Get Asymptotics in Statistics: Some Basic Concepts PDF

By Lucien Le Cam, Grace Lo Yang

ISBN-10: 146840377X

ISBN-13: 9781468403770

ISBN-10: 1468403796

ISBN-13: 9781468403794

The publication grew out of lectures given over a interval of approximately 30 to 35 years on Asymptotic equipment in sta- tistics. most present texts, other than the monographs through Le Cam (Springer-Verlag 1986) and Strasser (1985) emphasize a concept in line with greatest probability estimates whereas this article emphasizes approximation by means of Gaussian households of measures, in addition to quadratic expansions of log chance. The ebook provides in a quick shape a number of the major effects obtained long ago 20 years within the box of asymptotic statistical inference. The equipment can be utilized very broadly. the elemental theorems are offered at a degree that are meant to no longer disturb a starting graduate scholar. The authors have tried a unified method, in an easy atmosphere, to tips on how to be chanced on purely in papers or really good books.

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Extra info for Asymptotics in Statistics: Some Basic Concepts

Example text

These authors give what appears to be definite bounds for such ap- 4. Likelihood Ratios from Independent Observations 51 proximations. Another version, with much weaker bounds, appears in Le Cam [1986], Chapters 15 and 16. The final item of Chapter 4, about preservation of asymptotic normality under information loss, is taken from a paper of Le Cam and Yang [1988]. As explained there, the result has numerous applications. 1 Introduction The classical theory of asymptotics in Statistics relies heavily on certain local quadratic approximations to the logarithms of likelihood ratios.

1. Introduction Sections 7 and 9 are in the nature of discussions about what conditions should be used if one wants to obtain more general results. They are included here to make the reader aware of the fact that much of the theory has already been extended to cases that are not restricted to the open subsets of R k or to the use of single numbers 8n that tend to zero. 2 Locally asymptotically quadratic families For each integer n let {PO,n : 0 E 8} be a family of probability measures on some space (Xn, An).

The final statement is also important in that it will allow us to state limit theorems that do not involve at all the truncated expectations that make the standard results (see for instance Loeve [1977] or Gnedenko and Kolmogorov [1954]) difficult to read. As an application of Lemma 1 consider the following situation. Let PD[0,2] be the class of measurable functions 1 defined on [-1,00) and taking values in [-00, +00) that are such that 1(0) = 0 and such that there are two numbers 1'(0) and 1"(0) such that ~j/(x) x2 1(0) - xl'(O) - x 2 1"(0)j 2 tends to zero as x ---* O.

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Asymptotics in Statistics: Some Basic Concepts by Lucien Le Cam, Grace Lo Yang


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