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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 83 (13 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 30 (13 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.
Iterated hard shrinkage for minimization problems with sparsity constraints
 SIAM Journal on Scientific Computing
, 2006
"... Abstract. A new iterative algorithm for the solution of minimization problems in infinitedimensional Hilbert spaces which involve sparsity constraints in form of ℓ ppenalties is proposed. In contrast to the wellknown algorithm considered by Daubechies, Defrise and De Mol, it uses hard instead of s ..."
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Cited by 27 (15 self)
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Abstract. A new iterative algorithm for the solution of minimization problems in infinitedimensional Hilbert spaces which involve sparsity constraints in form of ℓ ppenalties is proposed. In contrast to the wellknown algorithm considered by Daubechies, Defrise and De Mol, it uses hard instead of soft shrinkage. It is shown that the hard shrinkage algorithm is a special case of the generalized conditional gradient method. Convergence properties of the generalized conditional gradient method with quadratic discrepancy term are analyzed. This leads to strong convergence of the iterates with convergence rates O(n −1/2) and O(λ n) for p = 1 and 1 < p ≤ 2 respectively. Numerical experiments on image deblurring, backwards heat conduction, and inverse integration are given. Key words. sparsity constraints, iterated hard shrinkage, generalized conditional gradient method, convergence analysis AMS subject classifications. 46N10, 49M05, 65K10 1. Introduction. This
On the ergodicity and mixing of maxstable processes
, 2007
"... Maxstable processes arise in the limit of componentwise maxima of independent processes, under appropriate centering and normalization. In this paper, we establish necessary and sufficient conditions for ergodicity and mixing of stationary maxstable processes. We do so in terms of their spectral rep ..."
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Cited by 13 (6 self)
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Maxstable processes arise in the limit of componentwise maxima of independent processes, under appropriate centering and normalization. In this paper, we establish necessary and sufficient conditions for ergodicity and mixing of stationary maxstable processes. We do so in terms of their spectral representations by using extremal integrals. The large classes of moving maxima and mixed moving maxima processes are shown to be mixing. Other examples of ergodic doubly stochastic processes and nonergodic processes are also given. The ergodicity conditions involve a certain measure of dependence. We relate this measure of dependence to the one of Weintraub (1991) and show that Weintraub's notion of '0mixing' is equivalent to mixing. Consistent estimators for the dependence function of an ergodic maxstable process are introduced and illustrated over simulated data.
Isometric factorization of weakly compact operators and the approximation property
 Israel J. Math
"... Abstract. Using an isometric version of the Davis, Figiel, Johnson, and Pe%lczyński factorization of weakly compact operators, we prove that a Banach space X has the approximation property if and only if, for every Banach space Y, the finite rank operators of norm ≤ 1 are dense in the unit ball of ..."
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Cited by 7 (3 self)
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Abstract. Using an isometric version of the Davis, Figiel, Johnson, and Pe%lczyński factorization of weakly compact operators, we prove that a Banach space X has the approximation property if and only if, for every Banach space Y, the finite rank operators of norm ≤ 1 are dense in the unit ball of W(Y,X), the space of weakly compact operators from Y to X, in the strong operator topology. We also show that, for every finite dimensional subspace F of W(Y,X), there are a reflexive space Z, a norm one operator J: Y → Z, and an isometry Φ: F →W(Z,X) which preserves finite rank and compact operators so that T = Φ(T) ◦ J for all T ∈ F. This enables us to prove that X has the approximation property if and only if the finite rank operators form an ideal in W(Y,X) for all Banach spaces Y.
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 4 (3 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.
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 4 (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.
Wellposedness of a parabolic movingboundary problem in the setting of Wasserstein Gradient Flows
, 2010
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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
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.