Item Details

Asymptotic Statistical Methods for Stochastic Processes

Yu. N. Linʹkov
Format
Book
Published
Providence, RI : American Mathematical Society, c2001.
Language
English
Russian (translated from)
Uniform Title
Asimptoticheskie Metody Statistiki Sluchaĭnykh Prot͡sessov English
Series
Translations of Mathematical Monographs
ISBN
0821811835 (alk. paper)
Contents
  • Chapter 1. Local Densities of Measures and Limit Theorems for Stochastic Processes 1
  • 1.1. Basic notions of the theory of stochastic processes 1
  • 1.2. Statistic experiments generated by stochastic processes 9
  • 1.3. Limit theorems for semimartingales 21
  • Chapter 2. Asymptotic Distinguishing between Simple Hypotheses in the Scheme of General Statistical Experiments 33
  • 2.1. Statistical hypotheses and tests 33
  • 2.2. Types of asymptotic distinguishability between families of hypotheses and their characterization 36
  • 2.3. Complete asymptotic distinguishability under the conditions of the law of large numbers 45
  • 2.4. Complete asymptotic distinguishability under conditions of weak convergence 52
  • 2.5. Contiguous families of hypotheses 58
  • 2.6. Case of asymptotic expansion of the likelihood ratio 65
  • 2.7. Reduction of the problem of testing hypotheses 70
  • Chapter 3. Asymptotic Behavior of the Likelihood Ratio in Problems of Distinguishing between Simple Hypotheses for Semimartingales 79
  • 3.1. Hellinger integrals and Hellinger processes 79
  • 3.2. Limit theorems for the likelihood ratio 83
  • 3.3. Asymptotic decomposition of the likelihood ratio in parametric formulation 88
  • 3.4. Observations of diffusion-type processes 98
  • 3.5. Observations of counting processes 112
  • Chapter 4. Asymptotic Estimation of Parameters 137
  • 4.1. Formulation of the problem 137
  • 4.2. Properties of the normalized likelihood ratio for semimartingales 140
  • 4.3. Observations of diffusion-type processes 151
  • 4.4. Observations of counting processes 156
  • Chapter 5. Asymptotic Information-Theoretic Problems in Parameter Estimation 161
  • 5.1. Asymptotic behavior of the Shannon information in observations with respect to an unknown parameter 161
  • 5.2. Lower bounds for the information about a statistical estimate of a parameter 174
  • 5.3. Bounds for risk functions of consistent estimates 186
  • 5.4. Observations of semimartingales 194.
Description
xvi, 216 p. : ill. ; 26 cm.
Notes
Includes bibliographical references and index.
Series Statement
Translations of mathematical monographs 0065-9282 ; v. 196
Technical Details
  • Access in Virgo Classic

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    g| Chapter 1. t| Local Densities of Measures and Limit Theorems for Stochastic Processes g| 1 -- g| 1.1. t| Basic notions of the theory of stochastic processes g| 1 -- g| 1.2. t| Statistic experiments generated by stochastic processes g| 9 -- g| 1.3. t| Limit theorems for semimartingales g| 21 -- g| Chapter 2. t| Asymptotic Distinguishing between Simple Hypotheses in the Scheme of General Statistical Experiments g| 33 -- g| 2.1. t| Statistical hypotheses and tests g| 33 -- g| 2.2. t| Types of asymptotic distinguishability between families of hypotheses and their characterization g| 36 -- g| 2.3. t| Complete asymptotic distinguishability under the conditions of the law of large numbers g| 45 -- g| 2.4. t| Complete asymptotic distinguishability under conditions of weak convergence g| 52 -- g| 2.5. t| Contiguous families of hypotheses g| 58 -- g| 2.6. t| Case of asymptotic expansion of the likelihood ratio g| 65 -- g| 2.7. t| Reduction of the problem of testing hypotheses g| 70 -- g| Chapter 3. t| Asymptotic Behavior of the Likelihood Ratio in Problems of Distinguishing between Simple Hypotheses for Semimartingales g| 79 -- g| 3.1. t| Hellinger integrals and Hellinger processes g| 79 -- g| 3.2. t| Limit theorems for the likelihood ratio g| 83 -- g| 3.3. t| Asymptotic decomposition of the likelihood ratio in parametric formulation g| 88 -- g| 3.4. t| Observations of diffusion-type processes g| 98 -- g| 3.5. t| Observations of counting processes g| 112 -- g| Chapter 4. t| Asymptotic Estimation of Parameters g| 137 -- g| 4.1. t| Formulation of the problem g| 137 -- g| 4.2. t| Properties of the normalized likelihood ratio for semimartingales g| 140 -- g| 4.3. t| Observations of diffusion-type processes g| 151 -- g| 4.4. t| Observations of counting processes g| 156 -- g| Chapter 5. t| Asymptotic Information-Theoretic Problems in Parameter Estimation g| 161 -- g| 5.1. t| Asymptotic behavior of the Shannon information in observations with respect to an unknown parameter g| 161 -- g| 5.2. t| Lower bounds for the information about a statistical estimate of a parameter g| 174 -- g| 5.3. t| Bounds for risk functions of consistent estimates g| 186 -- g| 5.4. t| Observations of semimartingales g| 194.
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