampling technique are used for parameter estimation in addition methods to assess the adequacy of model fit are introduced robustness of the model is studied by considering a multivariate t distribution the proposed method is applied to analyze the presidential election data of the american psychological association apa

the matrix inequality m b mb abstract the loewner partial ordering between nonnegative definite matrices m and b mb is considered the main result obtained generalizes a number of results known in the literature the rank subtractivity partial ordering and the drazin partial ordering between m and b mb are also discussed