Showing papers on "Bounded function published in 1996"

Weak Convergence and Empirical Processes: With Applications to Statistics

Abstract: 1.1. Introduction.- 1.2. Outer Integrals and Measurable Majorants.- 1.3. Weak Convergence.- 1.4. Product Spaces.- 1.5. Spaces of Bounded Functions.- 1.6. Spaces of Locally Bounded Functions.- 1.7. The Ball Sigma-Field and Measurability of Suprema.- 1.8. Hilbert Spaces.- 1.9. Convergence: Almost Surely and in Probability.- 1.10. Convergence: Weak, Almost Uniform, and in Probability.- 1.11. Refinements.- 1.12. Uniformity and Metrization.- 2.1. Introduction.- 2.2. Maximal Inequalities and Covering Numbers.- 2.3. Symmetrization and Measurability.- 2.4. Glivenko-Cantelli Theorems.- 2.5. Donsker Theorems.- 2.6. Uniform Entropy Numbers.- 2.7. Bracketing Numbers.- 2.8. Uniformity in the Underlying Distribution.- 2.9. Multiplier Central Limit Theorems.- 2.10. Permanence of the Donsker Property.- 2.11. The Central Limit Theorem for Processes.- 2.12. Partial-Sum Processes.- 2.13. Other Donsker Classes.- 2.14. Tail Bounds.- 3.1. Introduction.- 3.2. M-Estimators.- 3.3. Z-Estimators.- 3.4. Rates of Convergence.- 3.5. Random Sample Size, Poissonization and Kac Processes.- 3.6. The Bootstrap.- 3.7. The Two-Sample Problem.- 3.8. Independence Empirical Processes.- 3.9. The Delta-Method.- 3.10. Contiguity.- 3.11. Convolution and Minimax Theorems.- A. Appendix.- A.1. Inequalities.- A.2. Gaussian Processes.- A.2.1. Inequalities and Gaussian Comparison.- A.2.2. Exponential Bounds.- A.2.3. Majorizing Measures.- A.2.4. Further Results.- A.3. Rademacher Processes.- A.4. Isoperimetric Inequalities for Product Measures.- A.5. Some Limit Theorems.- A.6. More Inequalities.- A.6.1. Binomial Random Variables.- A.6.2. Multinomial Random Vectors.- A.6.3. Rademacher Sums.- Notes.- References.- Author Index.- List of Symbols.

Econometric methods for fractional response variables with an application to 401 (k) plan participation rates

Abstract: SUMMARY We develop attractive functional forms and simple quasi-likelihood estimation methods for regression models with a fractional dependent variable. Compared with log-odds type procedures, there is no difficulty in recovering the regression function for the fractional variable, and there is no need to use ad hoc transformations to handle data at the extreme values of zero and one. We also offer some new, robust specification tests by nesting the logit or probit function in a more general functional form. We apply these methods to a data set of employee participation rates in 401 (k) pension plans. I. INTRODUCTION Fractional response variables arise naturally in many economic settings. The fraction of total weekly hours spent working, the proportion of income spent on charitable contributions, and participation rates in voluntary pension plans are just a few examples of economic variables bounded between zero and one. The bounded nature of such variables and the possibility of observing values at the boundaries raise interesting functional form and inference issues. In this paper we specify and analyse a class of functional forms with satisfying econometric properties. We also synthesize and expand on the generalized linear models (GLM) literature from statistics and the quasi-likelihood literature from econometrics to obtain robust methods for estimation and inference with fractional response variables. We apply the methods to estimate a model of employee participation rates in 401 (k) pension plans. The key explanatory variable of interest is the plan's 'match rate,' the rate at which a firm matches a dollar of employee contributions. The empirical work extends that of Papke (1995), who studied this problem using linear spline methods. Spline methods are fiexible, but they do not ensure that predicted values lie in the unit interval. To illustrate the methodological issues that arise with fractional dependent variables, suppose that a variable y, O^y^l, is to be explained by a 1 x/^ vector of explanatory variables \ = {Xi,X2 XK), with the convention that Xi = l. The population model

Supervisory control of families of linear set-point controllers - Part I. Exact matching

Abstract: This paper describes a simple "high-level" controller called a "supervisor" which is capable of switching into feedback with a SISO process, a sequence of linear positioning or set-point controllers from a family of candidate controllers so as to cause the output of the process to approach and track a constant reference input. The process is assumed to be modeled by a SISO linear system whose transfer function is in the union of a number of subclasses, each subclass being small enough so that one of the candidate controllers would solve the positioning problem if the transfer function of the process were to be one of the subclasses' members. Each subclass contains a "nominal process model transfer function" about which the subclass is centred. It is shown that in the absence of unmodeled process dynamics, the proposed supervisor can successfully perform its function even if process disturbances are present, provided they are bounded and constant.

