Contiguity of Probability Measures: Some Applications in Statistics (Cambridge Tracts in Mathematics, Series Number 63)

This Tract Presents An Elaboration Of The Notion Of 'contiguity', Which Is A Concept Of 'nearness' Of Sequences Of Probability Measures. It Provides A Powerful Mathematical Tool For Establishing Certain Theoretical Results With Applications In Statistics, Particularly In Large Sample Theory Problems, Where It Simplifies Derivations And Points The Way To Important Results. The Potential Of This Concept Has So Far Only Been Touched Upon In The Existing Literature, And This Book Provides The First Systematic Discussion Of It. Alternative Characterizations Of Contiguity Are First Described And Related To More Familiar Mathematical Ideas Of A Similar Nature. A Number Of General Theorems Are Formulated And Proved. These Results, Which Provide The Means Of Obtaining Asymptotic Expansions And Distributions Of Likelihood Functions, Are Essential To The Applications Which Follow. On The Concept Of Contiguity And Related Theorems -- Asymptotic Expansions And Asymptotic Distribution Of Likelihood Functions -- Approximation Of A Given Family Of Probability Measures By An Exponential Family, Asymptotic Sufficiency -- Some Statistical Applications, Aump And Aumpu Test For Certain Testing Hypotheses Problems -- Some Statistical Applications, Asymptotic Efficiency Of Estimates -- Multiparameter Asymptotically Optimal Tests. [by] George G. Roussas. Bibliography: P. 240-245. Cover......Page 1 Frontmatter......Page 2 Contents......Page 8 Acknowledgement......Page 11 Preface......Page 12 Summary......Page 16 1 Some preliminary definitions and results......Page 17 2 Contiguity and its relation to other concepts of `nearness' of sequences of probability measures......Page 22 3 Alternative characterizations of contiguity......Page 25 4 Some auxiliary results......Page 33 5 Proof of Proposition 3.1......Page 40 6 An additional characterization of contiguity......Page 46 7 Some results following from contiguity......Page 48 Exercises......Page 54 Summary......Page 56 1 Preliminaries......Page 57 2 Assumptions......Page 60 3 Some examples......Page 62 4 Asymptotic expansion and asymptotic normality of likelihood functions......Page 67 5 Some lemmas......Page 69 6 Proof of theorems of Section 4......Page 78 Exercise......Page 81 Summary......Page 82 1 Formulation of the problem and some preliminary results......Page 83 2 Some auxiliary results......Page 87 3 The proof of the theorem......Page 91 4 Differential equivalence of sequences of probability measures and differential sufficiency......Page 94 5 Some statistical implications of Theorem 1.1......Page 96 Exercises......Page 99 Summary......Page 100 1 Additional assumptions -- Examples......Page 101 2 Some lemmas......Page 111 3 Testing a simple hypothesis against one-sided alternatives......Page 114 4 AUMP tests for the examples of Section 1......Page 120 5 Testing a simple hypothesis against two-sided alternatives......Page 122 6 Testing a one-sided hypothesis against one-sided alternatives......Page 138 Exercises......Page 142 1 W-efficiency -- preliminaries......Page 143 2 Some lemmas......Page 146 3 A representation theorem......Page 150 4 W-efficiency of estimates: upper bounds via Theorem 3.1......Page 156 5 W-efficiency of estimates: upper bounds......Page 162 6 Asymptotic efficiency of estimates: the classical approach......Page 172 7 Classical efficiency of estimates: the multiparameter case......Page 175 Exercise......Page 181 1 Some notation and preliminary results......Page 182 2 Formulation of some of the main results......Page 184 3 Restriction to the class of tests [SCRIPT CAPITAL F]......Page 186 4 Proof of the first main result......Page 192 5 Proof of the second main result......Page 199 6 Formulation and proof of the third main result......Page 203 7 Behaviour of the power under non-local alternatives......Page 211 Exercises......Page 212 1 Some theorems employed in Chapter 1......Page 214 2 Some theorems employed in Chapter 2......Page 219 Exercise......Page 238 3 A theorem employed in Chapter 5......Page 239 4 Some theorems employed in Chapter 6......Page 240 Exercises......Page 254 Bibliography......Page 255 Index......Page 261

This Tract presents an elaboration of the notion of 'contiguity', which is a concept of 'nearness' of sequences of probability measures. It provides a powerful mathematical tool for establishing certain theoretical results with applications in statistics, particularly in large sample theory problems, where it simplifies derivations and points the way to important results. The potential of this concept has so far only been touched upon in the existing literature, and this book provides the first systematic discussion of it. Alternative characterizations of contiguity are first described and related to more familiar mathematical ideas of a similar nature. A number of general theorems are formulated and proved. These results, which provide the means of obtaining asymptotic expansions and distributions of likelihood functions, are essential to the applications which follow.