Topic
Asymptotology
About: Asymptotology is a research topic. Over the lifetime, 1319 publications have been published within this topic receiving 35831 citations.
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01 Jan 2005
TL;DR: In this article, a software package called RobASt is developed by means of the statistics software R. This software package is based on the asymptotic theory of robustness.
Abstract: In the framework of this dissertation a software package – the R bundle RobASt – by means of the statistics software R has been developed. It includes all robust procedures introduced throughout the thesis. The dissertation itself consists of five parts and starts with a brief motivation, which makes precise why robust statistics is necessary. After that a detailed summary in German and English is given. Part I provides a description of the asymptotic theory of robustness (Chapter 1) which forms the basis of this thesis. It is based on Chapters 4 and 5 of Rieder (1994). Chapter 2 provides supplements to the asymptotic theory of robustness which have proved necessary for this thesis. More precisely, it contains results about: properties of the optimally robust influence curves (ICs), how one should proceed in an optimal way if the neighborhood radius is unknown – as mostly in practice, and the construction of estimates by means of the one-step method. At the end of Chapter 2 convergence of robust models is introduced which is related to the concept of convergence of experiments of Le Cam. Part II deals with optimally robust estimators for some non-standard models in robust statistics. These models are covered by the R package ROptEst which makes use of S4 classes and methods and is part of the R bundle RobASt. More precisely, the binomial (Chapter 3) and Poisson (Chapter 4) model, the exponential scale and Gumbel location model (Chapter 5) as well as the Gamma model (Chapter 6) are investigated. In particular, the binomial and Poisson model are used to study convergence of robust models. Using exponential scale and Gumbel location one can show that there is a connection between certain scale and location models via a log-transformation which also holds for the corresponding optimally robust ICs. Finally, the Gamma model is used to demonstrate how differentiable parameter transformations can be estimated in an optimally robust way. In Part III robust regression with random regressor and unknown error scale (Chapter 7) is treated where it is distinguished between simultaneous and separate estimation. In both cases the optimally robust estimators as well as robust estimators for several narrower classes of M estimators are considered. All these estimators are implemented in the R packages ROptRegTS and RobRex which are part of the R bundle RobASt. Numerical comparisons for several regressor distributions show that the various suboptimal M estimators may have very small but also huge efficiency losses. A further comparison of these and several other well-known robust estimators in case of normal location and scale is made in Chapter 8. These location and scale estimators are implemented in the R package RobLox which is part of the R bundle RobASt. In Part IV (Chapter 9) robust adaptivity in terms of two asymptotic MSE problems is defined. Hence, adaptivity is no longer only a dichotomous criterion but can be evaluated quantitatively in terms of efficiency loss. The various regression and time series models considered…
34 citations
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01 Apr 1954TL;DR: In this article, the authors obtained an asymptotic formula for the number of partitions of a large positive integer n into m parts λr, where the number m becomes large with n and the numbers λ1, λ2,… form a sequence of positive integers.
Abstract: It is the object of this paper to obtain an asymptotic formula for the number of partitions pm(n) of a large positive integer n into m parts λr, where the number m becomes large with n and the numbers λ1, λ2,… form a sequence of positive integers. The formula is proved by using the classical method of contour integration due to Hardy, Ramanujan and Littlewood. It will be necessary to assume certain conditions on the sequence λr, but these conditions are satisfied in most of the cases of interest. In particular, we shall be able to prove the asymptotic formula in the cases of partitions into positive integers, primes and kth powers for any positive integer k.
34 citations
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TL;DR: The good cut equation for a specific asymptotic shear was solved in this article, and the metric of the associated H•space was found to be type N, asmptotically flat and positive frequency.
Abstract: The good cut equation for a specific asymptotic shear is solved and the metric of the associated H‐space is found to be type N, asymptotically flat and positive frequency.
33 citations
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33 citations
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TL;DR: In this article, two methods for computing the coefficients of the asymptotic series near the transition point are discussed, and auxiliary functions that can be computed more efficiently than the coefficients in the first method, and do not need the tabulation of many coefficients.
Abstract: Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we do not need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.
32 citations