A Smooth Converse Lyapunov Theorem for Robust Stability

Abstract: This paper presents a converse Lyapunov function theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems. In a unified and natural manner, it (1) allows arbitrary bounded time-varying parameters in the system description, (2) deals with global asymptotic stability, (3) results in smooth (infinitely differentiable) Lyapunov functions, and (4) applies to stability with respect to not necessarily compact invariant sets.

Induced L2-norm control for LPV systems with bounded parameter variation rates

Abstract: A linear, finite-dimensional plant, with state-space parameter dependence, is controlled using a parameter-dependent controller. The parameters whose values are in a compact set, are known in real time. Their rates of variation are bounded and known in real time also. The goal of control is to stabilize the parameter-dependent closed-loop system, and provide disturbance/error attenuation as measured in induced L2 norm. Our approach uses a bounding technique based on a parameter-dependent Lyapunov function, and then solves the control synthesis problem by reformulating the existence conditions into a semi-infinite dimensional convex optimization. We propose finite dimensional approximations to get sufficient conditions for successful controller design.

Bounding Approaches to System Identification

Abstract: Overview of the Volume J. Norton. Optimal Estimation Theory for Dynamic System with Set Membership Uncertainty: An Overview M. Milanese, A. Vicino. Solving Linear Problems in the Presence of Bounded Data Perturbations B.Z. Kacewicz. A Review and a Comparison of Ellipsoidal Bounding Algorithms G. Favier, L.V.R. Arruda. On the Deadzone in System Identification K. Forsman, L. Ljung. Recursive Estimation Algorithms for Linear Models with Set Membership Error G. Belforte, T.T. Tay. Transfer Function Parameter Interval Estimation Using Recursive Least Squares in the Time and Frequency Domains P.O. Gutman. Volume-optimal Inner and Outer Ellipsoids L. Pronzato, E. Walter. Linear Interpolation and Estimation Using Interval Analysis S.M. Markov, E.D. Popova. Adaptive Approximation of Uncertainty Sets for Linear Regression Models A. Vicino, G. Zappa. Worstcase l1 Identification M. Milanese. Recursive Robust Minimax Estimation E. Walter, H. Piet-Lahanier. Robustness to Outliers of Bounded-error Estimators, Consequences on Experiment Design L. Pronzato, E. Walter. Ellipsoidal State Estimation for Uncertain Dynamical Systems T.F. Filipova, et al. Set-valued Estimation of State and Parameter Vectors within Adaptive Control-Systems V.M. Kuntsevich. Limited-complexity Polyhedric Tracking H. Piet-Lahanier, E. Walter. Parameterbounding Algorithms for Linear Errors in Variables Models S.M. Veres, J.P. Norton. Errors-invariables Models in Parameter Bounding V. Cerone. Identification of Linear Objects with Bounded Disturbances in Both Input and Output Channels Yu.A. Merkuryev. Identification of Nonlinear Statespace Models by Deterministic Search J.P. Norton, S.M. Veres. Robust Identification and Prediction for Nonlinear State-Space Models with Bounded Output Error K.J. Keesman. Estimation Theory for Nonlinear Models and Set Membership Uncertainty M. Milanese, A. Vicino. Guaranteed Nonlinear Set Estimation via Interval Analysis L. Jaulin, E. Walter. On Adaptive Control of Systems Subjected to Bounded Disturbances L.S. Zhitecki. Predictive Selftuning Control by Parameter Bounding and Worstcase Design S.M. Veres, J.P. Norton. Estimation of a Mobile Robot Localization: Geometric Approaches D. Meizel, et al. Improved Image Compression Using Bounded Error Parameter Estimation Concepts A.K. Rao. Application of OBE Algorithms to Speech Analysis, Recognition and Coding J.R. Deller Jr., et al. 2 additional articles. Index.

On the Cahn-Hilliard equation with degenerate mobility

Abstract: An existence result for the Cahn–Hilliard equation with a concentration dependent diffusional mobility is presented. In particular, the mobility is allowed to vanish when the scaled concentration takes the values $ \pm 1$, and it is shown that the solution is bounded by 1 in magnitude. Finally, applications of our method to other degenerate fourth-order parabolic equations are discussed.

Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms

Abstract: We develop results on geometric ergodicity of Markov chains and apply these and other recent results in Markov chain theory to multidimensional Hastings and Metropolis algorithms. For those based on random walk candidate distributions, we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution π. By phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly-used statistical forms. From these results we develop central limit theorems for the Metropolis algorithm. Converse results, showing non-geometric convergence rates for chains where the rejection rate is not bounded away from unity, are also given ; these show that the negative-definiteness property is not redundant.

