Contiguity of Probability Measures: Some Applications in Statistics (Cambridge Tracts in Mathematics, Series Number 63) 0521083540, 9780521083546
This Tract presents an elaboration of the notion of 'contiguity', which is a concept of 'nearness' o
143 84 2MB
English Pages 262 [263] Year 1972
Table of contents :
Cover
Frontmatter
Contents
Acknowledgement
Preface
1 - On the concept of contiguity and related theorems
Summary
1 Some preliminary definitions and results
2 Contiguity and its relation to other concepts of "nearness' of sequences of probability measures
3 Alternative characterizations of contiguity
4 Some auxiliary results
5 Proof of Proposition 3.1
6 An additional characterization of contiguity
7 Some results following from contiguity
Exercises
2 - Asymptotic expansion and asymptotic distribution of likelihood functions
Summary
1 Preliminaries
2 Assumptions
3 Some examples
4 Asymptotic expansion and asymptotic normality of likelihood functions
5 Some lemmas
6 Proof of theorems of Section 4
Exercise
3 - Approximation of a given family of probability measures by an exponential family -- asymptotic sufficiency
Summary
1 Formulation of the problem and some preliminary results
2 Some auxiliary results
3 The proof of the theorem
4 Differential equivalence of sequences of probability measures and differential sufficiency
5 Some statistical implications of Theorem 1.1
Exercises
4 - Some statistical applications: AUMP and AUMPU tests for certain testing hypotheses problems
Summary
1 Additional assumptions -- Examples
2 Some lemmas
3 Testing a simple hypothesis against one-sided alternatives
4 AUMP tests for the examples of Section 1
5 Testing a simple hypothesis against two-sided alternatives
6 Testing a one-sided hypothesis against one-sided alternatives
Exercises
5 - Some statistical applications: asymptotic efficiency of estimates
Summary
1 W-efficiency -- preliminaries
2 Some lemmas
3 A representation theorem
4 W-efficiency of estimates: upper bounds via Theorem 3.1
5 W-efficiency of estimates: upper bounds
6 Asymptotic efficiency of estimates: the classical approach
7 Classical efficiency of estimates: the multiparameter case
Exercise
6 - Multiparameter asymptotically optimal tests
1 Some notation and preliminary results
2 Formulation of some of the main results
3 Restriction to the class of tests [SCRIPT CAPITAL F]
4 Proof of the first main result
5 Proof of the second main result
6 Formulation and proof of the third main result
7 Behaviour of the power under non-local alternatives
Exercises
Appendix
1 Some theorems employed in Chapter 1
2 Some theorems employed in Chapter 2
Exercise
3 A theorem employed in Chapter 5
4 Some theorems employed in Chapter 6
Exercises
Bibliography
Index
Cover
Frontmatter
Contents
Acknowledgement
Preface
1 - On the concept of contiguity and related theorems
Summary
1 Some preliminary definitions and results
2 Contiguity and its relation to other concepts of "nearness' of sequences of probability measures
3 Alternative characterizations of contiguity
4 Some auxiliary results
5 Proof of Proposition 3.1
6 An additional characterization of contiguity
7 Some results following from contiguity
Exercises
2 - Asymptotic expansion and asymptotic distribution of likelihood functions
Summary
1 Preliminaries
2 Assumptions
3 Some examples
4 Asymptotic expansion and asymptotic normality of likelihood functions
5 Some lemmas
6 Proof of theorems of Section 4
Exercise
3 - Approximation of a given family of probability measures by an exponential family -- asymptotic sufficiency
Summary
1 Formulation of the problem and some preliminary results
2 Some auxiliary results
3 The proof of the theorem
4 Differential equivalence of sequences of probability measures and differential sufficiency
5 Some statistical implications of Theorem 1.1
Exercises
4 - Some statistical applications: AUMP and AUMPU tests for certain testing hypotheses problems
Summary
1 Additional assumptions -- Examples
2 Some lemmas
3 Testing a simple hypothesis against one-sided alternatives
4 AUMP tests for the examples of Section 1
5 Testing a simple hypothesis against two-sided alternatives
6 Testing a one-sided hypothesis against one-sided alternatives
Exercises
5 - Some statistical applications: asymptotic efficiency of estimates
Summary
1 W-efficiency -- preliminaries
2 Some lemmas
3 A representation theorem
4 W-efficiency of estimates: upper bounds via Theorem 3.1
5 W-efficiency of estimates: upper bounds
6 Asymptotic efficiency of estimates: the classical approach
7 Classical efficiency of estimates: the multiparameter case
Exercise
6 - Multiparameter asymptotically optimal tests
1 Some notation and preliminary results
2 Formulation of some of the main results
3 Restriction to the class of tests [SCRIPT CAPITAL F]
4 Proof of the first main result
5 Proof of the second main result
6 Formulation and proof of the third main result
7 Behaviour of the power under non-local alternatives
Exercises
Appendix
1 Some theorems employed in Chapter 1
2 Some theorems employed in Chapter 2
Exercise
3 A theorem employed in Chapter 5
4 Some theorems employed in Chapter 6
Exercises
Bibliography
Index
- Author / Uploaded
- George G. Roussas
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