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19
Consistent Specification Testing With Nuisance Parameters Present Only Under The Alternative
, 1995
"... . The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly ex ..."
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Cited by 55 (10 self)
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. The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly extend the nuisance parameter approach. How and why the nuisance parameter approach works and how it can be extended bears closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banachvalued random elements and applying the Banach Central Limit Theorem and Law of Iterated Logarithm, leading to simple procedures that can be used as a guide to when computationally more elaborate procedures may be warranted. 1. Introduction In testing whether or not a parametric statistical model is correctly specified, there are a number of apparently distinct approaches one might take. T...
Banach spaces with the Daugavet property
 Trans. Amer. Math. Soc
"... Abstract. A Banach space X is said to have the Daugavet property if every operator T: X → X of rank 1 satisfies ‖Id+T ‖ = 1+‖T ‖. We show that then every weakly compact operator satisfies this equation as well and that X contains a copy of ℓ1. However, X need not contain a copy of L1. We also study ..."
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Cited by 22 (12 self)
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Abstract. A Banach space X is said to have the Daugavet property if every operator T: X → X of rank 1 satisfies ‖Id+T ‖ = 1+‖T ‖. We show that then every weakly compact operator satisfies this equation as well and that X contains a copy of ℓ1. However, X need not contain a copy of L1. We also study pairs of spaces X ⊂ Y and operators T: X → Y satisfying ‖J + T ‖ = 1 + ‖T ‖, where J: X → Y is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with ‖Id + T ‖ = 1 + ‖T ‖ is as small as possible and give characterisations in terms of a smoothness condition. 1.
Control Theory for a Class of 2D ContinuousDiscrete Linear Systems
"... This paper considers a general class of 2D continuousdiscrete linear systems of both systems theoretic and applications interest. The focus is on the development of a comprehensive control systems theory for members of this class in a unified manner based on analysis in an appropriate algebraic and ..."
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Cited by 3 (0 self)
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This paper considers a general class of 2D continuousdiscrete linear systems of both systems theoretic and applications interest. The focus is on the development of a comprehensive control systems theory for members of this class in a unified manner based on analysis in an appropriate algebraic and operator setting. In particular, important new results are developed on stability, controllability, stabilization, and optimal control. 1
Preference for equivalent random variables: A price for unbounded utilities.*
, 2008
"... When realvalued utilities for outcomes are bounded, or when all variables are simple, it is consistent with expected utility to have preferences defined over probability distributions or lotteries. That is, under such circumstances two variables with a common probability distribution over outcomes ..."
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Cited by 2 (1 self)
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When realvalued utilities for outcomes are bounded, or when all variables are simple, it is consistent with expected utility to have preferences defined over probability distributions or lotteries. That is, under such circumstances two variables with a common probability distribution over outcomes – equivalent variables – occupy the same place in a preference ordering. However, if strict preference respects uniform, strict dominance in outcomes between variables, and if indifference between two variables entails indifference between their difference and the status quo, then preferences over rich sets of unbounded variables, such as variables used in the St. Petersburg paradox, cannot preserve indifference between all pairs of equivalent variables. In such circumstances, preference is not a function only of probability and utility for outcomes. Then the preference ordering is not defined in terms of lotteries.
Tentropy and Variational principle for the spectral radius of weighted shift operators
, 2008
"... The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation ..."
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Cited by 2 (1 self)
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The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or it is a topological Markov chain. As the main summands these principles contain the integrals over invariant measures and the Kolmogorov–Sinai entropy. In the article we derive the Variational Principle for an arbitrary dynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov– Sinai entropy but a new invariant of entropy type — the tentropy.
A Comparison of Entropies in the Underdetermined Moment Problem
, 1993
"... The problem of function reconstruction from a small number of accurate moments is reviewed and studied numerically. Since this problem is underdetermined, a number of selections have been proposed to pick out a single function. We consider those selections of the "entropic" variety, i.e. those of th ..."
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Cited by 1 (1 self)
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The problem of function reconstruction from a small number of accurate moments is reviewed and studied numerically. Since this problem is underdetermined, a number of selections have been proposed to pick out a single function. We consider those selections of the "entropic" variety, i.e. those of the form f(x) = Z OE(x(t))dt for some convex function OE. We present a summary of our attempts to determine which "entropy" is best for reconstructing x in a variety of model problems. This presumes we have a measure of "best", which in itself is a difficult issue. We include a review of the pertinent theory and the details of our numerical implementation. We give numerous numerical examples to support our conclusions. 1. Introduction The problem of reconstructing a function from a finite set of data is very common in applied problems. Usually the finite data is not sufficient to uniquely determine the underlying function and some sort of "best" function must be chosen from the collection ...
PerronFrobenius spectrum for random maps and its approximation
, 2001
"... To study the convergence to equilibrium in random maps we developed the spectral theory of the corresponding transfer (PerronFrobenius) operators acting in a certain Banach space of generalized functions. The random maps under study in a sense ll the gap between expanding and hyperbolic systems si ..."
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Cited by 1 (1 self)
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To study the convergence to equilibrium in random maps we developed the spectral theory of the corresponding transfer (PerronFrobenius) operators acting in a certain Banach space of generalized functions. The random maps under study in a sense ll the gap between expanding and hyperbolic systems since among their (deterministic) components there are both expanding and contracting ones. We prove stochastic stability of the PerronFrobenius spectrum and developed its nite rank operator approximations by means of a \stochastically smoothed" Ulam approximation scheme. A counterexample to the original Ulam conjecture about the approximation of the SBR measure and the discussion of the instability of spectral approximations by means of the original Ulam scheme are presented as well.
Dynamical Systems Method (DSM) for solving nonlinear operator equations in Banach spaces
"... Let F (u) = h be an operator equation in a Banach space X with Gateaux differentiable norm, ‖F ′ (u) − F ′ (v) ‖ ≤ ω(‖u − v‖), where ω ∈ C([0, ∞)), ω(0) = 0, ω(r) is strictly growing on [0, ∞). Denote A(u):= F ′ (u), where F ′ (u) is the Fréchet derivative of F, and Aa: = A + aI. Assume that (*) ..."
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Cited by 1 (0 self)
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Let F (u) = h be an operator equation in a Banach space X with Gateaux differentiable norm, ‖F ′ (u) − F ′ (v) ‖ ≤ ω(‖u − v‖), where ω ∈ C([0, ∞)), ω(0) = 0, ω(r) is strictly growing on [0, ∞). Denote A(u):= F ′ (u), where F ′ (u) is the Fréchet derivative of F, and Aa: = A + aI. Assume that (*) ‖A −1 a (u) ‖ ≤ c1 a  b, a > 0, b> 0, a ∈ L. Here a may be a complex number, and L is a smooth path on the complex aplane, joining the origin and some point on the complex a−plane, 0 < a  < ɛ0, where ɛ0> 0 is a small fixed number, such that for any a ∈ L estimate (*) holds. It is proved that the DSM (Dynamical Systems Method) ˙u(t) = −A −1 a(t) (u(t))[F (u(t)) + a(t)u(t) − f], du u(0) = u0, ˙u = dt, converges to y as t → +∞, where a(t) ∈ L, F (y) = f, r(t): = a(t), and r(t) = c4(t + c2) −c3, where cj> 0 are some suitably chosen constants, j = 2, 3, 4. Existence of a solution y to the equation F (u) = f is assumed. It is also assumed that the equation F (wa) + awa − f = 0 is uniquely solvable for any f ∈ X, a ∈ L, and lima→0,a∈L ‖wa − y ‖ = 0.
NonLocal Dispersal
"... Equations with nonlocal dispersal have been used extensively as models in material science, ecology and neurology. We discuss the scalar bistable case u t (x) = ae fi(x; y)u(y)dy \Gamma u(x) + f(u) and contrast it with the corresponding reactiondiffusion equation. We show that for large ..."
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Equations with nonlocal dispersal have been used extensively as models in material science, ecology and neurology. We discuss the scalar bistable case u t (x) = ae fi(x; y)u(y)dy \Gamma u(x) + f(u) and contrast it with the corresponding reactiondiffusion equation. We show that for large dispersal rate ae the asymptotic dynamics is determined by the ODE u = f(u).