an exact likelihood analysis of the multinomial probit model abstract we develop new methods for conducting a finite sample likelihood based analysis of the multinomial probit model using a variant of the gibbs sampler an algorithm is developed to draw from the exact posterior of the multinomial probit model with correlated errors this approach avoids direct evaluation of the likelihood and thus avoids the problems associated with calculating choice probabilities which affect both the standard likelihood and method of simulated moments approaches both simulated and actual consumer panel data are used to fit six dimensional choice models we also develop methods for analyzing random coefficient and multiperiod probit models

a computationally efficient quadrature procedure for the one factor multinomial probit model a problem of estimation that has long confronted many economists is the difficulty of estimating the parameters of equations with limited dependent variables on cross section time series i e panel data while there are widely available packaged computer programs for estimating either a cross section probit and tobit models or b simple permanent transitory random effects panel models with continuous dependent variables there are no available computationally feasible methods of combining these two models this is because the likelihood function that arises in such a combined model contains multivariate normal integrals whose evaluation is quite difficult if not impossible with conventional approximation methods there is a widespread feeling among those working in the area that one possible method of evaluation the use of quadrature techniques is in principle possible but is in practice computationally too burdensome to consider e g albright et al 2 p 13 hausman and wise 6 p 12 in this note we point out that this is true only of standard quadrature techniques such as trapezoidal integration or its improved variants gaussian quadrature on the other hand is extremely efficient and is well within the bounds of computational feasibility on modern computers in what follows we state the nature of the integrals that need to be evaluated provide a brief exposition of gaussian quadrature and provide a numerical illustration of its use in

optimal robust stabilization and dissipativity synthesis by behavioral interconnection given a nominal plant together with a fixed neighborhood of this plant the problem of robust stabilization is to find a controller that stabilizes all plants in that neighborhood in an appropriate sense if a controller achieves this design objective we say that it robustly stabilizes the nominal plant in this paper we formulate the robust stabilization problem in a behavioral framework with control as interconnection we also formulate a relevant behavioral mathcal h infty synthesis problem which will be instrumental in solving the robust stabilization problem we use both rational and polynomial representations for the behaviors under consideration necessary and sufficient conditions for the existence of robustly stabilizing controllers are obtained using the theory of dissipative systems we will also find the optimal stability radius i e the smallest upper bound on the radii of the neighborhoods for which there exists a robustly stabilizing controller this smallest upper bound is expressed in terms of certain storage functions associated with nominal control system

state maps for linear systems modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential algebraic equations as such state space models are not a natural starting point for modeling while they have utmost importance in the simulation and control phase the paper addresses the problem of computing state variables for systems of linear differential algebraic equations of various forms the point of view from which the problem is considered is the behavioral one as put forward in j c willems automatica j ifac 22 1986 pp 561 580 dynamics reported 2 1989 pp 171 269 ieee trans automat control 36 1991 pp 259 294

every storage function is a state function it is shown that for linear dynamical systems with quadratic supply rates a storage function can always be written as a quadratic function of the state of an associated linear dynamical system this dynamical system is obtained by combining the dynamics of the original system with the dynamics of the supply rate

simulated latent variable estimation of models with ordered categorical data this paper addresses the problem that occurs when estimating a linear regression model in which some of the explanatory variables are represented by categorical indicators we replace the categorical variable with a simulated latent variable slv drawn from a truncated multivariate normal distribution using the gibbs sampler consistent and efficient parameter estimates are derived using generalized two stage least squares where the instrument set contains a second slv drawn from the same distribution consistent estimates of the parameter covariance matrix are derived using a two stage procedure we illustrate this methodology using a monte carlo simulation and also investigate the consequences of incorrect distributional assumptions the simulated latent variable procedure is shown to be superior to conventional methods involving dummy variables or conditional means

estimating linear models with ordinal qualitative regressors abstract the theory and application of ordinal qualitative dependent variable models have been given considerable attention in the social science and statistical literatures linear models with ordinal qualitative regressors have however been neglected in this paper a simple specification for such models is developed and a consistent asymptotically normal estimator is offered the estimator is compared to the conventional dummy variable approach using simulated data the simulation results indicate that substantial gains with regard to bias and efficiency can be achieved relative to estimation using a conventional dummy variable scheme said gains appear to increase as the number of ordinal categories increases the asymptotic properties of the estimator are detailed in the appendices

