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Asymptotic methods in analysis

01 Jan 1958-
About: The article was published on 1958-01-01 and is currently open access. It has received 1315 citations till now. The article focuses on the topics: Asymptotic analysis & Asymptotology.
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Journal ArticleDOI
TL;DR: A new discussion of the complex branches of W, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing W are presented.
Abstract: The LambertW function is defined to be the multivalued inverse of the functionw →we w . It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches ofW, an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containingW.

5,591 citations


Cites background from "Asymptotic methods in analysis"

  • ...Now return to the special values of cr and 7- as functions of z. For z sufficiently large, we have 11 ])....

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  • ...Log in (4.11) (which all differ by multiples of 27ri) there is exactly one number v E [([ 11 ]....

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  • ...This display of the expansion corrects a typographical error in equation (2.4.4) in [ 11 ]....

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Journal ArticleDOI
TL;DR: These approximations to the posterior means and variances of positive functions of a real or vector-valued parameter, and to the marginal posterior densities of arbitrary parameters can also be used to compute approximate predictive densities.
Abstract: This article describes approximations to the posterior means and variances of positive functions of a real or vector-valued parameter, and to the marginal posterior densities of arbitrary (ie, not necessarily positive) parameters These approximations can also be used to compute approximate predictive densities To apply the proposed method, one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima Nevertheless, the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third-order expansions of the likelihood and requiring the evaluation of third derivatives The approximate marginal posterior densities behave very much like saddle-point approximations for sampling distributions The principal regularity condition required is that the likelihood times prior be unimodal

2,081 citations

Book
04 Dec 1998
TL;DR: The most useful parts of large-sample theory are accessible to scientists outside statistics and certainly to master's-level statistics students who ignore most of measure theory as discussed by the authors, which constitutes a coherent body of concepts and results that are central to both theoretical and applied statistics.
Abstract: This introductory book on the most useful parts of large-sample theory is designed to be accessible to scientists outside statistics and certainly to master’s-level statistics students who ignore most of measure theory. According to the author, “the subject of this book, first-order large- sample theory, constitutes a coherent body of concepts and results that are central to both theoretical and applied statistics.” All of the other existing books published on the subject over the last 20 years, from Ibragimov and Has’minskii in 1979 to the most recent by Van der Waart in 1998 have a common prerequisite in mathematical sophistication (measure theory in particular) that do not make the concepts available to a wide audience.

1,182 citations


Cites background from "Asymptotic methods in analysis"

  • ...For some general discussion of asymptotics, see, for example, DeBruijn (1958)....

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Journal ArticleDOI
TL;DR: This work presents a class of methods by which one can translate, on a term-by-term basis, an asymptotic expansion of a function around a dominant singularity into a corresponding asymPTotic expansion for the Taylor coefficients of the function.
Abstract: This work presents a class of methods by which one can translate, on a term-by-term basis, an asymptotic expansion of a function around a dominant singularity into a corresponding asymptotic expans...

922 citations

Journal ArticleDOI
TL;DR: This extended abstract describes and analyses a near-optimal probabilistic algorithm, HYPERLOGLOG, dedicated to estimating the number of \emphdistinct elements (the cardinality) of very large data ensembles, and makes it possible to estimate cardinalities well beyond $10^9$ with a typical accuracy of 2% while using a memory of only 1.5 kilobytes.
Abstract: This extended abstract describes and analyses a near-optimal probabilistic algorithm, HYPERLOGLOG, dedicated to estimating the number of \emphdistinct elements (the cardinality) of very large data ensembles. Using an auxiliary memory of m units (typically, "short bytes''), HYPERLOGLOG performs a single pass over the data and produces an estimate of the cardinality such that the relative accuracy (the standard error) is typically about $1.04/\sqrt{m}$. This improves on the best previously known cardinality estimator, LOGLOG, whose accuracy can be matched by consuming only 64% of the original memory. For instance, the new algorithm makes it possible to estimate cardinalities well beyond $10^9$ with a typical accuracy of 2% while using a memory of only 1.5 kilobytes. The algorithm parallelizes optimally and adapts to the sliding window model.

694 citations