Stability of nonlinear hawkes processes

Abstract: We address the problem of the convergence to equilibrium of a general class of point processes, containing, in particular, the nonlinear mutually exciting point processes, an extension of the linear Hawkes processes, and give general conditions guaranteeing the existence of a stationary version and the convergence to equilibrium of a nonstationary version, both in distribution and in variation. We also give a new proof of a result of Kerstan concerning point processes with bounded intensity and general nonlinear dynamics satisfying a Lipschitz condition.

High dimensional integration of smooth functions over cubes

Abstract: We construct a new algorithm for the numerical integration of functions that are defined on a \(d\)-dimensional cube. It is based on the Clenshaw-Curtis rule for \(d=1\) and on Smolyak's construction. This way we make the best use of the smoothness properties of any (nonperiodic) function. We prove error bounds showing that our algorithm is almost optimal (up to logarithmic factors) for different classes of functions with bounded mixed derivative. Numerical results show that the new method is very competitive, in particular for smooth integrands and \(d \ge 8\).

The law of the euler scheme for stochastic differential equations: i. convergence rate of the distribution function

Abstract: We study the approximation problem ofE f(X T ) byE f(X ), where (X t ) is the solution of a stochastic differential equation, (X ) is defined by the Euler discretization scheme with stepT/n, andf is a given function. For smoothf's, Talay and Tubaro have shown that the errorE f(X T ) −f(X ) can be expanded in powers of 1/n, which permits to construct Romberg extrapolation precedures to accelerate the convergence rate. Here, we prove that the expansion exists also whenf is only supposed measurable and bounded, under an additional nondegeneracy condition of Hormander type for the infinitesimal generator of (X t ): to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law ofX and compare it to the density of the law ofX T .

A generalized Drazin inverse

Abstract: The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point ofthe operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).

Superfaces: polygonal mesh simplification with bounded error

Abstract: The algorithm presented simplifies polyhedral meshes within prespecified tolerances based on a bounded approximation criterion. The vertices in the simplified mesh are a proper subset of the original vertices. The algorithm, called Superfaces, makes two major contributions to the research in this area: it uses a bounded approximation approach, which guarantees that a simplified mesh approximates the original mesh to within a prespecified tolerance (that is, every vertex v in the original mesh will lie within a user specified distance /spl epsiv/ of the simplified mesh); its face merging procedure is efficient and greedy-that is, it does not backtrack or undo any merging once completed and thus, the algorithm is practical for simplifying very large meshes.

Finding a small root of a bivariate integer equation; factoring with high bits known

Abstract: We present a method to solve integer polynomial equations in two variables, provided that the solution is suitably bounded. As an application, we show how to find the factors of N = PQ if we are given the high order ((1/4) log2 N) bits of P. This compares with Rivest and Shamit's requirement of ((1/3) log2 N) bits.

Asymptotic Equivalence of Density Estimation and Gaussian White Noise

Abstract: Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance $\Delta$ would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression. We consider the analogous problem for the experiment given by n i.i.d. observations having density f on the unit interval. Our basic result concerns the parameter space of densities which are in a Holder ball with exponent $\alpha > 1/2$ and which are uniformly bounded away from zero. We show that an i. i. d. sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift $f^{1/2}$ and variance $1/4 n^{-1}$. This represents a nonparametric analog of Le Cam's heteroscedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques related to the Hungarian construction. White noise models on f and log f are also considered, allowing for various "automatic" asymptotic risk bounds in the i.i.d. model from white noise.

Bounded scheduling of process networks

Abstract: We present a scheduling policy for complete, bounded execution of Kahn process network programs. A program is a set of processes that communicate through a network of first-in first-out queues. In a complete execution, the program terminates if and only if all processes block attempting to consume data from empty communication channels. We are primarily interested in programs that operate on infinite streams of data and never terminate. In a bounded execution, the number of data elements buffered in each of the communication channels remains bounded. The Kahn process network model of computation is powerful enough that the questions of termination and bounded buffering are undecidable. No finite-time algorithm can decide these questions for all Kahn process network programs. Fortunately, because we are interested in programs that never terminate, our scheduler has infinite time and can guarantee that programs execute forever with bounded buffering whenever possible. Our scheduling policy has been implemented using Ptolemy, an object-oriented simulation and prototyping environment.

