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Statistical Estimation

Asymptotic Theory

  • I. A. Ibragimov
  • R. Z. Has’minskii

Part of the Applications of Mathematics book series (SMAP, volume 16)

Table of contents

  1. Front Matter
    Pages i-vii
  2. I. A. Ibragimov, R. Z. Has’minskii
    Pages 1-2
  3. I. A. Ibragimov, R. Z. Has’minskii
    Pages 3-9
  4. I. A. Ibragimov, R. Z. Has’minskii
    Pages 10-112
  5. I. A. Ibragimov, R. Z. Has’minskii
    Pages 113-172
  6. I. A. Ibragimov, R. Z. Has’minskii
    Pages 173-213
  7. I. A. Ibragimov, R. Z. Has’minskii
    Pages 214-240
  8. I. A. Ibragimov, R. Z. Has’minskii
    Pages 241-280
  9. I. A. Ibragimov, R. Z. Has’minskii
    Pages 321-361
  10. Back Matter
    Pages 363-403

About this book

Introduction

when certain parameters in the problem tend to limiting values (for example, when the sample size increases indefinitely, the intensity of the noise ap­ proaches zero, etc.) To address the problem of asymptotically optimal estimators consider the following important case. Let X 1, X 2, ... , X n be independent observations with the joint probability density !(x,O) (with respect to the Lebesgue measure on the real line) which depends on the unknown patameter o e 9 c R1. It is required to derive the best (asymptotically) estimator 0:( X b ... , X n) of the parameter O. The first question which arises in connection with this problem is how to compare different estimators or, equivalently, how to assess their quality, in terms of the mean square deviation from the parameter or perhaps in some other way. The presently accepted approach to this problem, resulting from A. Wald's contributions, is as follows: introduce a nonnegative function w(0l> ( ), Ob Oe 9 (the loss function) and given two estimators Of and O! n 2 2 the estimator for which the expected loss (risk) Eown(Oj, 0), j = 1 or 2, is smallest is called the better with respect to Wn at point 0 (here EoO is the expectation evaluated under the assumption that the true value of the parameter is 0). Obviously, such a method of comparison is not without its defects.

Keywords

Asymptotische Wirksamkeit Estimation Theory Estimator Parameter Schätzung (Statistik)

Authors and affiliations

  • I. A. Ibragimov
    • 1
  • R. Z. Has’minskii
    • 2
  1. 1.LOMILeningradUSSR
  2. 2.Doz., Institut Problem Peredači Inf.MoscowUSSR

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4899-0027-2
  • Copyright Information Springer Science+Business Media New York 1981
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4899-0029-6
  • Online ISBN 978-1-4899-0027-2
  • Series Print ISSN 0172-4568
  • Buy this book on publisher's site