distribution of poles for scattering on the real line abstract the density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential in the case of a potential with finite singularities at the endpoints of the support asymptotic formulae for the poles are given while in the c 0 case an example of a potential with infinitely many scattering poles on i r is constructed the scattering amplitude of a compactly supported potential is also characterized

explaining individual response using aggregated data abstract empirical analysis of individual response behavior is sometimes limited due to the lack of explanatory variables at the individual level in this paper we put forward a new approach to estimate the effects of covariates on individual response where the covariates are unknown at the individual level but observed at some aggregated level this situation may for example occur when the response variable is available at the household level but covariates only at the zip code level we describe the missing individual covariates by a latent variable model which matches the sample information at the aggregate level parameter estimates can be obtained using maximum likelihood or a bayesian analysis we illustrate the approach estimating the effects of household characteristics on donating behavior to a dutch charity donating behavior is observed at the household level while the covariates are only observed at the zip code level

the strict dissipativity synthesis problem and the rank of the coupling qdf the problem of existence of a controlled behavior that is strictly dissipative with respect to a quadratic supply rate is studied the relation between strictness and the rank of a suitable coupling condition that combines the dissipativity properties of the hidden behavior and the orthogonal complement of the plant behavior is analyzed

a note on the dixon formula for a finite hypergeometric series this is a note on the paper 4 b m brown m s p eastham extensions of the kummer evaluation of a finite hypergeometric series j comput appl math to appear doi 10 1016 j cam 2005 06 018

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

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

on the location of resonances of exponentially decaying sturm liouville potentials the paper is concerned with analytic continuation into the unphysical sheet for sturm liouville potentials whose decay at infinity is no faster than o e ax a resonance free region is found and the existence of a natural boundary is discussed an example related to the bessel equation is examined in some detail

median regression for ordered discrete response abstract we introduce a semiparametric median regression approach to ordered discrete responses generalizing the maximum score estimation of manski 1985 for binary response the identification and a s consistency are proved we assume that there exists a latent continuous variable and several variations of the median regression approach depending on the restriction on the thresholds for the latent variable are examined for some cases bounds for the scale are derived we also derive bounds for the finite sample breakdown point of the median regression method to show its robustness

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

the singular control problem with dynamic measurement feedback this paper is concerned with the h infty problem with measurement feedback the problem is to find a dynamic feedback from the measured output to the control input such that the closed loop system has an h infty norm strictly less than some a priori given bound gamma and such that the closed loop system is internally stable necessary and sufficient conditions are given under which such a feedback exists the only assumption that must be made is that there are no invariant zeros on the imaginary axis for two subsystems contrary to recent publications no assumptions are made on the direct feedthrough matrices of the plant it turns out that this problem can be reduced to an almost disturbance decoupling problem with measurement feedback and internal stability i e the problem in which we can make the h infty norm arbitrarily small

bounds on scattering poles in one dimension for the class of super exponentially decaying potentials on the real line sharp upper bounds on the counting function of the poles in discs are derived and the density of the poles in strips is estimated in the case of nonnegative potentials explicit estimates for the width of a pole free strip are obtained

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

simulation estimation of dynamic switching regression and dynamic disequilibrium models some monte carlo results abstract this article considers the estimation of dynamic exogenous switching regression models and dynamic endogenous switching models with autocorrelation in disturbances or latent lagged dependent variables likelihood functions of such models involve high dimensional integrals and a huge number of summations over unobserved regime paths simulated likelihood methods and simulated methods of moments are available these approaches simulate both continuous and discrete latent dependent variables by monte carlo experiments it is found that the performances of various approaches depend crucially on how discrete state variables are simulated the valuable approach is to simulate regime paths with regime probabilities based on the current and past sample information

analytic continuation and resonance free regions for sturm liouville potentials with power decay the analytic continuation of the jost solution into a sector of the unphysical sheet is established for analytic potentials with power decay the method also yields resonance free regions which in the case of super exponential decay are confirmed to be the best possible of their kind by the known asymptotic distribution of resonances

