minimum distance estimation in random coefficient regression models random coefficient regression models are important in modeling heteroscedastic multivariate linear regression in econometrics the analysis of panel data is one example in statistics the random and mixed effects models of anova deconvolution models and affine mixture models are all special cases of random coefficient regression some inferential problems such as constructing prediction regions for the modeled response require a good nonparametric estimator of the unknown coefficient distribution this paper introduces and studies a consistent nonparametric minimum distance method for estimating the coefficient distribution our estimator translates the difficult problem of estimating an inverse radon transform into a minimization problem

local linear regression smoothers and their minimax efficiencies in this paper we introduce a smooth version of local linear regression estimators and address their advantages the mse and mise of the estimators are computed explicitly it turns out that the local linear regression smoothers have nice sampling properties and high minimax efficiency they are not only efficient in rates but also nearly efficient in constant factors in the nonparametric regression context the asymptotic minimax lower bound is developed via the heuristic of the hardest onedimensional subproblem of donoho and liu connections of the minimax risk with the modulus of continuity are made the lower bound is also applicable for estimating conditional mean regression and conditional quantiles for both fixed and random design regression problems

eigenvalue assignment by minimal state feedback gain in lti multivariable systems in this article an improved method for eigenvalue assignment via state feedback in the linear time invariant multivariable systems is proposed this method is based on elementary similarity operations and involves mainly utilisation of vector companion forms and thus is very simple and easy to implement on a digital computer in addition to the controllable systems the proposed method can be applied for the stabilisable ones and also systems with linearly dependent inputs moreover two types of state feedback gain matrices can be achieved by this method 1 the numerical one which is unique and 2 the parametric one in which its parameters are determined in order to achieve a gain matrix with minimum frobenius norm the numerical examples are presented to demonstrate the advantages of the proposed method

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

estimates of transition densities for brownian motion on nested fractals we obtain upper and lower bounds for the transition densities of brownian motion on nested fractals compared with the estimate on the sierpinski gasket the results require the introduction of a new exponent dj related to the shortest path metric and chemical exponent on nested fractals further holder order of the resolvent densities sample paths and local times are obtained the results are obtained using the theory of multi type branching processes

dead beat control of linear discrete time systems the paper considers the problem of determining the various classes of state feed back dead beat controllers for linear multivariable discrete time systems unlike earlier work attention is not restricted to the special case of minimum time dead beat controllers a general dead beat control problem is posed and solved the solution is based principally on an approach of closed loop eigenstructure assignment the minimum of free parameters in each class are specified and used in an explicit parametric form for the feedback gain matrix the concept of the greatest minimum of free parameters is introduced finally the method is illustrated by a numerical example

curve estimation when the design density is low in problems where a high dimensional design is projected into a lower number of dimensions the density of the new design is typically not bounded away from zero over its support even if the original one was contexts where this problem arises include projection pursuit regression estimation in single index models and application of the projection slice method of radon transform inversion theoretical work in these settings typically involves ignoring data toward the ends of the support of the projected design but in practice that waste of information is not an attractive option motivated by these difficulties we analyze the way in which local linear smoothing is affected by unboundedly sparse design and apply the conclusions of that study to develop empirical adaptive bandwidth choice methods our results even add to knowledge in the familiar case of a design density that is bounded away from zero where they provide adaptive bandwidth selectors that are optimal right to the ends of the design interval

alternative em methods for nonparametric finite mixture models summary this research focuses on a general class of maximum likelihood problems in which it is desired to maximise a nonparametric mixture likelihood with finitely many known component densities over the set of unknown weight parameters convergence of the conventional em algorithm for this problem is extremely slow when the component densities are poorly separated and when the maximum likelihood estimator requires some of the weights to be zero as the algorithm can never reach such a boundary point alternative methods based on the principles of em are developed using a two stage approach first a new data augmentation scheme provides improved convergence rates in certain parameter directions secondly two cyclic versions of this data augmentation are created by changing the missing data formulation between the em steps these extend the acceleration directions to the whole parameter space giving another order of magnitude increase in convergence rate examples indicate that the new cyclic versions of the data augmentation schemes can converge up to 500 times faster than the conventional em algorithm for fitting nonparametric finite mixture models

smoothly mixing regressions this paper extends the conventional bayesian mixture of normals model by permitting state probabilities to depend on observed covariates the dependence is captured by a simple multinomial probit model a conventional and rapidly mixing mcmc algorithm provides access to the posterior distribution at modest computational cost this model is competitive with existing econometric models as documented in the paper s illustrations the first illustration studies quantiles of the distribution of earnings of men conditional on age and education and shows that smoothly mixing regressions are an attractive alternative to nonbayesian quantile regression the second illustration models serial dependence in the s p 500 return and shows that the model compares favorably with arch models using out of sample likelihood criteria

estimating coefficient distributions in random coefficient regressions random coefficient regression models are important in representing linear models with heteroscedastic errors and in unifying the study of classical fixed effects and random effects linear models for prediction intervals and for bootstrapping in random coefficient regressions it is necessary to estimate the distributions of the random coefficients consistently we show that this is often possible and provide practical representative estimators of these distributions

estimation of semiparametric models when the criterion function is not smooth we provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated our results extend existing theories such as those of pakes and pollard 1989 andrews 1994a and newey 1994 we also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters we apply our results to two examples a hit rate and a partially linear median regression with some endogenous regressors

semiparametric estimation of a proportional hazard model with unobserved heterogeneity the proportional hazard model with unobserved heterogeneity gives the hazard function of a random variable conditional on covariates and a second random variable representing unobserved heterogeneity this paper shows how to estimate the baseline hazard function and the distribution of the unobserved heterogeneity nonparametrically the baseline hazard function and heterogeneity distribution are assumed to satisfy smoothness conditions but are not assumed to belong to known finite dimensional parametric families existing estimators assume that the baseline hazard function or heterogeneity distribution belongs to a known parametric family thus the estimators presented here are more general than existing ones

the random coefficients logit model is identified the random coecients multinomial choice logit model also known as the mixed logit has been widely used in empirical choice analysis for the last thirty years we prove that the distribution of random coecients in the multinomial logit model is nonparametrically identified

on random tomography with unobservable projection angles we formulate and investigate a statistical inverse problem of a random tomographic nature where a probability density function on mathbb r 3 is to be recovered from observation of finitely many of its two dimensional projections in random and unobservable directions such a problem is distinct from the classic problem of tomography where both the projections and the unit vectors normal to the projection plane are observable the problem arises in single particle electron microscopy a powerful method that biophysicists employ to learn the structure of biological macromolecules strictly speaking the problem is unidentifiable and an appropriate reformulation is suggested hinging on ideas from kendall s theory of shape within this setup we demonstrate that a consistent solution to the problem may be derived without attempting to estimate the unknown angles if the density is assumed to admit a mixture representation

a simple nonparametric approach to estimating the distribution of random coefficients in structural models abstract we explore least squares and likelihood nonparametric mixtures estimators of the joint distribution of random coefficients in structural models the estimators fix a grid of heterogeneous parameters and estimate only the weights on the grid points an approach that is computationally attractive compared to alternative nonparametric estimators we provide conditions under which the estimated distribution function converges to the true distribution in the weak topology on the space of distributions we verify most of the consistency conditions for three discrete choice models we also derive the convergence rates of the least squares nonparametric mixtures estimator under additional restrictions we perform a monte carlo study on a dynamic programming model

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

the geometry of mixture likelihoods a general theory in this paper certain fundamental properties of the maximum likelihood estimator of a mixing distribution are shown to be geometric properties of the likelihood set the existence support size likelihood equations and uniqueness of the estimator are revealed to be directly related to the properties of the convex hull of the likelihood set and the support hyperplanes of that hull it is shown using geometric techniques that the estimator exists under quite general conditions with a support size no larger than the number of distinct observations analysis of the convex dual of the likelihood set leads to a dual maximization problem a convergent algorithm is described the defining equations for the estimator are compared with the usual parametric likelihood equations for finite mixtures sufficient conditions for uniqueness are given part ii will deal with a special theory for exponential family mixtures

efficient greedy learning of gaussian mixture models this article concerns the greedy learning of gaussian mixtures in the greedy approach mixture components are inserted into the mixture one after the other we propose a heuristic for searching for the optimal component to insert in a randomized manner a set of candidate new components is generated for each of these candidates we find the locally optimal new component and insert it into the existing mixture the resulting algorithm resolves the sensitivity to initialization of state of the art methods like expectation maximization and has running time linear in the number of data points and quadratic in the final number of mixture components due to its greedy nature the algorithm can be particularly useful when the optimal number of mixture components is unknown experimental results comparing the proposed algorithm to other methods on density estimation and texture segmentation are provided

weak uncertainty principle for fractals graphs and metric measure spaces we develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by okoudjou and strichartz we assume either an appropriate measure growth condition with respect to the effective resistance metric or in the absence of such a metric we assume the poincare inequality and reverse volume doubling property we also consider the weak uncertainty inequality in the context of nash type inequalities our results can be applied to a wide variety of metric measure spaces including graphs fractals and manifolds

parametric time optimal control of linear discrete time systems by state feedback part 2 irregular kronecker invariants in an earlier paper part 1 a parametric state feedback time optimal control law was devised for the regular kronecker invariants on the basis of parametrization of the vector companion form the present paper describes a straight forward technique for obtaining the parametric feedback matrix by exploiting the parametric vector companion forms in the case of irregular kronecker invariants hidden properties of vector companion forms are also given which result in entirely different and interesting techniques for parametrization of the control law an illustrative example is presented

estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals this paper studies nonparametric estimation of conditional moment restrictions in which the generalized residual functions can be nonsmooth in the unknown functions of endogenous variables this is a nonparametric nonlinear instrumental variables iv problem we propose a class of penalized sieve minimum distance psmd estimators which are minimizers of a penalized empirical minimum distance criterion over a collection of sieve spaces that are dense in the infinite dimensional function parameter space some of the psmd procedures use slowly growing finite dimensional sieves with flexible penalties or without any penalty others use large dimensional sieves with lower semicompact and or convex penalties we establish their consistency and the convergence rates in banach space norms such as a sup norm or a root mean squared norm allowing for possibly noncompact infinite dimensional parameter spaces for both mildly and severely ill posed nonlinear inverse problems our convergence rates in hilbert space norms such as a root mean squared norm achieve the known minimax optimal rate for the nonparametric mean iv regression we illustrate the theory with a nonparametric additive quantile iv regression we present a simulation study and an empirical application of estimating nonparametric quantile iv engel curves

convex optimization shape constraints compound decisions and empirical bayes rules estimation of mixture densities for the classical gaussian compound decision problem and their associated empirical bayes rules is considered from two new perspectives the first motivated by brown and greenshtein introduces a nonparametric maximum likelihood estimator of the mixture density subject to a monotonicity constraint on the resulting bayes rule the second motivated by jiang and zhang proposes a new approach to computing the kiefer wolfowitz nonparametric maximum likelihood estimator for mixtures in contrast to prior methods for these problems our new approaches are cast as convex optimization problems that can be efficiently solved by modern interior point methods in particular we show that the reformulation of the kiefer wolfowitz estimator as a convex optimization problem reduces the computational effort by several orders of magnitude for typical problems by comparison to prior em algorithm based methods and thus greatly expands the practical applicability of the resulting method

parameterization of the class of deadbeat controllers via the theory of decoupling a novel approach is presented for parameterizing the class of minimum time deadbeat controllers mtdc through the minimum number of parameters the approach is based on the theory of decoupling and the properties of square decouplable systems the main result is a compact parametric form for the class of mtdc of a discrete time system contrary to many existing techniques no special assumption on the invertibility of the transition matrix is required

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

time optimal control of disturbance rejection tracking systems for discrete time time delayed systems by state feedback in this paper we solve the disturbance rejection and tracking problem for linear multivariable discrete time systems with time delayed controlled inputs a set of necessary and sufficient conditions under which the proposed problem is controllable is defined also the nilpotency properties of such systems is established and used as the basis of a comprehensive design procedure this general procedure is illustrated by designing a time optimal disturbance rejection tracking system for a stirred tank with time delayed control inputs

the triangular model with random coefficients abstract the triangular model is a very popular way to allow for causal inference in the presence of endogeneity in this model an outcome is determined by an endogenous regressor which in turn is first caused by an instrument we study the triangular model with random coefficients and additional exogenous regressors in both equations and establish non identification of the joint distribution of random coefficients this implies that counterfactual outcomes are not identified either non identification continues to hold if we confine ourselves to the joint distribution of coefficients in the outcome equation or indeed any marginal except the one on the endogenous regressor nonidentification prevails as well if we focus on means of random coefficients implying that iv is asymptotically biased based on these insights we derive bounds on the joint distribution of economically relevant functionals e g counterfactual outcomes and suggest an additional restriction that allows to point identify the distribution of random coefficients in the outcome equation we extend the model to cover the case where the regressors and instruments have limited support and analyze semi and nonparametric sample counterpart estimators in finite and large samples and we provide an application to consumer demand

parameterizations of the state feedback controllers for linear multivariable systems abstract a general framework is constructed upon which an explicit parametric formula can be derived for state feedback controllers containing all the possible combination of parameters the relation between the parameters is nonlinear in general and therefore many different constraint may be imposed by the designer to obtain desired performance criteria a couple of illustrative examples are presented

a smoothed em approach to indirect estimation problems with particular reference to stereology and emission tomography there are many practical problems where the observed data are not drawn directly from the density g of real interest but rather from another distribution derived from g by the application of an integral operator the estimation of g then entails both statistical and numerical difficulties a natural statistical approach is by maximum likelihood conveniently implemented using the em algorithm but this provides unsatisfactory reconstructions of g in this paper we modify the maximum likelihood em approach by introducing a simple smoothing step at each em iteration in our experience this algorithm converges in relatively few iterations to good estimates of g that do not depend on the choice of starting configuration some theoretical background is given that relates this smoothed em algorithm to a maximum penalized likelihood approach two applications are considered in detail the first is the classical stereology problem of determining particle size distributions from data collected on a plane section through a composite medium the second concerns the recovery of the structure of a section of the human body from external observations obtained by positron emission tomography for this problem we also suggest several technical improvements on existing methodology

simultaneous control of linear systems by state feedback in this paper a new method for finding a state feedback matrix in order to control simultaneously a collection of linear systems of the same size by using similarity operations is presented for stabilization of all the systems it is necessary that the eigenvalues of the closed loop systems lie inside a specified region in the left hand side of the complex plane this aim is achieved by solving linear and nonlinear parametric systems of equations using nonlinear programming the presented method is implemented in two examples and the results are verified in view of the norm of the state feedback matrix and stabilizability

maximum likelihood estimation of a binary choice model with random coefficients of unknown distribution abstract we consider a binary response model yi 1 xi i ei 0 with xi independent of the unobservables i e i no finite dimensional parametric restrictions are imposed on f0 the joint distribution of i e i a nonparametric maximum likelihood estimator for f0 is shown to be consistent we analyze some conditions under which f0 is or is not identified in particular we show that if the support of f0 is a subset of any half of the unit hypersphere then f0 is identified relative to all distributions on the unit hypersphere we also provide some monte carlo evidence on the small sample performance of our estimator

nonparametric maximum likelihood methods for binary response models with random coefficients the venerable method of maximum likelihood has found numerous recent applications in nonparametric estimation of regression and shape constrained densities for mixture models the nonparametric max

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

on nonparametric estimation of intercept and slope distributions in random coefficient regression an experiment records stimulus and response for a random sample of cases the relationship between response and stimulus is thought to be linear the values of the slope and intercept varying by case from such data we construct a consistent asymptotically normal nonparametric estimator for the joint density of the slope and intercept our methodology incorporates the radial projection slice theorem for the radon transform a technique for locally linear nonparametric regression and a tapered fourier inversion computationally the new density estimator is more feasible than competing nonparametric estimators one of which is based on moments and the other on minimum distance considerations

variability assessment in positron emission tomography and related generalized deconvolution models abstract the problem of variance assessment for positron emission tomography pet image reconstructions is considered in the context of generalized deconvolution a refinement of an approximate technique proposed by carson and colleagues is examined computational implications of representing the reconstruction kernel in terms of a weighted sum of gaussian densities are developed bias and variance characteristics of the resulting variance estimators are examined by numerical simulation for typical regions the error in estimated standard deviations is found to be on the order of 10 the use of smoothing to obtain more reliable pointwise variance estimators is described and some theoretical analysis of this technique is carried out for the pet application simulations suggest that the percent improvement in the root mean squared error accuracy of pointwise variance estimators obtained by smoothing can be on the order of 30 a practical application of the methodology to a pet study is presented

spectral dimension and bohr s formula for schrodinger operators on unbounded fractal spaces we establish an asymptotic formula for the eigenvalue counting function of the schrodinger operator for some unbounded potentials v on several types of unbounded fractal spaces we give sufficient conditions for bohr s formula to hold on metric measure spaces which admit a cellular decomposition and then verify these conditions for fractafolds and fractal fields based on nested fractals in particular we partially answer a question of fan khandker and strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow up of a sierpinski gasket

on parametric state feedback design an explicit parametric expression for the controller gain matrix in linear state variable feedback is derived it involves both the case where the set of closed loop eigenvalues does not include any open loop eigenvalue and the case where the two eigenvalue sets have elements in common some advantages of this new explicit parameterization of the controller gain matrix are illustrated by two numerical examples

nonparametric identification of the distribution of random coefficients in binary response static games of complete information this paper studies binary response static games of complete information allowing complex heterogeneity through a random coefficients specification the main result of the paper establishes nonparametric point identification of the joint density of all random coefficients except those on interaction effects under additional independence assumptions we identify the joint density of the interaction coefficients moreover we prove that in the presence of covariates that are common to both players the player specific coefficient densities are identified while the joint density of all random coefficients is not point identified however we do provide bounds on counterfactual probabilities that involve this joint density

the discrete linear time invariant time optimal control problem an overview a review of two decades of investigation into one of the fundamental problems of control theory the discrete linear time invariant time optimal control problem is presented two classes of multi input time optimal controller one based on system controllability and the other on eigenvalue assignment are considered in some detail also discussed are the dual state reconstruction problem the deadbeat control of linear systems with inaccessible state and other extensions the exposition is a unifying one and involves only that matrix algebra commonly encountered in control theory

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

waves on fractal like manifolds and effective energy propagation we prove that the average speed of effective energy propagation of the waves satisfying long wave length condition in some globally fractal like manifolds is asymptotically zero

transition densities for brownian motion on the sierpinski carpet upper and lower bounds are obtained for the transition densitiesp t x y of brownian motion on the sierpinski carpet these are of the same form as those which hold for the sierpinski gasket in addition the joint continuity ofp t x y is proved the existence of the spectral dimension is established and the einstein relation connecting the spectral dimension the hausdorff dimension and the resistance exponent is shown to hold

nonparametric errors in variables models with measurement errors on both sides of the equation abstract measurement errors are often correlated as in surveys where respondent s biases or tendencies to err affect multiple reported variables we extend schennach 2007 to identify moments of the conditional distribution of a true y given a true x when both are measured with error the measurement errors in y and x are correlated and the true unknown model of y given x has nonseparable model errors after showing nonparametric identification we provide a sieve generalized method of moments based estimator of the model and apply it to nonparametric engel curve estimation in our application measurement errors on the expenditures of a good y are by construction correlated with measurement errors in total expenditures x this problem which is present in many consumption data sets has been ignored in most demand applications we find that accounting for this problem casts doubt on hildenbrand s 1994 increasing dispersion assumption

pole assignment by state transition graph the pole assignment problem is considered using the graph representation of a matrix the parametrization of controllers which have a specified characteristic polynomial is given a simple algorithm based on graphs is presented and two examples are given