Topic

# Asymptotology

About: Asymptotology is a(n) research topic. Over the lifetime, 1319 publication(s) have been published within this topic receiving 35831 citation(s).

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01 Jan 1961

TL;DR: In this article, a wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering.

Abstract: : This book is devoted to the approximate asymptotic methods of solving the problems in the theory of nonlinear oscillations met in many fields of physics and engineering. It is intended for the wide circle of engineering-technical and scientific workers who are concerned with oscillatory processes. Contents include the following: Natural oscillations in quasi-linear systems; The method of the phase plane; The influence of external periodic forces; The method of the mean; Justification of the asymptotic methods.

2,259 citations

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TL;DR: In this paper, the authors analyze the recent development of the theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i).

Abstract: Abstract. Let X j denote the life length of the j th component of a machine. In reliability theory, one is interested in the life length Z n of the machine where n signifies its number of components. Evidently, Z n = min (X j : 1 ≤ j ≤ n). Another important problem, which is extensively discussed in the literature, is the service time W n of a machine with n components. If Y j is the time period required for servicing the j th component, then W n = max (Y j : 1 ≤ j ≤ n). In the early investigations, it was usually assumed that the X's or Y's are stochastically independent and identically distributed random variables. If n is large, then asymptotic theory is used for describing Z n or W n . Classical theory thus gives that the (asymptotic) distribution of these extremes (Z n or W n ) is of Weibull type. While the independence assumptions are practically never satisfied, data usually fits well the assumed Weibull distribution. This contradictory situation leads to the following mathematical problems: (i) What type of dependence property of the X's (or the Y's) will result in a Weibull distribution as the asymptotic law of Z n (or W n )? (ii) given the dependence structure of the X's (or Y's), what type of new asymptotic laws can be obtained for Z n (or W n )? The aim of the present paper is to analyze the recent development of the (mathematical) theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i). In regard to (ii), the following result holds: the class of limit laws of extremes for exchangeable variables is identical to the class of limit laws of extremes for arbitrary random variables. One can therefore limit attention to exchangeable variables. The basic references to this paper are the author's recent papers in Duke Math. J. 40 (1973), 581–586, J. Appl. Probability 10 (1973, 122–129 and 11 (1974), 219–222 and Zeitschrift fur Wahrscheinlichkeitstheorie 32 (1975), 197–207. For multivariate extensions see H. A. David and the author, J. Appl. Probability 11 (1974), 762–770 and the author's paper in J. Amer. Statist. Assoc. 70 (1975), 674–680. Finally, we shall point out the difficulty of distinguishing between several distributions based on data. Hence, only a combination of theoretical results and experimentations can be used as conclusive evidence on the laws governing the behavior of extremes.

1,943 citations

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01 Jan 1984

TL;DR: The Linear Model and Instrumental Variables Estimators as mentioned in this paper have been used to estimate Asymptotic Covariance Matrices, and Central Limit Theory has been applied to this problem.

Abstract: The Linear Model and Instrumental Variables Estimators. Consistency. Laws of Large Numbers. Asymptotic Normality. Central Limit Theory. Estimating Asymptotic Covariance Matrices. Functional Central Limit Theory and Applications. Directions for Further Study. Solution Set. References. Index.

1,709 citations

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29 Sep 2014

TL;DR: The basic concepts of asymptotic expansions, Mellin transform techniques, and the distributional approach are explained.

Abstract: Preface 1. Fundamental concepts of asymptotics 2. Classical procedures 3. Mellin transform techniques 4. The summability method 5. Elementary theory of distributions 6. The distributional approach 7. Uniform asymptotic expansions 8. Double integrals 9. Higher dimensional integrals Bibliography Symbol Index Author index Subject index.

1,008 citations