62,120 research outputs found

    Multiariate Wavelet-based sahpe preserving estimation for dependant observation

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    We present a new approach on shape preserving estimation of probability distribution and density functions using wavelet methodology for multivariate dependent data. Our estimators preserve shape constraints such as monotonicity, positivity and integration to one, and allow for low spatial regularity of the underlying functions. As important application, we discuss conditional quantile estimation for financial time series data. We show that our methodology can be easily implemented with B-splines, and performs well in a finite sample situation, through Monte Carlo simulations.Conditional quantile; time series; shape preserving wavelet estimation; B-splines; multivariate process

    Rearrangement inequalities for functionals with monotone integrands

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    The rearrangement inequalities of Hardy-Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands of the form F(u_1,..., u_m) where F is supermodular; in particular, they hold when F has nonnegative mixed second derivatives. This paper concerns the regularity assumptions on F and the equality cases. It is shown here that extended Hardy-Littlewood and Riesz inequalities are valid for supermodular integrands that are just Borel measurable. Under some nondegeneracy conditions, all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant (i.e., measure-preserving maps for the Hardy-Littlewood inequality, translations for the Riesz inequality). The proofs rely on monotone changes of variables in the spirit of Sklar's theorem.Comment: 20 pages. Slightly re-organized, four added reference

    Regularity Preserving but not Reflecting Encodings

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    Encodings, that is, injective functions from words to words, have been studied extensively in several settings. In computability theory the notion of encoding is crucial for defining computability on arbitrary domains, as well as for comparing the power of models of computation. In language theory much attention has been devoted to regularity preserving functions. A natural question arising in these contexts is: Is there a bijective encoding such that its image function preserves regularity of languages, but its pre-image function does not? Our main result answers this question in the affirmative: For every countable class C of languages there exists a bijective encoding f such that for every language L in C its image f[L] is regular. Our construction of such encodings has several noteworthy consequences. Firstly, anomalies arise when models of computation are compared with respect to a known concept of implementation that is based on encodings which are not required to be computable: Every countable decision model can be implemented, in this sense, by finite-state automata, even via bijective encodings. Hence deterministic finite-state automata would be equally powerful as Turing machine deciders. A second consequence concerns the recognizability of sets of natural numbers via number representations and finite automata. A set of numbers is said to be recognizable with respect to a representation if an automaton accepts the language of representations. Our result entails that there is one number representation with respect to which every recursive set is recognizable

    Függvényegyenletek, függvényalgebrák = Functional equations, function algebras

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    A támogatott kutatás keretében a függvényegyenletek és a függvényalgebrák területén stabilitási, regularitás-elméleti, konvexitási, reflexivitási, lineáris megőrzési, a Wigner-féle unitér-antiunitér tétellel kapcsolatos, valamint effekt-algebrákra vonatkozó problémákkal foglalkoztunk. Stabilitási vizsgálataink során általános struktúrán értelmezett függvényekre felírt függvényegyenletekre vonatkozó stabilitási tételeket, feltételes és aszimptotikus stabilitási eredményeket igazoltunk, valamint függvény-egyenlőtlenségek stabilitását bizonyítottuk. Új regularitás-elméleti eredmények segítségével egy, a haszonelméletből származó egyértelműségi problémát oldottunk meg. A konvex függvények elméletének egyik klasszikus eredménye, a Bernstein--Doetsch tétel megfelelőit igazoltuk általános (M,N)-konvex valamint kvázi-konvex függvényekre. A korábban kizárólag egyváltozós függvényekre vizsgált Beckenbach-konvexitást kiterjesztettük kétváltozós függvényekre, utat nyitva ezzel további vizsgálatok felé. Általánosított deriváltak segítségével karakterizáltuk magasabb rendben konvex függvények egy általános osztályát. Wigner Jenő híres unitér-antiunitér tételére új, elemi bizonyítást adtunk, s sikerült kiterjesztenünk a tételt. A lineáris megőrzési problémák terén több új eredményt igazoltunk. Meghatároztuk a fizikában jelentős szerepet játszó effekt-algebrák függvényalgebrás megfelelőinek rendezéstartó bijekcióit. | On the fields of functional equations and function algebras, we investigated stability, regularity, convexity and reflexivity problems, linear preserving maps, effect-algebras, and questions connected to Wigner's theorem. Related to the stability theory, we proved stability theorems for functional equations defined on and mapping into general structures. We obtained conditional asymptotic stability results and we studied the stability of certain functional inequalities. Proving new results in the regularity theory of composite functions, we solved a uniqueness problem arising from a comparison of utility representations. Investigating convexity properties of functions, we generalized the classical Bernstein-Doetsch theorem for (M,N)-convex as well as for quasi-convex functions. We extended the concept of Beckenbach-convexity of real functions to two-variable functions. We characterized a class of convex functions of higher order in terms of generalized derivatives. By the help of a new approach, we gave an elementary proof for Wigner's well-known theorem and we extended the classical theorem, too. We proved several new linear preserving theorems. We determined the order preserving bijections of function algebras related to effect-algebras having some important applications in physics

    Shape preserving C2C^2 interpolatory subdivision schemes

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    Stationary interpolatory subdivision schemes which preserve shape properties such as convexity or monotonicity are constructed. The schemes are rational in the data and generate limit functions that are at least C2C^2. The emphasis is on a class of six-point convexity preserving subdivision schemes that generate C2C^2 limit functions. In addition, a class of six-point monotonicity preserving schemes that also leads to C2C^2 limit functions is introduced. As the algebra is far too complicated for an analytical proof of smoothness, validation has been performed by a simple numerical methodology
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