13,133 research outputs found

    Best rotated minimax approximation

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    Thesis submitted 1970; degree awarded 1971.In this dissertation we consider the minimax approximation of functions f(x) E"C[O, l] rotated about the origin, and the characterization of the optimal rotation, a*, of f in the sense of least minimax error over all possible rotations. The paper divides naturally into two sections: a) Existence, uniqueness, and characterization for unisolvent minimax approximation for each rotation a of f. These results are applications of Dunham (1967). b) Existence, non-uniqueness, and com.putation of a*; derivation of necessary conditions for the minimax [TRUNCATED

    ЧСбишовськС наблиТСння Π·Π° нСповною ΡΠΈΡΡ‚Π΅ΠΌΠΎΡŽ стСпСнСвих Ρ„ΡƒΠ½ΠΊΡ†Ρ–ΠΉ

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    Розглянуто Π·Π°Π΄Π°Ρ‡Ρƒ Ρ‡Π΅Π±ΠΈΡˆΠΎΠ²ΡΡŒΠΊΠΎΠ³ΠΎ (Ρ€iΠ²Π½ΠΎΠΌiΡ€Π½ΠΎΠ³ΠΎ, ΠΌiΠ½iмаксного) наблиТСння Ρ„ΡƒΠ½ΠΊΡ†iΠΉ ΠΏΠΎΠ»iΠ½ΠΎΠΌΠΎΠΌ i Ρ€Π°Ρ†iональним Π²ΠΈΡ€Π°Π·ΠΎΠΌ Π·Π° нСповною ΡΠΈΡΡ‚Π΅ΠΌΠΎΡŽ стСпСнСвих Ρ„ΡƒΠ½ΠΊΡ†iΠΉ. ВстановлСно Π½Π΅ΠΎΠ±Ρ…iΠ΄Π½i ΠΉ достатнi ΡƒΠΌΠΎΠ²ΠΈ iснування Ρ‚Π°ΠΊΠΎΡ— апроксимацiΡ—. ΠžΠ΄Π΅Ρ€ΠΆΠ°Π½ΠΎ характСристичнi властивостi Ρ‡Π΅Π±ΠΈΡˆΠΎΠ²ΡΡŒΠΊΠΎΡ— апроксимацiΡ— Ρ„ΡƒΠ½ΠΊΡ†iΠΉ ΠΏΠΎΠ»iΠ½ΠΎΠΌΠΎΠΌ i Ρ€Π°Ρ†iональним Π²ΠΈΡ€Π°Π·ΠΎΠΌ Π·Π° нСповною ΡΠΈΡΡ‚Π΅ΠΌΠΎΡŽ базисних Ρ„ΡƒΠ½ΠΊΡ†iΠΉ iΠ· наймСншою Π°Π±ΡΠΎΠ»ΡŽΡ‚Π½ΠΎΡŽ ΠΉ Π²iдносною ΠΏΠΎΡ…ΠΈΠ±ΠΊΠΎΡŽ. Π—Π°ΠΏΡ€ΠΎΠΏΠΎΠ½ΠΎΠ²Π°Π½ΠΎ Π°Π»Π³ΠΎΡ€ΠΈΡ‚ΠΌΠΈ для визначСння ΠΏΠ°Ρ€Π°ΠΌΠ΅Ρ‚Ρ€iΠ² Ρ‚Π°ΠΊΠΈΡ… наблиТСнь.The problem of the Chebyshevian (uniform, minimax) approximation to a given function by a polynomial and a rational expression based on an incomplete system of basic power functions is considered. Both necessary and sufficient conditions of existence for such an approximation are established. The alternance property of polynomial and rational Chebyshevian approximations based on the aforementioned system of functions for both absolute and relative minimal errors are discussed. The algorithm for calculating the parameters of such an approximation is proposed

    On rate optimality for ill-posed inverse problems in econometrics

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    In this paper, we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildly ill-posed and severely ill-posed cases. We show that both a simple projection estimator for the NPIR model, and a sieve minimum distance estimator for the NPIV model, can achieve the minimax risk lower bounds, and are rate-optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.Comment: 27 page

    An application of a linear programing technique to nonlinear minimax problems

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    A differential correction technique for solving nonlinear minimax problems is presented. The basis of the technique is a linear programing algorithm which solves the linear minimax problem. By linearizing the original nonlinear equations about a nominal solution, both nonlinear approximation and estimation problems using the minimax norm may be solved iteratively. Some consideration is also given to improving convergence and to the treatment of problems with more than one measured quantity. A sample problem is treated with this technique and with the least-squares differential correction method to illustrate the properties of the minimax solution. The results indicate that for the sample approximation problem, the minimax technique provides better estimates than the least-squares method if a sufficient amount of data is used. For the sample estimation problem, the minimax estimates are better if the mathematical model is incomplete
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