extensions of the kummer evaluation of a finite hypergeometric series kummer s evaluation of a finite hypergeometric series f a b c 1 c b a 1 b 2m is extended in two directions to answer questions related to the positivity of jost functions arising from one dimensional schrodinger equations with exponentially decaying potentials

extended hurwitz results for hypergeometric functions arising in spectral theory a result of hurwitz is that the bessel function j v c v 0 c 0 has no zeros for 2n v 2n 1 with integer n here corresponding results for hypergeometric and confluent hypergeometric functions are given extensions are obtained where the power series are terminated after 2n 1 terms and larger zero free intervals 2n v are found

asymptotic distribution of resonances in one dimension abstract we determine the leading asymptotics of the resonance counting function for a class of schrodinger operators in one dimension whose potentials may have non compact support

the state feedback h problem at optimality abstract we discuss the infinite horizon h control problem by state feedback for a linear time invariant plant which is not restricted with respect to its finite or infinite zero structure we provide an algebraic characterization for the optimal value to be attained in terms of the solvability of a nonstrict riccati inequality and propose possibilities for verifying these criteria if the optimum is not achieved we give a complete test for the need of high gain feedback in order to approach the optimal value

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

the hurwitz theorem for bessel functions and antibound states in spectral theory an 1889 result of hurwitz is that the bessel function j v c v 0 c 0 has no zeros for 2 n v n 1 with integer n a new application of this result is given which describes the behaviour of jost functions and antibound states for one dimensional schrodinger equations with exponentially decaying potential a new hurwitz type theorem with similar applications is developed for hypergeometric functions extensions are obtained where the power series are terminated after 2 n 1 terms and larger zero free intervals 2 n v are found

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

synthesis of dissipative systems using quadratic differential forms part ii the problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative our formulation and solution of this problem is completely representation free the system dynamics are specified by a differential behavior and the performance is specified through a quadratic differential form we view control as interconnection a controller constrains a distinguished set of system variables the control variables the resulting behavior of the to be controlled variables is called the controlled behavior the constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior obtained by setting the control variables equal to zero and the plant behavior obtained by leaving the control variables unconstrained the main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications the essential requirement is a coupling condition an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior

dissipativity of uncontrollable systems storage functions and lyapunov functions dissipative systems have played an important role in the analysis and synthesis of dynamical systems the commonly used definition of dissipativity often requires an assumption on the controllability of the system in this paper we use a definition of dissipativity that is slightly different and less often used in the literature to study a linear time invariant possibly uncontrollable dynamical system we provide a necessary and sufficient condition for an uncontrollable system to be strictly dissipative with respect to a supply rate under the assumption that the uncontrollable poles are not mixed i e no pair of uncontrollable poles is symmetric about the imaginary axis this condition is known to be related to the solvability of a lyapunov equation we link lyapunov functions for autonomous systems to storage functions of an uncontrollable system the set of storage functions for a controllable system has been shown to be a convex bounded polytope in the literature we show that for an uncontrollable system the set of storage functions is unbounded and that the unboundedness arises precisely due to the set of lyapunov functions for an autonomous linear system being unbounded further we show that stabilizability of a system results in this unbounded set becoming bounded from below positivity of storage functions is known to be very important for stability considerations because the maximum stored energy that can be drawn out is bounded when the storage function is positive in this paper we establish the link between stabilizability of an uncontrollable system and existence of positive definite storage functions in most of the results in this paper we assume that no pair of the uncontrollable poles of the system is symmetric about the imaginary axis we explore the extent of necessity of this assumption and also prove some results for the case of single output systems regarding this necessity

resonances and analytic continuation for exponentially decaying sturm liouville potentials abstract the sturm liouville equation on the half line 0 is considered with a potential which decays exponentially at infinity it is known that green s function has an analytic continuation into part of the unphysical sheet im here it is shown that analytic continuation into the whole of this sheet is possible for certain types of decay the methods are constructive and they also provide a new and rigorous way of locating resonances

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea