local asymptotics for polynomial spline regression in this paper we develop a general theory of local asymptotics for least squares estimates over polynomial spline spaces in a regression problem the polynomial spline spaces we consider include univariate splines tensor product splines and bivariate or multivariate splines on triangulations we establish asymptotic normality of the estimate and study the magnitude of the bias due to spline approximation the asymptotic normality holds uniformly over the points where the regression function is to be estimated and uniformly over a broad class of design densities error distributions and regression functions the bias is controlled by the minimum l norm of the error when the target regression function is approximated by a function in the polynomial spline space that is used to define the estimate the control of bias relies on the stability in l norm of l 2 projections onto polynomial spline spaces asymptotic normality of least squares estimates over polynomial or trigonometric polynomial spaces is also treated by the general theory in addition a preliminary analysis of additive models is provided

newton interpolation in fejer and chebyshev points soit une courbe de jordan dans le plan complexe et soit l ensemble compact borne par soit f une fonction analytique sur l approximation de f sur par un polynome p de degre inferieur a n et qui interpole f en n points de est etudiee

structured squaring down and zero assignment the problem of zero assignment by squaring down is considered for a system of p inputs n outputs and n states m p where not all outputs are free variables for design we consider the case where a k subset of outputs is preserved in the new output set and the rest are recombined to produce a total new set of p outputs new invariants for the problem are introduced which include a new class of fixed zeros and the methodology of the global linearization developed originally for the output feedback pole assignment problem is applied to this restricted form of the squaring down problem it is shown that the problem can be solved generically if p k m p where k k p is the number of fixed outputs and is a system and compensation scheme invariant which is defined as the restricted forney degree

multivariate locally weighted least squares regression nonparametric regression using locally weighted least squares was first discussed by stone and by cleveland recently it was shown by fan and by fan and gijbels that the local linear kernel weighted least squares regression estimator has asymptotic properties making it superior in certain senses to the nadaraya watson and gasser muller kernel estimators in this paper we extend their results on asymptotic bias and variance to the case of multivariate predictor variables we are able to derive the leading bias and variance terms for general multivariate kernel weights using weighted least squares matrix theory this approach is especially convenient when analysing the asymptotic conditional bias and variance of the estimator at points near the boundary of the support of the predictors we also investigate the asymptotic properties of the multivariate local quadratic least squares regression estimator discussed by cleveland and devlin and in the univariate case higher order polynomial fits and derivative estimation

fast and stable rational interpolation in roots of unity and chebyshev points a new method for interpolation by rational functions of prescribed numerator and denominator degrees is presented when the interpolation nodes are roots of unity or chebyshev points the algorithm is particularly simple and relies on discrete fourier transform matrices which results in a fast implementation using the fast fourier transform the method is generalized for arbitrary grids which requires the construction of polynomials orthogonal on the set of interpolation nodes the appearance of common factors in the numerator and denominator due to finite precision arithmetic is explained by the behavior of the singular values of the linear system associated with the rational interpolation problem the new algorithm has connections with other methods particularly the work of jacobi and kronecker berrut and mittelmann and egecioglu and koc short matlab codes and numerical experiments are included

construction and parameterization of all static and dynamic h 2 optimal state feedback solutions for discrete time systems abstract this paper considers an h 2 optimization problem via state feedback for discrete time systems the class of problems dealt with here has a left invertible transfer matrix function from the control input to the controlled output the paper constructs and parameterizes all the static and dynamic h 2 optimal state feedback solutions moreover all the eigenvalues of an optimal closed loop system are characterized all optimal closed loop systems share a set of eigenvalues which are termed the optimal fixed modes every h 2 optimal controller must assign among the closed loop eigenvalues the set of optimal fixed modes this set of optimal fixed modes includes a set of optimal fixed decoupling zeros which shows the minimum absolutely necessary number and locations of pole zero cancellations present in any h 2 optimal design most of the results presented here are analogous to but not quite the same as those for continuous time systems in fact there are some fundamental differences between the continuous and discrete time systems reflecting mainly the inherent nature and characteristics of these systems

a non iterative method for computing the infimum in h optimization this paper presents a simple and non iterative procedure for the computation of the exact value of the infimum in the singular h optimization problem and is an extension of our earlier work the problem formulation is general and does not place any restriction on the direct feedthrough terms between the control input and the controlled output variables and between the disturbance input and the measurement output variables our method is applicable to a class of singular h optimization problems for which the transfer functions from the control input to the controlled output and from the disturbance input to the measurement output have no invariant zeros on the jw axis and also satisfy certain geometric conditions the computation of the infimum in our method involves solving two well defined riccati and two lyapunov equations

squaring down lti systems a geometric approach in this paper the problem of reducing a given lti system into a left or right invertible one is addressed and solved with the standard tools of the geometric control theory first it will be shown how an lti system can be turned into a left invertible system thus preserving key system properties like stabilizability phase minimality right invertibility relative degree and infinite zero structure moreover the additional invariant zeros introduced in the left invertible system thus obtained can be arbitrarily assigned in the complex plane by duality the scheme of a right inverter will be derived straightforwardly moreover the squaring down problem will be addressed in fact when the left and right reduction procedures are applied together a system with an unequal number of inputs and outputs is turned into a square and invertible system furthermore as an example it will be shown how these techniques may be employed to weaken the standard assumption of left invertibility of the plant in many optimization problems

root finding by divided differences a recursive method is presented for computing a simple zero of an analytic functionf from information contained in a table of divided differences of its reciprocalh 1 f a good deal of flexibility is permitted in the choice of ordinate and derivative values and in the choice of the number of previous points upon which to base the next estimate of the required zero the method is shown to be equivalent to a process of fitting rational functions with linear numerators to data sampled fromf asymptotic and regional convergence properties of such a process have already been studied in particular asymptotically quadratic convergence is easily obtained at the expense of only one function evaluation and a moderate amount of overhead computation per step in these respects the method is comparable with the newton form of iterated polynomial inverse interpolation while its regional convergence characteristics may be superior in certain circum stances it is also shown that the method is not unduly sensitive to round off errors

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

optimal convergence rates bahadur representation and asymptotic normality of partitioning estimators abstract this paper studies the asymptotic properties of partitioning estimators of the conditional expectation function and its derivatives mean square and uniform convergence rates are established and shown to be optimal under simple and intuitive conditions the uniform rate explicitly accounts for the effect of moment assumptions which is useful in semiparametric inference a general asymptotic integrated mean square error approximation is obtained and used to derive an optimal plug in tuning parameter selector a uniform bahadur representation is developed for linear functionals of the estimator using this representation asymptotic normality is established along with consistency of a standard error estimator the finite sample performance of the partitioning estimator is examined and compared to other nonparametric techniques in an extensive simulation study

a method for directly finding the denominator values of rational interpolants in this paper we give an algorithm for directly finding the denominator values of rational interpolants at the nodes and present an expression for the corresponding rational interpolant when the latter exists with these denominator values our method also provides information concerning the existence of the interpolant and the presence of unattainable points and poles

asymptotic normality of series estimators for nonparametric and semiparametric regression models this paper establishes the asymptotic normality of series estimators for nonparametric regression models gallant s fourier flexible form estimators trigonometric series estimators and polynomial series estimators are prime examples of the estimators covered by the results the results apply to a wide variety of estimates in the regression model under consideration including derivatives and integrals of the regression function the errors in the model may be homoskedastic or heteroskedastic the paper also considers series estimators for additive interactive regression semiparametric regression and semiparametric index regression models and shows them to be consistent and asymptotically normal copyright 1991 by the econometric society

divided differences shift transformations and larkin s root finding method for a one dimensional complex valued function f this paper deals with iterative root finding methods using divided differences of f assuming that f is given in a newtonian representation we show how homer like transformations shift transformations yield the divided differences needed in each iteration step in particular we consider an iteration method given by larkin 5 and derive an equivalent version of this method fitting into this context monotonic convergence to real roots of real polynomials is investigated both shift and nonshift versions of several root finding methods are tested and compared with respect to their numerical behavior

polynomial interpolation lagrange versus newton we show that the lagrangian form of the interpolating polynomial can be calculated with the same number of arithmetic operations as the newtonian form

mappings of the finite and the infinite zero structures and invertibility structures of general linear multivariable systems under the bilinear transformation this paper presents a comprehensive picture of the mapping of structural properties associated with general linear multivariable systems under bilinear transformation while the mapping of poles of linear multivariable systems under such a transformation is well known the question of how the structural invariant properties of a given system are mapped remains unanswered this paper addresses this question more specifically we investigate how the finite and infinite zero structures as well as invertibility structures of a general continuous time discrete time linear time invariant multivariable system are mapped to those of its discrete time continuous time counterpart under the bilinear inverse bilinear transformation we demonstrate that the structural invariant indices lists i2 and i3 of morse remain invariant under the bilinear transformation while the structural invariant indices lists i1 and i4 of morse are in general changed

squaring down by static and dynamic compensators given a system which is not necessarily invertible and which has an unequal number of inputs a method of squaring down that is a method of designing pre compensators and postcompensators such that the resulting system has an equal number of inputs and outputs and is invertible is presented the compensators are in general dynamic and have the property that the additional finite zeros induced by them are assignable to the open left half complex plan furthermore the compensators are asymptotically stable and hence do not deteriorate the robustness and performance of an eventual feedback design also the compensator design preserves the stabilizability detectability and minimum phase properties and the infinite zero structure of the original system thus a method of designing nonsquare systems by converting them to square invertible systems is introduced two applications of such a design philosophy 1 diagonal decoupling with state feedback and 2 almost disturbance decoupling with output feedback are pointed out

a note on convergence rates and asymptotic normality for series estimators uniform convergence rates this paper establishes improved uniform convergence rates for series estimators series estimators are least squares fits of a regression function where the number of regressors depends on sample size i will specialize my results to the cases of polynomials and regression splines these results improves upon results obtained earlier by newey yet fail to attain the optimal rates of convergence

robust nonparametric confidence intervals for regression discontinuity designs in the regression discontinuity rd design units are assigned to treatment based on whether their value of an observed covariate exceeds a known cutoff in this design local polynomial estimators are now routinely employed to construct confidence intervals for treatment effects the performance of these confidence intervals in applications however may be seriously hampered by their sensitivity to the specific bandwidth employed available bandwidth selectors typically yield a large bandwidth leading to data driven confidence intervals that may be biased with empirical coverage well below their nominal target we propose new theory based more robust confidence interval estimators for average treatment effects at the cutoff in sharp rd sharp kink rd fuzzy rd and fuzzy kink rd designs our proposed confidence intervals are constructed using a bias corrected rd estimator together with a novel standard error estimator for practical implementation we discuss mean squared error optimal bandwidths which are by construction not valid for conventional confidence intervals but are valid with our robust approach and consistent standard error estimators based on our new variance formulas in a special case of practical interest our procedure amounts to running a quadratic instead of a linear local regression more generally our results give a formal justification to simple inference procedures based on increasing the order of the local polynomial estimator employed we find in a simulation study that our confidence intervals exhibit close to correct empirical coverage and good empirical interval length on average remarkably improving upon the alternatives available in the literature all results are readily available in r and stata using our companion software packages described in calonico cattaneo and titiunik 2014d 2014b

a method of estimating the average derivative abstract we derive a simple semi parametric estimator of the direct average derivative e d m x where m x is the regression function and s the support of the density of x is compact we partition s into disjoint bins and the local slope d m x within these bins is estimated by using ordinary least squares our average derivative estimate a is then obtained by taking the weighted average of these least squares slopes we show that this estimator is asymptotically normally distributed we also propose a consistent estimator of the variance of a using monte carlo simulation experiments based on a censored regression model with tobit model as a special case we produce small sample results comparing our estimator with the hardle stoker 1989 investigating smooth multiple regression by the method of average derivatives journal of american statistical association 84 408 986 995 method we conclude that a performs better that the hardle stoker estimator for bounded and discontinuous covariates

optimal data driven regression discontinuity plots exploratory data analysis plays a central role in applied statistics and econometrics in the popular regression discontinuity rd design the use of graphical analysis has been strongly advocated because it provides both easy presentation and transparent validation of the design rd plots are nowadays widely used in applications despite its formal properties being unknown these plots are typically presented employing ad hoc choices of tuning parameters which makes these procedures less automatic and more subjective in this article we formally study the most common rd plot based on an evenly spaced binning of the data and propose several optimal data driven choices for the number of bins depending on the goal of the researcher these rd plots are constructed either to approximate the underlying unknown regression functions without imposing smoothness in the estimator or to approximate the underlying variability of the raw data while smoothing out the otherwise uninformative scatterplot of the da

matrices for the direct determination of the barycentric weights of rational interpolation abstract let x0 xn be n 1 interpolation points nodes and 0 n be n 1 interpolation data then every rational function r with numerator and denominator degress n interpolating these values can be written in its barycentric form completely determined by a vector u of n 1 barycentric weights uk finding u is therefore an alternative to the determination of the coefficients in the canonical form of r it is advantageous inasmuch as u contains information about unattainable points and poles in classical rational interpolation the numerator and the denominator of r are made unique up to a constant factor by restricting their respective degrees we determine here the corresponding vectors u by applying a stabilized elimination algorithm to a matrix whose kernel is the space spanned by the u s the method is of complexity o n3 in terms of the denominator degree n it seems on the other hand to be among the most stable ones

squaring down and the problems of almost zeros for continuous time systems abstract in this paper we construct precompensators which square down a system such that the resulting square system has the same unstable zeros and system gains we derive the minimal gain of the precompensator to achieve such a square system if the gain of this precompensator is too large then we derive an explicit trade off between the gain of the precompensator and the number of unstable zeros we allow the precompensator to introduce in the system

heteroscedasticity robust inference in linear regression models with many covariates we consider inference in linear regression models that is robust to heteroscedasticity and the presence of many control variables when the number of control variables increases at the same rate as

squaring down of non strictly proper systems in an earlier work a procedure for squaring down proper systems by static and dynamic compensators has been developed the compensators designed there have the following properties a they are asymptotically stable b the additional finite zeros induced by them are assignable to the open left half complex plane and c they preserve the fundamental properties such as stabilizability detectability infinite zero structure and minimum phase nature of the original system the earlier design procedure is extended to non strictly proper systems while retaining all the above mentioned properties

on the denominator values and barycentric weights of rational interpolants we improve upon the method of zhu and zhu a method for directly finding the denominator values of rational interpolants j comput appl math 148 2002 341 348 for finding the denominator values of rational interpolants reducing considerably the number of arithmetical operations required for their computation in a second stage we determine the points if existent which can be discarded from the rational interpolation problem furthermore when the interpolant has a linear denominator we obtain a formula for the barycentric weights which is simpler than the one found by berrut and mittelmann matrices for the direct determination of the barycentric weights of rational interpolation j comput appl math 78 1997 355 370 subsequently we give a necessary and sufficient condition for the rational interpolant to have a pole

inference in linear regression models with many covariates and heteroskedasticity the linear regression model is widely used in empirical work in economics statistics and many other disciplines researchers often include many covariates in their linear model specification in a