Asymptotic Theory of Testing Statistical Hypotheses: Efficient Statistics, Optimality, Power Loss and Deficiency

✍ Scribed by Vladimir E. Bening

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 304
• Category: Library

✦ 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