Global attractors for the three-dimensional Navier-Stokes equations

Abstract: In this paper we show that the weak solutions of the Navier-Stokes equations on any bounded, smooth three-dimensional domain have a global attractor for any positive value of the viscosity. The proof of this result, which bypasses the two issues of the possible nonuniqueness of the weak solutions and the possible lack of global regularity of the strong solutions, is based on a new point of view for the construction of the semiflow generated by these equations. We also show that, under added assumptions, this global attractor consists entirely of strong solutions.

The Cauchy integral, analytic capacity, and uniform rectifiability

Abstract: Several explanations concerning notation, terminology, and background are in order. First notation: by 7Hi we have denoted the one-dimensional Hausdorff measure (i.e. length), and A(z,r) stands for the closed disc with center z and radius r. A curve F is called AD-regular, that is, Ahlfors-David-regular, if it satisfies (1.2) (with E = F). Since the lower bound is automatic for curves, this means that 'Hl (r n A(z, r)) 0. General sets satisfying (1.2) are called AD-regular. It is simplest to define the L2-boundedness of the Cauchy singular integral operator via the truncated integrals: we say that CE is bounded in L2(E) (without really defining the operator CE itself) if there is M < ox such that

Statistical inverse estimation in Hilbert scales

Abstract: The recovery of signals from indirect measurements, blurred by random noise, is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable. For greater flexibility the general problem is embedded in an abstract Hilbert scale. In the applications Sobolev scales are used. For the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product, where the operator is bounded but not necessarily compact. A lower bound to certain minimax rates is included, and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate. Examples include errors-in-variables (deconvolution) and indirect nonparametric regression. Special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation.

On Weighted Sobolev Spaces

Abstract: We study density and extension problems for weighted Sobolev spaces on bounded (e, δ) domains 𝓓 when a doubling weight w satisfies the weighted Poincare inequality on cubes near the boundary of 𝓓 and when it is in the Muckenhoupt A p class locally in 𝓓. Moreover, when the weights w i (x) are of the form dist(x, M i )αi , α i ∈ ℝ, M i ⊂ 𝓓 that are doubling, we are able to obtain some extension theorems on (e, ∞) domains.

Tiling the line with translates of one tile

Abstract: A region $T$ is a closed subset of the real line of positive finite Lebesgue measure which has a boundary of measure zero. Call a region $T$ a tile if ${\Bbb R}$ can be tiled by measure-disjoint translates of $T$ . For a bounded tile all tilings of ${\Bbb R}$ with its translates are periodic, and there are finitely many translation-equivalence classes of such tilings. The main result of the paper is that for any tiling of ${\Bbb R}$ by a bounded tile, any two tiles in the tiling differ by a rational multiple of the minimal period of the tiling. From it we deduce a structure theorem characterizing such tiles in terms of complementing sets for finite cyclic groups.

Mosaic-skeleton approximations

Abstract: If a matrix has a small rank then it can be multiplied by a vector with many savings in memory and arithmetic. As was recently shown by the author, the same applies to the matrices which might be of full classical rank but have a smallmosaic rank. The mosaic-skeleton approximations seem to have imposing applications to the solution of large dense unstructured linear systems. In this paper, we propose a suitable modification of brandt's definition of an asymptotically smooth functionf(x,y). Then we considern×n matricesA n =[f(x ,y )] for quasiuniform meshes {x } and {y } in some bounded domain in them-dimensional space. For such matrices, we prove that the approximate mosaic ranks grow logarithmically inn. From practical point of view, the results obtained lead immediately toO(n logn) matrix-vector multiplication algorithms.

Heat-kernels and functional determinants on the generalized cone

Abstract: We consider zeta functions and heat-kernel expansions on the bounded, generalized cone in arbitrary dimensions using an improved calculational technique. The specific case of a global monopole is analysed in detail and some restrictions thereby placed on theA 5/2 coefficient. The computation of functional determinants is also addressed. General formulas are given and known results are incidentally, and rapidly, reproduced.

Linear time computable problems and first-order descriptions

Abstract: It is well known that every algorithmic problem definable by a formula of first-order logic can be solved in polynomial time, since all these problems are in L (see Aho and Ullman (1979) and Immerman (1987)). Using an old technique of Hanf (Hanf 1965) and other techniques developed to prove the decidability of formal theories in mathematical logic, it is shown that an arbitrary FO-problem over relational structures of bounded degree can be solved in linear time.

On the resolution of monotone complementarity problems

Abstract: A reformulation of the nonlinear complementarity problem (NCP) as an unconstrained minimization problem is considered. It is shown that any stationary point of the unconstrained objective function is a solution of NCP if the mapping F involved in NCP is continuously differentiable and monotone, and that the level sets are bounded if F is continuous and strongly monotone. A descent algorithm is described which uses only function values of F. Some numerical results are given.