The role of duality in optimization problems involving entropy functionals with applications to information theory

We consider infinite-dimensional optimization problems involving entropy-type functionals in the objective function as well as as in the constraints. A duality theory is developed for such problems and applied to the reliability rate function problem in information theory.This research was supported by ONR Contracts N00014-81-C-0236 and N00014-82-K-0295 with the Center for Cybernetics Studies, University of Texas, Austin, Texas. The first author was partly supported by NSF.     

Exponential smoothing for irregular data

Various types of exponential smoothing for data observed at irregular time intervals are surveyed. Double exponential smoothing and some modifications of Holt’s method for this type of data are suggested. A real data example compares double exponential smoothing and Wright’s modification of Holt’s method for data observed at irregular time intervals. This research has been supported by the Ministry of Education, Youth and Sports of the Czech Republic, Research Project MSM 0021620839, and by the Grant Agency of Charles University, Grant No. 342/2005.     

Asymptotic distribution theory for general statistical functionals

Summary In this paper an asymptotic theory is developed for a general statistical functional which includesL-estimators,M-estimators, andR-estimators as special cases. It is shown that a proof of the asymptotic normality ofL-estimators,M-estimators, andR-estimators can be based on one important fact, namely, that the inverse c.d.f. is compactly differentiable.     

Critical point theory for perturbations of symmetric functionals

Functionals which are invariant under the action of a compact transformation groupG often have many critical values. Here we consider functionals which are notG-invariant and give conditions for them to have infinitely many critical values; including a mountain pass theorem. We apply it to prove the existence of infinitely many solutions of a nonlinear Dirichlet problem with perturbedG-symmetries.     

Cohomology and Morse theory for strongly indefinite functionals

Supported in part by the Swedish Natural Science Research Council     

Exponential smoothing and credibility theory

Some years ago Gerber & Jones (1975a, 1975b) investigated simple recursion formulas for the linear credibility estimator. In the present paper we reexamine such formulas and show their connection to models of modern times series analysis.     

Pointwise convergence of Fourier regularization for smoothing data

The classical smoothing data problem is analyzed in a Sobolev space under the assumption of white noise. A Fourier series method based on regularization endowed with generalized cross validation is considered to approximate the unknown function. This approximation is globally optimal, i.e., the mean integrated squared error reaches the optimal rate in the minimax sense. In this paper the pointwise convergence property is studied. Specifically, it is proved that the smoothed solution is locally convergent but not locally optimal. Examples of functions for which the approximation is subefficient are given. It is shown that optimality and superefficiency are possible when restricting to more regular subspaces of the Sobolev space.     

Linear algorithms for recovering linear functionals from jittered information

The classical Smolyak Lemma proves the existence of linear algorithms for recovering linear functionals from exact information. This lemma remains valid for perturbed measurements of data. The goal of this paper is to show that the lemma does not hold for the case of jittered information, i.e., consisting of samples taken at slightly perturbed points.     

A new morse index theory for strongly indefinite functionals

In this paper, a new Morse index theory for strongly indefinite functionals was developed via Gălerkin approximation. In particular, the abstract theory is valid for those kinds of strongly indefinite functionals corresponding to wave equation and beam equation.     

Some remarks on critical point theory for nondifferentiable functionals

In this paper we study the existence of critical points of nondifferentiable functionals J of the kind with A (x,z) a Carathéodory function bounded between positive constant and with bounded derivative respect to the variable z, and F (x,z) is the primitive of a (Carathéodory) nonlinearity f (x,z) satisfying suitable hipotheses. Since J is just differentible along bounded directions, a suitable compactness condition is introduced. Its connection with coercivity is discussed. In addition, the case of concave-convex nonlinearities f (x,z), unbounded coefficients A (x,z) and related problems are also studied. Received December 19, 1997     

Asymptotic distribution theory of statistical functionals: The compact derivative approach for robust estimators

Summary Derivatives of statistical functionals have been used to derive the asymptotic distributions ofL-,M- andR-estimators. This approach is often heuristic because the types of derivatives chosen have serious limitations. The Gateaux derivative is too weak and the Fréchet derivative is too strong. In between lies the compact derivative. This paper obtains strong results in a rigorous manner using the compact derivative onC ₀(R). This choice of space allows results for a broader class of functionals than previous choices, and the fact that is often tight provides the compact set required. A major result is the derivation of the compact derivative of the inverse c.d.f. when the range space is endowed with the uniform norm. It has applications to the asymptotic theory ofL-,M- andR-estimators. We illustrate the power of this result by applications toL-estimators in settings including the one sample problem, data grouped by quantiles, and censored survival time data.     

Calculation of cubic smoothing splines for equally spaced data

Summary A method is described for fitting cubic smoothing splines to samples of equally spaced data. The method is based on the canonical decomposition of the linear transformation from the data to the fitted values. Techniques for estimating the required amount of smoothing, including generalized cross validation, may easily be integrated into the calculations. For large samples the method is fast and does not require prohibitively large data storage.     

On use of the Kalman filter for spatial smoothing

A state-space model to perform discrete thin plate smoothing for data on a two-dimensional rectangular lattice is proposed with the use of the Kalman filter. The use of the Kalman filter reduces computational difficulties in the maximum likelihood estimation of a smoothing parameter. A procedure to reduce computational difficulties in the estimation of trend is given also. Numerical illustration is provided using two sets of artificial data.     

Intuitionistic theory of functionals of a high type

Translated from Matematicheskie Zametki, Vol. 45, No. 3, pp. 66–79, March, 1989.     

Approximation by smoothing variational vector splines for noisy data

This paper addresses the problem of constructing some free-form curves and surfaces from given to different types of data: exact and noisy data. We extend the theory of D^m${\mathrm{D}}^m$-splines over a bounded domain for noisy data to the smoothing variational vector splines. Both results of convergence for respectively the exact and noisy data are established, as soon as some estimations of errors are given.     

Besov regularization,thresholding and wavelets for smoothing data

Smoothing of data is a problem very important for many applications ranging from 1-D signals (e.g., speech) to 2-D and 3-D signals (e.g., images). Many methods exist in the literature for facing the problem; in the present paper we point our attention on regularization. We shall treat regularization methods in a general framework which is well suited for wavelet analysis; in particular we shall investigate on the relation existing between thresholding methods and regularization. We shall also introduce a new regularization method (Besov regularization), which includes some known regularization and thresholding methods as particular cases. Numerical experiments based on some test problems will be performed in order to compare the performance of some methods of smoothing data.

AMS (MOS) Subject Classifications: 65R30, 41A60.     

Sensitivity functionals in contact problems of elasticity theory

The sensitivity functional constructed for the variational elasticity problem with given friction is proved to be lower semicontinuous. An analysis based on this property is conducted for a duality scheme with the modified Lagrangian functional.