Best rotated minimax approximation

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

Чебишовське наближення за неповною системою степеневих функцій

Розглянуто задачу чебишовського (рiвномiрного, мiнiмаксного) наближення функцiй полiномом i рацiональним виразом за неповною системою степеневих функцiй. Встановлено необхiднi й достатнi умови iснування такої апроксимацiї. Одержано характеристичнi властивостi чебишовської апроксимацiї функцiй полiномом i рацiональним виразом за неповною системою базисних функцiй iз найменшою абсолютною й вiдносною похибкою. Запропоновано алгоритми для визначення параметрiв таких наближень.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

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

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