Asymptotic Efficiency of Statistical Estimators: Concepts and Higher Order Asymptotic Efficiency

1. Front Matter

   Pages I-VII

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2. General Discussion

   • Masafumi Akahira, Kei Takeuchi

   Pages 1-20

3. Consistency of Estimators and Order of Consistency

   • Masafumi Akahira, Kei Takeuchi

   Pages 21-53

4. Asymptotic Efficiency

   • Masafumi Akahira, Kei Takeuchi

   Pages 54-80

5. Higher Order Asymptotic Efficiency

   • Masafumi Akahira, Kei Takeuchi

   Pages 81-135

6. Second Order and Third Order Asymptotic Efficiency of the Maximum Likelihood Estimator and other Estimators

   • Masafumi Akahira, Kei Takeuchi

   Pages 136-187

7. Discretized Likelihood Methods

   • Masafumi Akahira, Kei Takeuchi

   Pages 188-203

8. Higher Order Asymptotic Efficiency and Asymptotic Completeness

   • Masafumi Akahira, Kei Takeuchi

   Pages 204-230

9. Back Matter

   Pages 231-247

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This monograph is a collection of results recently obtained by the authors. Most of these have been published, while others are awaitlng publication. Our investigation has two main purposes. Firstly, we discuss higher order asymptotic efficiency of estimators in regular situa­ tions. In these situations it is known that the maximum likelihood estimator (MLE) is asymptotically efficient in some (not always specified) sense. However, there exists here a whole class of asymptotically efficient estimators which are thus asymptotically equivalent to the MLE. It is required to make finer distinctions among the estimators, by considering higher order terms in the expansions of their asymptotic distributions. Secondly, we discuss asymptotically efficient estimators in non­ regular situations. These are situations where the MLE or other estimators are not asymptotically normally distributed, or where l 2 their order of convergence (or consistency) is not n / , as in the regular cases. It is necessary to redefine the concept of asympto­ tic efficiency, together with the concept of the maximum order of consistency. Under the new definition as asymptotically efficient estimator may not always exist. We have not attempted to tell the whole story in a systematic way. The field of asymptotic theory in statistical estimation is relatively uncultivated. So, we have tried to focus attention on such aspects of our recent results which throw light on the area.

• Asymptotische Wirksamkeit
• Estimator
• Likelihood
• Schätzung (Statistik)
• linear regression

• Department of Mathematics, University of Electro-Communications, Chofu, Tokyo, Japan

  Masafumi Akahira

• Faculty of Economics, University of Tokyo, Hongo, Bunkyo-ky, Tokyo, Japan

  Kei Takeuchi

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