Asymptotic Theory of Testing Statistical Hypotheses

Efficient Statistics, Optimality, Power Loss and Deficiency

Vladimir E. Bening

• Publisher: De Gruyter
• Year: 2011
• Print ISBN:
  9783110622775
• Edition: 1
• eBook ISBN:
  9783110935998
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

No detailed description available for "Asymptotic Theory of Testing Statistical Hypotheses".

Table of contents

• Foreword
• Preface
• Notations
• 1 Asymptotic test theory
• 1.1 First-order asymptotic theory
• 1.2 Second order efficiency
• 1.3 On efficiency of first and second order
• 1.4 Power loss
• 1.5 Efficiency and deficiency
• 1.6 Deficiency results for the symmetry problem
• 2 Asymptotic expansions under alternatives
• 2.1 Introduction
• 2.2 A formal rule
• 2.3 General Theorem
• 2.4 Proof of General Theorem
• 2.5 L-, R-, and U-statistics
• 2.6 Auxiliary lemmas
• 3 Power loss
• 3.1 Introduction
• 3.2 General theorem
• 3.3 Tests based on L-, R-, and U-statistics
• 3.4 Proof of General Theorem: Lemmas
• 3.5 Proof of Lemmas
• 3.6 Power loss for L-, R-, and U-tests
• 3.7 Proofs of Theorems
• 3.8 Combined L-tests
• 3.9 Other statistics
• 4 Edgeworth expansion for the likelihood ratio
• 4.1 Introduction
• 4.2 Moment conditions
• 4.3 Case of independent but not identically distributed terms
• A LeCam’s Third Lemma
• B Convergence rate under alternatives
• B.1 General theorem
• B.2 Proof of Theorem B.1.1
• B.3 L-, R-, and U-statistics
• B.4 Proof of Theorem B.3.1
• C Proof of Theorem 1.3.1
• D The Neyman-Pearson Lemma
• E Edgeworth expansions
• F Proof of Lemmas 2.6.1–2.6.5
• F.1 Proof of Lemma 2.6.1
• F.2 Proof of Lemma 2.6.2
• F.3 Proof of Lemma 2.6.3
• F.4 Proof of Lemma 2.6.4
• F.5 Proof of Lemma 2.6.5
• G Proof of Lemmas 3.7.1–3.7.5
• G.1 Proof of Lemma 3.7.1
• G.2 Proof of Lemma 3.7.2
• G.3 Proof of Lemma 3.7.3
• G.4 Proof of Lemma 3.7.4
• G.5 Proof of Lemma 3.7.5
• H Asymptotically complete classes
• H.1 Non-asymptotic theorem on complete classes
• H.2 Asymptotic theorem on complete classes
• H.3 Power functions of complete classes
• I Higher order asymptotics for R-, L-, and U-statistics
• I.1 R-statistics
• 1.2 L-statistics
• 1.3 U-statistics
• 1.4 Symmetric statistics
• Bibliography
• Subject Index
• Author Index